Data-Oblivious Stream Productivity

Data-Oblivious Stream Pro ductivit y J¨ org Endrullis 1 , Clemens Grabmay er 2 , and Dimitri Hendriks 1 1 V rij e Universiteit Amsterdam, Department of Computer Science De Bo elelaan 1081a, 1081 H V Amsterdam, The Neth erlands joerg@few. vu.nl diem@cs.vu .nl 2 Universiteit Utrec ht, Department of Philosoph y Heidelb erglaan 8, 3584 CS Utrecht, The Netherlands clemens@ph il.uu.nl Abstract. W e are concerned with demonstrating pro ductiv ity of sp ec- iﬁcations of inﬁnite streams of data, b ased on orthogonal rewrite rules. In general, this prop erty is undecidable, but for restricted formats com- putable suﬃcient conditions can b e obtained. The usu al analysis, also adopted here, disregards the iden tity of data, thus leading to approac hes that we call data-oblivious. W e present a metho d that is p ro va b ly opti- mal among all such data-ob livious approaches. This means that in order to impro ve on our algorithm one h as to p roceed in a data-aw are fashion. 3 1 In tro duction F or prog ramming with inﬁnite s tructures, pr o ductivity is what termination is fo r progra mming with ﬁnite structures. P ro ductivit y captures the intuitiv e notion of unlimited prog ress, of ‘working’ progr ams pro ducing deﬁned v alues indeﬁnitely . In functional languag es, usage of inﬁnite structures is common pra ctice. F o r the correctness of programs dealing with such structures one must guarantee that every ﬁnite part of the inﬁnite structure can b e ev aluated, that is, the sp eciﬁcation of the inﬁnite structure m ust b e pr o ductive. data-obliviously recognizable pure F ¬ F P ¬ P P = pro ductive F = ﬂat Our contribution: = automated recognition = decision Fig. 1: Ma p of s tr eam s peciﬁca tions W e inv estigate this notion for stream sp eciﬁcations, formalized as orthogo nal term rewr iting systems. Common to all previous appro aches for re c ognizing pro ductivity is a quan- titative ana lysis that abstr acts aw ay from the concrete v alues of stream elements. W e formalize this by a notion of ‘data-oblivious’ rewriting, and int r o duce the concept of data- oblivious pr oductivity . Data-oblivio us (non-)pro ductivity implies (non-)pr o ductivity , but neither o f the conv er se im- plications holds. Fig. 1 shows a V enn dia gram of stream speciﬁca tions, high- lighting the subs e t of ‘da ta-obliviously recogniz a ble’ sp eciﬁcations where (non-) 3 This research has b een partially funded by the N etherlands Organisation for S cien- tiﬁc R esearc h (NWO) under FOCUS/BRICKS grant num b er 642.000.502. 2 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks pro ductivity can b e r ecognized by a data-oblivio us a na lysis. W e ident ify tw o syntactical classes of strea m sp eciﬁcations: ‘ﬂat’ and ‘pure’ sp eciﬁcations, see the description b elow. F or the ﬁrst we devise a decisio n algo - rithm for da ta-oblivious (d-o) pro ductivity . This gives rise to a computable, d-o optimal, criter ion for pr o ductivity: every ﬂat strea m sp eciﬁcation that can b e es- tablished to be pro ductive by whatever d-o argument is re c ognized a s pro ductive by this criterion (see Fig. 1). F or the sub class of pure spec iﬁca tions, w e establish that d-o pro ductivity coincides with pro ductivity , and thereby obtain a decision algorithm for productivity o f this class. Additionally , w e extend our criterion beyond the class o f ﬂat stream s p eciﬁca tio ns, allowing for ‘frie ndly nesting’ in the sp eciﬁcation of str eam functions; here d-o optimality is not pre s erved. In deﬁning the diﬀerent formats of stream sp eciﬁcations, we distinguish b e- t ween rules for strea m co nstant s , and rules for stream functions. Only the la tter are sub jected to syntactic restrictions . In ﬂat s tream s p eciﬁca tions the deﬁning rules fo r the strea m functions do not hav e nesting of str eam function symbols; how ever, in deﬁning r ules for str eam co nstant s nesting of stre a m function sym- bo ls is allo wed. This format makes us e of exhaustive pattern matching on data to deﬁne s tream functions, allo wing for multiple deﬁning rules for an individ- ual stream function sy m b ol. Since the qua n titative consumption/pro duction b e- haviour of a symbol f might diﬀer among its deﬁning rules, in a d-o a nalysis one has to settle for the us e of lower b ounds when tr ying to rec o gnize pro ductivity . If for all stream function symbols f in a ﬂat sp eciﬁcation T the deﬁning rules for f coincide, disre g arding the iden tity of da ta-elements, then T is called pure. Our decis ion algo rithm for d-o pro ductivity deter mines the tight d-o low er bo und on the production b ehaviour of every stream function, and uses these bo unds to calc ula te the d-o pro duction of str eam constants. W e br ieﬂy explain bo th a s pects. Consider the strea m sp eciﬁcation A → 0 : f ( A ) together with the rules f ( 0 : σ ) → 1 : 0 : 1 : f ( σ ), and f ( 1 : σ ) → 0 : f ( σ ), deﬁning the strea m 0 : 1 : 0 : 1 : . . . of alterna ting bits. The tight d-o low er b ound for f is the function id : n 7→ n . F urther note that su c : n 7→ n + 1 captur e s the quantit ative b ehaviour of the function prep ending a data ele ment to a s tream ter m. Therefor e the d-o pro duction of A can b e computed as lfp( suc ◦ id ) = ∞ , where lfp( f ) is the least ﬁxed point of f : N → N and N := N ∪ { ∞} ; henc e A is pro ductive. As a co mparison, only a ‘data-aware’ a pproach is able to establish pr o ductivity of B → 0 : g ( B ) with g ( 0 : σ ) → 1 : 0 : g ( σ ), a nd g ( 1 : σ ) → g ( σ ). The d-o lo wer bo und of g is n 7→ 0, due to the latter rule. This makes it imp ossible for any conceiv able d-o approach to recogniz e pr o ductivity of B . W e obta in the following results: (i) F o r the class of ﬂat stream sp eciﬁcations we give a computable, d-o optimal, suﬃcient co ndition for pro ductivity . (ii) W e show decida bilit y of pro ductivity for the class of pur e stream s peciﬁca - tions, a n extension o f the format in [2]. (iii) Disr e g arding d-o optimality , we extend (i) to the big ger class of friendly nesting s tream sp eciﬁcations. (iv) A tool automating (i), (ii), a nd (iii), which can be downloaded from, and used via a web int erface a t: http://infinit y.few.vu.nl/productivity . Data-Oblivious Stream Pro ductivity 3 R elate d work. Previo us appro aches [8, 3, 9 , 1] employ ed d-o r easoning (without using this name for it) to ﬁnd suﬃcient criter ia ensur ing pr o ductivit y , but did not a im at optimality . The d-o pro duction b ehaviour of a stream function f is b ounded from b elow b y a ‘mo dulus of pro duction’ ν f : N k → N with the prop erty that the ﬁrs t ν f ( n 1 , . . . , n k ) element s of f ( t 1 , . . . , t k ) can b e computed whenever the ﬁrst n i elements of t i are deﬁned. Sijtsma develops an a ppr oach allowing arbitrar y pr o ductio n moduli ν : N k → N , which, while providing an adequate mathematical description, are less amenable to automa tion. T elford and T ur ner [9] employ pro duction mo duli o f the form ν ( n ) = n + a with a ∈ Z . Hughes, Pareto and Sabry [3] use ν ( n ) = max { c · x + d | x ∈ N , n ≥ a · x + b } ∪ { 0 } with a, b, c , d ∈ N . Both c lasses of pro duction mo duli are strictly contained in the class o f ‘p erio dica lly incre a sing’ functions which we employ in our analysis. W e show that the set of d-o low er b ounds of ﬂat s tream function sp eciﬁcations is exa ctly the set of p erio dically increasing functions. Buc hho lz [1] presents a t y pe system for pro ductivity , us ing unrestric ted pro duction mo duli. T o obtain an automatable metho d for a res tricted class of stream sp eciﬁcations, he dev ises a syntactic cr iterion to ensur e pr o ductivity . This cr iter ion easily handles all the examples of [9], but fails to deal with functions that have a neg a tive eﬀect like odd deﬁned by o dd ( x : y : σ ) → y : o dd ( σ ) w ith a (p erio dically increas ing) mo dulus ν o dd ( n ) = ⌊ n 2 ⌋ . Overview. In Sec. 2 we deﬁne the notion of stream sp eciﬁcation, and the syntactic format of ﬂa t and pur e sp eciﬁcations. In Sec. 3 we formalize the notion of d-o rewriting. In Sec. 4 we introduce a pro duction ca lculus as a mea ns to compute the pro duction of the data -abstracted stream sp eciﬁcatio ns, based on the set of p erio dically increasing functions. A translatio n o f strea m sp eciﬁca tio ns into pro duction terms is deﬁned in Sec. 5. O ur main results, mentioned ab ov e, ar e collected in Sec. 6. W e conclude and discuss some future res earch topics in Sec. 8. 2 Stream Sp eciﬁcations W e int r o duce the notio n of stre a m sp eciﬁcation. An example is given in Fig. 2, a pro ductive sp eciﬁcation of Pascal’s triang le where the rows are separated by P → 0 : s ( 0 ) : f ( P ) f ( s ( x ) : y : σ ) → a ( s ( x ) , y ) : f ( y : σ ) str e am layer f ( 0 : σ ) → 0 : s ( 0 ) : f ( σ ) a ( s ( x ) , y ) → s ( a ( x, y )) data layer a ( 0 , y ) → y Fig. 2: Example of a ﬂat stream sp eciﬁcation. 4 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks zeros. Indeed, ev aluating this sp eciﬁcation, we get: P ։ ։ 0 : 1 : 0 : 1 : 1 : 0 : 1 : 2 : 1 : 0 : 1 : 3 : 3 : 1 : . . . , where, for n ∈ N , n is the numer al for n , deﬁned by n := s n ( 0 ). Stream sp eciﬁcatio ns consist of a st r e am layer (top) where stream constants and functions are sp eciﬁed and a data layer (b ottom). The hierarchical setup of stream spe ciﬁcations is motiv ated as follows. In order to abstrac t from the terminatio n problem when investigating pro ductivity , the data lay er is a term rewriting system o n its own, and is required to be strongly normalizing. Moreover, an isolated data layer preven ts the use of stream symbols by data rules . The stream lay er may use symbols of the data lay er , but no t vice- versa. Stream dep enden t data functions, like head ( x : σ ) → x , might cause the output of undeﬁned da ta terms. Co ns ider the fo llowing exa mples fro m [8]: S → 0 : S (6) : S T → 0 : T (7) : T , where σ ( n ) := head ( tail n ( σ )); here S is well-deﬁned, wherea s T is not. Note that this is not a se vere restr iction, since str e a m dep endent data functions usua lly can be replaced b y pattern matching: f ( σ, τ ) → ( head ( σ ) + head ( τ )) : f ( tail ( σ ) , tail ( τ )), for example, can b e r eplaced by the b etter r eadable f ( x : σ , y : τ ) → ( x + y ) : f ( σ, τ ). Stream speciﬁca tions are forma lized a s many-sorted, o rthogonal, constructor term rew r iting systems [10]. W e distinguish be tw e en str e am terms and data terms . F or the sake of simplicity we consider only o ne sort S for strea m terms and one sor t D for data terms. Without any complica tion, our results ex tend to stream sp eciﬁcatio ns with multiple sorts for data terms and for s tream terms. Let U b e a ﬁnite s et of sorts . A U -sorte d set A is a family o f sets { A u } u ∈ U ; for V ⊆ U we deﬁne A V := S v ∈ V A v . A U -sorte d signatur e Σ is a U -sor ted set of function sy m b ols f , eac h equipped with an ar it y ar ( f ) = h u 1 · · · u n , u i ∈ U ∗ × U where u is the s ort of f ; we wr ite u 1 × . . . × u n → u for h u 1 · · · u n , u i . Let X b e a U -sorted set of v aria bles. The U -sorte d set of terms T er ( Σ , X ) is inductively deﬁned by: for a ll u ∈ U , X u ⊆ T er ( Σ , X ) u , and f ( t 1 , . . . , t n ) ∈ T er ( Σ , X ) u if f ∈ Σ , ar ( f ) = u 1 × . . . × u n → u , and t i ∈ T er ( Σ , X ) u i . T er ∞ ( Σ , X ) denotes the set of (p ossibly) inﬁnite terms over Σ and X (see [10]). Usually we keep the s et of v a riables implicit and write T er ( Σ ) and T er ∞ ( Σ ). A U -sorte d term re writing syst em (TRS) is a pair h Σ , R i cons isting of a U -sorted sig nature Σ a nd a U -sorted set R of ru les that satisfy well-sortedness , for all u ∈ U : R u ⊆ T er ( Σ , X ) u × T er ( Σ , X ) u , as well as the sta nda rd TRS requirements. Let T = h Σ , R i b e a U -sorted TRS. F or a ter m t ∈ T er ( Σ ) u where u ∈ U we denote the ro ot symbol of t by r oot ( t ). W e say that tw o o ccur r ences of s y m b ols in a term ar e n est e d if the p osition [10 , p.29] of one is a preﬁx of the position of the other. W e deﬁne D ( Σ ) := { r oot ( l ) | l → r ∈ R } , the set o f deﬁne d symb ols , and C ( Σ ) := Σ \ D ( Σ ), the set of c onstru ctor symb ols . Then T is calle d a c onstructor TRS if for every rewrite rule ρ ∈ R , the left-hand side is of the form f ( t 1 , . . . , t n ) with t i ∈ T er ( C ( Σ )); then ρ is called a deﬁning ru le for f . W e call T ex haust ive for f ∈ Σ if every ter m f ( t 1 , . . . , t n ) with closed constructor terms t i is a r edex. Data-Oblivious Stream Pro ductivity 5 A str e am TRS is a ﬁnite { S , D } -so rted, o rthogonal, constructo r TRS h Σ , R i such tha t ‘:’ ∈ Σ S , the stre am c onst ructor symb ol , with arity D × S → S is the single constr uctor symbo l in Σ S . Elements o f Σ D and Σ S are called the data symb ols and the st r e am symb ols , res p ectively . W e let Σ s := Σ S \ { ‘:’ } , and, for all f ∈ Σ s , we ass ume w.l.o.g. that the s tream arguments are in fro n t: ar ( f ) ∈ S ar s ( f ) × D ar d ( f ) → S , where ar s ( f ) and ar d ( f ) ∈ N ar e ca lled the str e am arity and the data arity of f , r esp ectively . B y Σ sc on we denote the set of s y m b ols in Σ s with stream ar it y 0 , called the str e am c onstant symb ols , and Σ sfun := Σ s \ Σ sc on the set o f symbols in Σ s with stream ar it y unequal to 0 , called the str e am funct ion symb ols . By R sc on we mean the deﬁning rules for the s y m b ols in Σ sc on . W e r epea t that the r estriction to a sing le data sort D is so lely for keeping the presentation s imple; all of the deﬁnitions and res ults in the sequel gener alize to multip le data and stre a m sor ts. F or str eam TRSs we assume non-emptyness of data s o rts, that is, for every data sor t there exists a ﬁnite, closed, contructor term of this s ort. In ca se there is only one data sort D , then the r equirement bo ils down to the existence of a nullary cons tructor symbol of this so r t. W e co me to the deﬁnition of stream sp eciﬁcations. Deﬁnition 2.1 . A str e am sp e ciﬁc ation T is a stream TRS T = h Σ , R i such that the following conditio ns hold: (i) Ther e is a designated symbol M 0 ∈ Σ sc on with ar d ( M 0 ) = 0, the r o ot of T . (ii) h Σ D , R D i is a terminating, D - sorted TRS; R D is called the data-layer of T . (iii) T is exhaustive (for all deﬁned sy m b ols in Σ = Σ S ⊎ Σ D ). In the s e q uel we restrict to stream sp eciﬁca tio ns in which all stream constants in Σ sc on are r eachable fr o m the r o ot: M ∈ Σ sc on is re achable if there is a ter m s such that M 0 → ∗ s and M o ccurs in s . Note that reachabilit y of stream constants is decidable, and that unr e achable symbols may b e neglected for in vestigating (non-)pro ductivity . Note tha t Def. 2.1 indeed imp oses a hierarchical setup; in particular, stream depe ndent da ta functions ar e excluded by item (ii). Exhaustivity for Σ D together with str ong normaliza tion o f R D guarantees that closed data terms rewrite to constructor nor mal forms, a prop erty known as suﬃcient completeness [4]. W e a re interested in productivity of recur sive s tream sp eciﬁcatio ns that make use of a libr ary of ‘manag eable’ strea m functions. By this we mean a cla ss of stream functions deﬁned by a syntactic format with the prop erty that their d-o low er b ounds are computable and contained in a se t o f pro duction mo duli that is e ﬀectiv ely closed under comp osition, p o in twise inﬁmum and where least ﬁxe d po in ts ca n b e computed. As suc h a format w e deﬁne the class of ﬂat stre a m sp eciﬁcations (Def. 2 .2) for which d- o low er b o unds ar e precisely the set o f ‘p e- rio dically incr easing’ functions (see Sec. 4). Thu s o nly the strea m function rules are sub ject to syntactic r estrictions. No co nditio n other than well-sortedness is impo sed on the deﬁning rules of stream c o nstant symbols. In the seq uel let T = h Σ , R i b e a stream sp eciﬁcation. W e deﬁne the relatio n on rules in R S : for all ρ 1 , ρ 2 ∈ R S , ρ 1 ρ 2 ( ρ 1 dep ends on ρ 2 ) holds if and only if ρ 2 is the deﬁning r ule of a stream function sym b ol on the righ t-hand 6 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks side of ρ 1 . F urthermo re, for a bina ry relation → ⊆ A × A on a set A we de ﬁne ( a → ) := { b ∈ A | a → b } for all a ∈ A , and w e denote by → + and → ∗ the tr ansitive closur e and the r eﬂexive and tra ns itive closur e of → , resp ectively . Deﬁnition 2.2 . A rule ρ ∈ R S is calle d n esting if its rig h t- ha nd side contains nested o ccurr ences of stream sy m b ols fro m Σ s . W e use R nest to denote the subset of nesting r ule s of R and deﬁne R ¬ nest := R S \ R nest , the se t o f non-nesting rules . A rule ρ ∈ R S is called ﬂat if all r ules in ( ρ ∗ ) are non- nesting. A symbo l f ∈ Σ s is called ﬂat if all deﬁning rules of f a re ﬂat; the set o f ﬂat symbols is denoted Σ ﬂat . A s tream sp eciﬁcation T is called ﬂat if Σ s ⊆ Σ ﬂat ∪ Σ sc on , that is, a ll symbols in Σ s are either ﬂat o r stream cons tan t symbols. The sp eciﬁcation given in Fig. 2 is an example o f a ﬂat sp eciﬁcation. A s econd example is given in Fig. 3. This is a pro ductive spe c iﬁcation that deﬁnes the Q → a : Q ′ Q ′ → b : c : f ( Q ′ ) f ( a : σ ) → a : b : c : f ( σ ) str e am layer f ( b : σ ) → a : c : f ( σ ) f ( c : σ ) → b : f ( σ ) data layer Fig. 3: Example of a ﬂat stream sp eciﬁcation. ternary Thue–Morse sequence, a square - free w or d o ver { a , b , c } ; see, e.g ., [7]. Indeed, ev alua ting this speciﬁc a tion, we get: Q ։ ։ a : b : c : a : c : b : a : b : c : b : a : c : . . . . As the basis of d- o r ewriting (see Def. 3 .3) we deﬁne the data a bs traction of terms as the res ults of r eplacing all data-subterms by the symbol • . Deﬁnition 2.3 . Let L Σ M := {•} ⊎ Σ S . F or strea m terms s ∈ T er ( Σ ) S , the data abstr action L s M ∈ T er ( L Σ M ) S is deﬁned b y : L σ M = σ , L u : s M = • : L s M , L f ( s 1 , . . . , s n , u 1 , . . . , u m ) M = f ( L s 1 M , . . . , L s n M , • , . . . , • ) . Based on this deﬁnition of data abstracted terms, w e deﬁne the class of pure stream sp eciﬁcatio ns, an extensio n of the equally named class in [2]. Deﬁnition 2.4 . A stream sp eciﬁcation T is ca lle d pur e if it is ﬂat and if for every s ym b ol f ∈ Σ s the data abstractions L ℓ M → L r M o f the de ﬁning rules ℓ → r of f coincide (mo dulo renaming of v ar iables). Data-Oblivious Stream Pro ductivity 7 Q → diﬀ ( M ) M → 0 : zip ( inv ( M ) , tail ( M )) zip ( x : σ , τ ) → x : zip ( τ , σ ) str e am layer inv ( x : σ ) → i ( x ) : inv ( σ ) tail ( x : σ ) → σ diﬀ ( x : y : σ ) → x ( x, y ) : diﬀ ( y : σ ) i ( 0 ) → 1 i ( 1 ) → 0 x ( 0 , 0 ) → b x ( 0 , 1 ) → a data layer x ( 1 , 0 ) → c x ( 1 , 1 ) → b Fig. 4: Example of a pure stream sp eciﬁcation. Fig. 4 shows a n alternative sp eciﬁcation of the ternary Th ue– Morse sequence, this time constructed fro m the binar y Thue–Morse s e quence sp eciﬁed by M . This sp eciﬁcation belo ngs to the subcla ss of pure speciﬁca tio ns, whic h is ea sily inferr ed by the shap e of the stream layer: for ea c h s ym b ol in Σ sfun = { zip , inv , tail , diﬀ } there is precisely one deﬁning r ule. Another exa mple of a pure strea m sp eciﬁcation is given in Fig. 5. Both deﬁn- M → 0 : M ′ M ′ → 1 : h ( M ′ ) str e am layer h ( 0 : σ ) → 0 : 1 : h ( σ ) h ( 1 : σ ) → 1 : 0 : h ( σ ) data layer Fig. 5: Example of a pure stream sp eciﬁcation. ing rules fo r h consume one, and pro duce tw o stream elements, that is, their data abstractions co incide. Def. 2.4 generalize s the s peciﬁca tions called ‘pure’ in [2 ] in four wa ys con- cerning the deﬁning rules of stream functions: First, the requirement of right- linearity of s tream v aria bles is dropp ed, allowing for rules like f ( σ ) → g ( σ, σ ). Second, ‘additional supply’ to the stream arg umen ts is allow ed. F or instance, in the deﬁning r ule for the function diﬀ in Fig. 4 , the v aria ble y is ‘supplied’ to the recursive c all o f diﬀ . Third, the use of non-pro ductive str eam functions is allow ed now, relaxing a n earlier r equirement on stream function symbols to b e ‘weakly g uarded’. Finally , deﬁning r ules for str eam function symbols may use a restricted form of pattern matching as long as, for every str eam function f , the 8 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks d-o consumption/ pr o duction be haviour (see Sec. 3) o f all deﬁning rules for f is the sa me. Next we deﬁne friendly ne s ting stream spec iﬁc a tions, an e xtension of the class o f ﬂa t strea m sp eciﬁcations. Deﬁnition 2.5 . A rule ρ ∈ R S is called friend ly if for all rules γ ∈ ( ρ ∗ ) we hav e: (1) γ consumes in each ar gument a t most one stream elemen t, and (2) it pro duces a t least o ne. The set of friend ly nesting ru les R fnest is the la rgest extension o f the set of friendly rules by non-nesting rules fr o m R S that is closed under . A symbol f ∈ Σ s is friend ly nest ing if all deﬁning rules of f are friendly nesting. A strea m spec iﬁc a tion T is ca lled friend ly nest ing if Σ s ⊆ Σ fnest ∪ Σ sc on , that is, all sym b ols in Σ s are either friendly nesting or stream constant symbols. An example o f a friendly nesting stream speciﬁca tio n is given in Fig. 6 . The r oo t nats → 0 : × ( ones , ones ) ones → s ( 0 ) : ones × ( x : σ , y : τ ) → m ( x, y ) : add ( times ( τ , x ) , × ( σ, y : τ )) times ( x : σ , y ) → m ( x, y ) : times ( σ , y ) add ( x : σ, y : τ ) → a ( x, y ) : add ( σ , τ ) str e am layer a ( 0 , y ) → y a ( s ( x ) , y ) → s ( a ( x, y )) m ( 0 , y ) → 0 m ( s ( x ) , y ) → a ( y , m ( x, y )) data layer Fig. 6: Example of a friendly nesting stream sp eciﬁcation. of this sp eciﬁcation, the constant na ts , ev alua tes to the stre a m 0 : 1 : 2 : . . . . The stream lay er sp eciﬁes a parameterized stream function times , which multiplies every element of a stream with the par ameter, and the binary str eam functions add for compo nent wise addition, a nd × , the c onvolution pr o duct of streams (see [6 ]), mathematically deﬁned as a n op eration h σ , τ i 7→ σ × τ : ( σ × τ )( i ) = i X j =0 σ ( j ) · τ ( i − j ) (for all i ∈ N ). The rewrite rule for × is nesting, b ecause its right-hand has nested stream func- tion sy m b ols, b oth ti m e s and × are nested within add . Beca us e of the pr esence of a nesting rule in the stream la yer, which is not a deﬁning rule of a stream con- stant symbol, this stream speciﬁca tion is not ﬂa t. How ever, the deﬁning rules for add , times , and × ar e friendly . F or instance for the rule for × we check: in b oth arguments it consumes (at most) o ne stream element ( x and y ), it pro duces Data-Oblivious Stream Pro ductivity 9 (at least) one strea m element ( m ( x, y )), and the deﬁning r ules of the stream function s ym b ols in the r ight - hand side ( add , times , and × ) ar e aga in friendly . Thu s the function s tr eam la yer consists of one friendly nesting rule, t wo ﬂat (and friendly) rules, and one deﬁning rule for a strea m constant. Ther efore this stream sp eciﬁcatio n is frie ndly nesting. Deﬁnition 2.6 . Let A = h T er ( Σ ) S , →i b e an abstract reduction sy stem (ARS) on the set of terms ov er a stream TRS signature Σ . The pr o duction function Π A : T er ( Σ ) S → N of A is deﬁned fo r all s ∈ T er ( Σ ) S by: Π A ( s ) := sup { n ∈ N | s → ∗ A u 1 : . . . : u n : t } . W e ca ll A pr o ductive for a str e am term s if Π A ( s ) = ∞ . A strea m sp eciﬁcation T is ca lled pr o ductive if T is pro ductive for its r o ot M 0 . The following pr opo sition justiﬁes this deﬁnition o f pro ductivit y by stating an easy consequence: the r o ot o f a pro ductive stream sp eciﬁcation can be ev aluated contin ually in such a w ay that a uniquely determined stream in constructor normal for m is obtained as the limit. This follows easily from the fact that a stream sp eciﬁcatio n is an orthogonal TRS, and hence ha s a conﬂuent r ewrite relation that enjoys the prop erty UN ∞ (uniqueness of inﬁnite no rmal form) [5]. Prop osition 2.7. A str e am sp e ciﬁc ation is pr o ductive if and only if its r o ot has an inﬁnite c onstru ctor term of the form u 1 : u 2 : u 3 : . . . as its un ique inﬁnite normal form. 3 Data-Oblivious Analysis W e forma lize the notion of d- o r ewriting and introduce the concept of d- o pro- ductivit y . The idea is a qua n titative r easoning where all knowledge ab out the concrete v alues of data elements during a n ev aluation seq uence is ignored. F or example, cons ider the fo llowing stream s peciﬁca tion: M → f ( 0 : 1 : M ) (1) f ( 0 : x : σ ) → 0 : 1 : f ( σ ) (2) f ( 1 : x : σ ) → x : f ( σ ) The sp eciﬁcation of M is pro ductive: M → 2 0 : 1 : f ( M ) → 3 0 : 1 : 0 : 1 : f ( f ( M )) → ∗ . . . . During the rewrite sequence (2) is nev er a pplied. Disregar ding the iden tity of data, how ever, (2) b ecomes applicable a nd allows for the rewrite sequence: M → f ( • : • : M ) → (2) • : f ( M ) → ∗ • : f ( • : f ( • : f ( . . . ))) , pro ducing only one element. Hence M is not d-o pr o ductiv e. D-o term rewriting can b e thought of as a tw o-play e r ga me b et ween a r ewrite player R which p erfo r ms the usual ter m rewriting, and an op p onent G which befo re ev er y rewrite step is allow ed to arbitra rily exchange da ta elemen ts for (sort-res pecting) da ta terms in constructo r no r mal form. The opp onent can ei- ther handicap or supp ort the rewrite player. Resp ectively , the d-o low er (upp er) 10 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks bo und o n the pro duction of a stream term s is the inﬁm um (supr em um) o f the pro duction of s with resp ect to a ll p ossible strategie s for the o pp onent G . Fig. 7 depicts d-o rewr iting of the ab ov e strea m sp eciﬁcation M ; by exchanging data elements, the opp onent G enforces the applica tion o f (2). R emark 3.1. A t ﬁrs t glance it might app ear natural to mo del the o ppo nen t using a function G : T er ( Σ D ) → T er ( Σ D ) from data terms to da ta ter ms. How ever, such a p er-element deterministic exc ha nge strateg y preserves equality of data elements. The n the following sp eciﬁcation of W : W → g ( Z , Z ) Z → 0 : Z g ( 0 : σ, 0 : τ ) → 0 : g ( σ, τ ) g ( 0 : σ, 1 : τ ) → g ( σ , τ ) g ( 1 : σ, 1 : τ ) → 0 : g ( σ, τ ) g ( 1 : σ, 0 : τ ) → g ( σ , τ ) would be pro ductive for every such G , which is c le arly no t what one would exp ect of a d-o ana lysis. The opp onent can b e mo delled b y an o per ation on stream terms: T er ( Σ ) S → T er ( Σ ) S . Howev er, it can b e shown that it is suﬃcient to qua n tify over stra tegies for G for which G ( s ) is inv ar ia n t under exchange of data elements in s for all terms s . Therefor e w e ﬁrst a bstract from the data elements in fav o ur of symbols • and then deﬁne the opp onent G on the abstracted terms, G : T er ( L Σ M ) S → T er ( Σ ) S . Deﬁnition 3.2 . Let T = h Σ , R i b e a stream sp eciﬁcation. A data-exchange function on T is a function G : T er ( L Σ M ) S → T er ( Σ ) S such that ∀ r ∈ T er ( L Σ M ) S it holds: L G ( r ) M = r , and G ( r ) is in data-co nstructor normal for m. Deﬁnition 3.3 . W e deﬁne the ARS A T , G ⊆ T er ( L Σ M ) S × T er ( L Σ M ) S for every data-exchange function G , as follows: A T , G := { s → L t M | s ∈ T er ( L Σ M ) , t ∈ T er ( Σ ) with G ( s ) → T t } . The d-o pr o duction ra n ge do T ( s ) of a da ta abstracted stream term s ∈ T er ( L Σ M ) S is deﬁned as follows: do T ( s ) := { Π A T , G ( s ) | G a data- exchange function on T } . M M f ( 0 : 1 : M ) f ( 1 : 0 : M ) 0 : f ( M ) 0 : f ( M ) 0 : f ( f ( 0 : 1 : M )) 0 : f ( f ( 1 : 0 : M )) 0 : f ( 0 : f ( M )) . . . G R G R G R G R G Fig. 7: Data-oblivious rewriting Data-Oblivious Stream Pro ductivity 11 The d-o lower and upp er b ound on the pr o duction of s are deﬁned by do T ( s ) := inf ( do T ( s )) , and do T ( s ) : = sup( do T ( s )) , resp ectively . These notio ns c arry ov er to strea m terms s ∈ T er ( Σ ) S by taking their da ta abstractio n L s M . A str eam sp eciﬁcation T is d-o pr o ductive ( d-o non-pr o ductive ) if do T ( M 0 ) = ∞ (if do T ( M 0 ) < ∞ ) holds. Prop osition 3.4. F or T = h Σ , R i a str e am sp e ciﬁc ation and s ∈ T er ( Σ ) S : do T ( s ) ≤ Π T ( s ) ≤ do T ( s ) . Henc e d-o pr o duct ivity implies pr o ductivity and d-o n on- pr o ductivity implies non- pr o ductivity. W e deﬁne low er and upp er b ounds on the d-o consumption/pro duction b e- haviour of str eam functions. The s e b ounds are us e d to rea son ab out d-o (non-) pro ductivity of stream c o nstants, see Sec. 6. Deﬁnition 3.5 . Let T = h Σ , R i be a stream sp eciﬁcation, g ∈ Σ s , k = ar s ( g ), and ℓ = ar d ( g ). The d-o pr o duction r ange do T ( g ) : N k → P ( N ) of g is : do T ( g )( n 1 , . . . , n k ) := do T ( g (( • n 1 : σ ) , . . . , ( • n k : σ ) , • , . . . , • | {z } ℓ -times ) ) , where • m : σ := m ti mes z }| { • : . . . : • : σ . The d-o lower and upp er b ounds on t he pr o duction of g are deﬁned by do T ( g ) := inf ( do T ( g )) and do T ( g ) := sup( do T ( g )), resp ectively . Even simple stream function sp eciﬁcatio ns ca n exhibit a complex d-o b e- haviour. F or example, co nsider the ﬂat function sp eciﬁcation: f ( σ ) → g ( σ, σ ) g ( 0 : y : σ , x : τ ) → 0 : 0 : g ( σ , τ ) g ( 1 : σ , x 1 : x 2 : x 3 : x 4 : τ ) → 0 : 0 : 0 : 0 : 0 : g ( σ , τ ) Fig. 8 (left) shows a (small) selection of the p ossible function-call tra ces fo r f . In particular, it depicts the traces that contribute to the d-o lower b o und do T ( f ). The low er b ound do T ( f ), shown on the r ight, is a sup erp osition of multiple traces of f . In g e neral do T ( f ) ca n even b e a s up erp ositio n of inﬁnitely many trace s. First Obs erv ations on Data-Oblivious Rewriti ng F or a stream function symbol g , we de ﬁne its o ptimal pr o duction mo dulus ν g , the data-aware, qua ntitative lower b ound on the pr o duction of g , a nd c ompare it to its d-o low er b ound do T ( g ). 12 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks input 5 10 15 output 5 10 15 input 5 10 15 output 5 10 15 Fig. 8: T races Deﬁnition 3.6 . Let T = h Σ , R i b e a str eam s peciﬁca tion. W e deﬁne the set S n of n -deﬁne d strea m ter ms, i.e. ﬁnite strea m terms with a constructor -preﬁx of leng th n ∈ N : S n := { u 1 : . . . : u n : σ | u 1 , . . . , u n ∈ T er ( C ( Σ D ) , ∅ ) } , Moreov e r, let g ∈ Σ sfun with k = ar s ( g ) a nd ℓ = ar d ( g ), a nd deﬁne, for all n = n 1 , . . . , n k , the set G n of a pplica tions of g to n i -deﬁned arg umen ts: G n := { g ( s , v ) | ∀ i. s i ∈ S n i , ∀ j. v j ∈ T er ( C ( Σ D ) , ∅ ) } , where s = s 1 , . . . , s k and v = v 1 , . . . , v ℓ . Then, the optimal pr o duction m o dulus ν g : N k → N of g is deﬁned by: ν g ( n ) := inf { Π T ( t ) | t ∈ G n } . T o illustra te the diﬀerence betw e en the optimal pro duction mo dulus ν h and the d- o lo wer b ound do T ( h ), co nsider the follo wing str e am function sp eciﬁcation: h ( 0 : x : σ ) → x : x : h ( 0 : σ ) ( ρ h 0 ) h ( 1 : x : σ ) → x : h ( 0 : σ ) ( ρ h 1 ) with Σ D = { 0 , 1 } . T he n ν h ( n ) = 2 n . − 3 is the optimal pr o duction mo dulus o f the s tream function h . T o obtain this b ound one has to take into account that the data element 0 is supplied to the recur sive call and c onclude tha t ( ρ h 1 ) is only applicable in the ﬁrst s tep of a rewrite seq uence h ( u 1 : . . . : u n : σ ) → . . . . How ever, the d-o lower b ound is do T ( h )( n ) = n . − 1, derived from r ule ( ρ h 1 ). The fo llowing lemmas state an observ ation a b out the ro le of the o pponent and the r ewrite play er. Basica lly , the oppo nen t G can sele ct the rule w hich is applicable for each p osition in the term; the rewr ite play er R can choos e which po sition to rewr ite. W e use subs c ripts for p ebbles • , for exa mple • w , to introduce ‘names’ for referring to these p ebbles. Data-Oblivious Stream Pro ductivity 13 Deﬁnition 3.7 (Instan tiation with resp ect to a rule ρ ). F or s ∈ T er ( L Σ M ) S with s ≡ f ( s 1 , . . . , s ar s ( f ) , • 1 , . . . , • ar d ( f ) ) and ρ ∈ R a deﬁning rule of f , we deﬁne a data-e x c ha nge function G s,ρ as follows. Note that ρ is of the form: ρ : f ( u 1 : σ 1 , . . . , u n : σ n , v 1 , . . . , v ar d ( f ) ) → r ∈ R . F or i = 1 , . . . , ar s ( f ) let n i ∈ N be maxima l such that s i ≡ • i, 1 : . . . • i,n i : s ′ i . Let G s,ρ for all 1 ≤ i ≤ ar d ( f ) instantiate the p ebbles • i with closed instances of the data patters v i , a nd for all 1 ≤ i ≤ ar s ( f ), 1 ≤ j ≤ min( n i , | u i | ) insta n tia te the p ebble • i,j with clos ed instances of the data pattern u i j , resp ectively . Lemma 3.8 . L et s ∈ T er ( L Σ M ) S , s ≡ f ( s 1 , . . . , s ar s ( f ) , • 1 , . . . , • ar d ( f ) ) and ρ ∈ R a deﬁning rule of f ; we use the notation fr om Def. 3.7. Then G s,ρ ( s ) is a r e dex if and only if for al l 1 ≤ i ≤ ar s ( f ) we have: u i ≤ n i , that is, ther e is ‘enough supply for the applic ation of ρ ’. F urthermor e if G s,ρ ( s ) is a r e dex, then it is a r e dex with re s p e ct to ρ (and no other rule fr om R by ortho gonality). Pr o of. A consequence of the fa ct that T is an orthogo nal constr uctor TRS. ⊓ ⊔ Lemma 3.9 . L et G b e a data-exchange function, t ∈ T er ( L Σ M ) a t erm and deﬁn e P = { p | p ∈ Pos ( t ) , r oot ( t | p ) ∈ Σ } . F or every ς : P → R such that ς ( p ) is a deﬁning ru le for r oot ( t | p ) for al l p ∈ P , ther e is a data-exchange function G t,ς such that for al l p ∈ P if the term G t,ς ( t ) | p is a r e dex, then it is an ς ( p ) -re dex, and G t,ς ( s ) ≡ G ( s ) for al l s 6≡ t . Pr o of. F or all p ∈ P a nd ς ( p ) ≡ ℓ → r we alter G to obtain a data-gues s function G t,ς as follows. If q ∈ Pos ( ℓ ) such that ℓ | q is a data term and none of the t ( q · p ′ ) with p ′ a non empty preﬁx of p is a symbol fro m Σ sfun , then instantiate the data term at po sition q · p in t with an instance of the data pattern ℓ | q . Then if G t,ς ( t ) is a redex, then by o rthogonality of T it can o nly b e a ℓ → r -redex. ⊓ ⊔ History Aw are Data-Exc hange Above we hav e mo delled the o pponent using history free data -exchange strate- gies and the rewr ite play er was omniscient, that is, she alwa y s chooses the b est po ssible r ewrite s equence, which pro duces the maximum po ssible num b er of el- ement s . Now we inv estig a te the robustness of our deﬁnition. W e streng then the op- po nen t, allowing for history aw a re data-exchange stra tegies, and we weaken the rewrite play er, dr opping omnisc ie nce, a s suming only an outermost-fair rewrite strategy . How ever it turns o ut that these changes do not aﬀect the d- o pr o duc- tion rang e do T ( s ), in this w ay providing evidence for the robustness of our ﬁrst deﬁnition. Deﬁnition 3.1 0. Let T = h Σ , R i b e a stre am sp eciﬁcation. A history in T is a ﬁnite list in ( T er ( Σ ) S × R × N ∗ × T er ( Σ ) S ) ∗ of the fo r m: h s 0 , ℓ 0 → r 0 , p 0 , t 0 ih s 1 , ℓ 1 → r 1 , p 1 , t 1 i . . . h s n , ℓ n → r n , p n , t n i 14 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks such that ∀ 0 ≤ i ≤ n : s i → t i by a n a pplication of r ule ℓ i → r i at p o sition p i . W e use H T to deno te the set of all historie s o f T . A function G : H T → ( T er ( L Σ M ) S → T er ( Σ ) S ) from histor ies in T to data - exchange functions on T is c alled a history-awar e data-exchange funct ion on T . F or such a function we write G h as shor thand for G ( h ) for all h ∈ H T . Deﬁnition 3.1 1. Let T = h Σ , R i be a stream sp eciﬁcation. Let G b e a history- aw are da ta -exchange function. W e deﬁne the ARS H T , G ⊆ ( T er ( L Σ M ) S × H T ) × ( T er ( L Σ M ) S × H T ) as follows: H T , G := {h L s M , h i → h L t M , h h s, ℓ → r , p, t ii | s, t ∈ T er ( Σ ), h ∈ H T , with G h ( L s M ) = s a nd s → T t } . F or B ⊆ H T , G we deﬁne the pr o duction function Π B : T er ( Σ ) S → N by: Π B ( s ) := sup { n ∈ N | h s, ǫ i → ∗ B h u 1 : . . . : u n : t, h i } , that is, the pro duction of s starting with empty history ǫ . In an ARS H T , G we a llow to write s 0 → H T , G s 1 → H T , G s 2 → H T , G . . . when we will a ctually mea n a rewrite seq uence in H T , G of the form h s 0 , h 0 i → H T , G h s 1 , h 1 i → H T , G h s 2 , h 2 i → H T , G . . . with some histor ie s h 0 , h 1 , h 2 , . . . ∈ H T . The notion of p ositions of r ewrite steps and r e dexes extends to the ARS H T , G in the obvious w ay , s ince the steps in H T , G are pro jections of steps in T . No te that the o ppo nen t can ch ange the da ta elements in H T , G at any time, therefor e the rules might change with resp ect to which a cer tain p osition is a redex. F or this reason we adapt the notion of outermo st-fairness for o ur purp oses. Intiutiv ely , a rewr ite sequenc e is outermost-fair if e v e r y p osition which is always ev e ntually an outermost redex p osition, is r ewritten inﬁnitely often. F or mally , w e deﬁne a rewrite sequence h s 1 , h 1 i → h s 2 , h 2 i → . . . in H T , G to b e outermost-fair if it is ﬁnite, o r it is inﬁnite and whenever p is an outer mo st redex p osition for inﬁnitely many s n , n ∈ N , then ther e exist inﬁnitely man y rewrite steps s m → s m +1 , m ∈ N , at p osition p . Moreov er , a stra teg y B ⊆ H T , G is outermost-fair if every rewrite se q uence with r espec t to this strateg y is outermost-fair . Prop osition 3.12 . F or every str e am sp e ciﬁc ation T = h Σ , R i t her e exists a c omputable, deterministic om-fair stra t e gy on H T , G . Deﬁnition 3.1 3. The history awar e d-o pr o duction r ange do H T ( s ) of s ∈ T er ( L Σ M ) S is deﬁned as follows: do H T ( s ) :=  Π B ( s ) | G a histor y-aw ar e da ta-exchange function o n T , B ⊆ H T , G an outermo s t-fair strategy  . F or s ∈ T er ( Σ ) S deﬁne do H T ( s ) := do H T ( L s M ). Note that also the strategy of the rewr ite play er B ⊆ H T , G is histor y aw are since the elements o f the ARS H T , G hav e histor y a nnotations. Data-Oblivious Stream Pro ductivity 15 Lemma 3.1 4. L et T = h Σ , R i b e a str e am sp e ciﬁc ation. L et f ∈ Σ sfun with ar s ( f ) = n and ar ( f ) = n ′ , and let s 1 , . . . , s n ∈ T er ( Σ ) S , u 1 , . . . , u n ′ ∈ T er ( Σ ) D . Then it holds: do H T ( f ( s 1 , . . . , s n , u 1 , . . . , u n ′ )) = do H T ( f ( • m 1 : σ , . . . , • m n : σ , u 1 , . . . , u n ′ )) , (1) wher e m i := do H T ( s i ) for al l i ∈ { 1 , . . . , n } . Pr o of. Let T = h Σ , R i b e a str eam sp eciﬁcation. Let f ∈ Σ sfun with ar s ( f ) = n , and s 1 , . . . , s n ∈ T er ( Σ ) S . W e assume that the data ar it y of f is zer o ; in the presence o f da ta arg uments the pro of pr o ce e ds analog o usly . F urther mo re we let m i := do H T ( s i ) for i ∈ { 1 , . . . , n } . W e show (1) by demo nstrating the inequalities “ ≤ ” and “ ≥ ” in this eq uation. ≥ : Supp ose that do H T ( f ( s 1 , . . . , s n )) = Π B ( L f ( s 1 , . . . , s n ) M ) = N ∈ N for some om-fair strategy B ⊆ H T , G and so me history- aw are data-exchange function G on T . (In case that do H T ( f ( s 1 , . . . , s n )) = ∞ , nothing remains to b e shown.) Then there exists an om-fair r ewrite sequence ξ : L f ( s 1 , . . . , s n ) M → H T , G u 1 → H T , G u 2 → H T , G . . . . . . → H T , G u l → H T , G u l +1 → H T , G . . . in H T , G with lim l →∞ # : ( u l ) = N . Due to the deﬁnition of the num b e r s m i , ah history- aw are data- e x c ha nge function G ′ from H T to data-exchange functions ca n be deﬁned that ena bles a re write sequence ξ ′ : f ( • m 1 : σ , . . . , • m n : σ ) → = H T , G ′ u ′ 1 → = H T , G ′ u ′ 2 → = H T , G ′ . . . . . . → = H T , G ′ u ′ l → = H T , G ′ u ′ l +1 → = H T , G ′ . . . with the pr o per ties that # : ( u l ) = # : ( u ′ l ) ho lds for all l ∈ N , and that ξ ′ is the pro jection of ξ to a rewr ite sequence with source f ( • m 1 : σ , . . . , • m n : σ ). More precisely , ξ ′ arises a s follows: Steps u l → H T , G u l +1 in ξ that co ntract redexes in descendants of the outermost sym b ol f in the source o f ξ give rise to s teps u ′ l → H T , G ′ u ′ l +1 that co ntract redex e s at the same p ositio n and apply the same rule. This is p ossible b ecause, due to the deﬁnition of the nu mber s m i , always enoug h pebble- supply is gua ranteed to ca rry out steps in ξ ′ corres p onding to such steps in ξ . Steps u l → H T , G u l +1 that contract redexes in descendants o f one of the subterms s 1 , . . . , s n in the source of ξ pr o ject to empty steps u ′ l = u ′ l +1 in ξ ′ . (On his tories h ∈ H T and terms r ∈ T er ( L Σ M ) S o ccurring in ξ , G ′ is deﬁned in such a wa y as to make this pro jection p oss ible. On histor ies h and ter ms r that do not o ccur in ξ ′ , G ′ is deﬁned arbitrarily with the only r estriction L G h ( r ) M ensuring that G ′ behaves as a history-aware da ta-exchange function on these terms.) By its co nstruction, ξ ′ is ag ain om-fair. Now let B ′ be the extension of the sub-ARS o f H T , G induced by ξ ′ to a deterministic om-fair strategy for H T , G ′ . (Cho ose an a r bitrary deterministic om-fair s trategy ˜ B on H T , G , which is 16 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks po ssible by Prop osition 3.12. On term-histor y pa irs that do not o ccur in ξ ′ , deﬁne B ′ according to ˜ B .) Then ξ ′ is also a rewrite sequence in B ′ that witnesses Π B ( f ( • m 1 : σ, . . . , • m n : σ )) = lim l →∞ # : ( u ′ l ) = lim l →∞ # : ( u l ) = N = Π B ( L f ( s 1 , . . . , s n ) M ). Now it follows: do H T ( f ( • m 1 : σ , . . . , • m n : σ )) ≤ Π B ′ ( f ( • m 1 : σ , . . . , • m n : σ )) = Π B ( L f ( s 1 , . . . , s n ) M ) = do H T ( f ( s 1 , . . . , s n )) , establishing “ ≥ ” in (1). ≤ : Supp ose that do H T ( f ( • m 1 : σ, . . . , • m n : σ )) = Π B ( f ( • m 1 : σ , . . . , • m n : σ )) = N for some N ∈ N , and an om- fa ir stra tegy B ⊆ H T , G and a histo ry-aw ar e data- exchange function G on T . (In cas e that do H T ( f ( • m 1 : σ, . . . , • m n : σ )) = ∞ , nothing remains to be shown.) Then there ex ists an om-fair rewr ite sequence ξ : f ( • m 1 : σ , . . . , • m n : σ ) → H T , G u 1 → H T , G u 2 → H T , G . . . → H T , G u l → H T , G u l +1 → H T , G . . . in H T , G with lim l →∞ # : ( u l ) = N . Let l 0 ∈ N b e minimal such tha t # : ( u l 0 ) = N . F urthermore, let, for all i ∈ { 1 , . . . , n } , m ′ i ≤ m i be minimal such that each of the topmost m ′ i symbols “ • ” of the subterm • m i : σ of the source f ( • m 1 : σ, . . . , • m n : σ ) of ξ is part o f a redex pa ttern at so me step among the ﬁrst l 0 steps o f ξ . Since by assumption, we have m i := do H T ( s i ) for i ∈ { 1 , . . . , n } , there exis t, for ea c h i ∈ { 1 , . . . , n } , histor y-aw ar e data-guess functions G i , a nd om-fair strategies B i ⊆ H T , G i such tha t Π B i ( L s i M ) = m i ; let all B i and G i be chosen as such. Then there exist, for each i , om-fair rewrite sequences o f the form: ξ i : L s i M → H T , G i u i, 1 → H T , G i . . . → H T , G i u i,l i = • m ′ i : ˜ u i,l i → H T , G i . . . in H T , G with lim l →∞ # : ( u i,l ) = m i . Now it is p ossible to combine the histor y-aw are data-ex change functions G , G 1 , . . . , G n int o a function G ′ that makes it po ssible to combine the re w r ite sequences ξ and ξ 1 , . . . , ξ n int o a rewrite s e quence of the form: ξ ′ : L f ( s 1 , . . . , s n ) M → ∗ H T , G ′ f ( • m ′ 1 : u 1 ,l 1 , . . . , • m ′ n : u n,l n ) (carry out the ﬁr st l i steps in ξ i in the context f ( 2 1 , . . . , 2 n )) → H T , G ′ u ′ 1 → H T , G ′ u ′ 2 → H T , G ′ . . . → H T , G ′ u ′ l 0 = • N : ˜ u ′ l 0 (parallel to the ﬁrs t l 0 steps o f ξ ) → H T , G ′ w 1 → H T , G ′ w 2 → H T , G ′ . . . (fair interleaving of the rests of the ξ i put in context and of steps para llel to the rest of ξ ′ ) By its construction, ξ ′ is ag a in o m-fair and it holds: lim l →∞ w l = N . Now let B ′ be the extensio n of the s ub- ARS of H T , G induced by ξ ′ to a deterministic Data-Oblivious Stream Pro ductivity 17 om-fair strategy for H T , G ′ . (Aga in choose an arbitrar y deterministic om-fair strategy ˜ B o n H T , G , which is p ossible by Pro p os itio n 3.1 2. On term-history pairs that do not o ccur in ξ ′ , deﬁne B ′ according to ˜ B .) Then ξ ′ is also a rewrite sequence in B ′ that w itnes s es Π B ( L f ( s 1 , . . . , s n ) M ) = lim l →∞ # : ( w l ) = N = Π B ( f ( • m 1 : σ , . . . , • m n : σ )). Now it follows: do H T ( f ( s 1 , . . . , s n )) ≤ Π B ( L f ( s 1 , . . . , s n ) M ) = Π B ′ ( f ( • m 1 : σ , . . . , • m n : σ )) = do H T ( f ( • m 1 : σ , . . . , • m n : σ )) , establishing “ ≤ ” in (1). ⊓ ⊔ Lemma 3.1 5. L et s ∈ T er ( L Σ M ) S b e a str e am t erm and φ : V ar S → T er ( L Σ M ) S a substitu tion of str e am variables. Then do H T ( s φ ) = do H T ( s ϕ ) wher e ϕ : V ar S → T er ( L Σ M ) S is deﬁne d by ϕ ( τ ) = • m τ : σ , and m τ := do H T ( φ ( τ )) . Pr o of. Induction using Le m. 3.14 . ⊓ ⊔ Corollary 3.1 6. L et s ∈ T er ( L Σ M ) S and φ, ϕ : V ar S → T er ( L Σ M ) S substitutions of str e am variables such that do H T ( φ ( τ )) = do H T ( ϕ ( τ )) for al l τ ∈ V ar S . Then do H T ( s φ ) = do H T ( s ϕ ) . Pr o of. Two applications of Lem. 3.15. ⊓ ⊔ W e deﬁne a histor y free data-exchange function G T such that no reduction with res pect to G T pro duces mor e than do H T ( s ) elements. Deﬁnition 3.1 7. Let a strea m sp eciﬁcation T = h Σ , R i b e given, and let > b e a w ell-founded order on R . W e deﬁne the lower b ou n d data-exchange fun ction G T : T er ( L Σ M ) S → T er ( Σ ) S as follows. Let s ∈ T er ( L Σ M ) S and f ( s 1 , . . . , s ar s ( f ) , • 1 , . . . , • ar d ( f ) ) a subterm o ccurrence of s . F or i = 1 , . . . , ar s ( f ) let n i ∈ N b e maximal such that s i ≡ • i, 1 : . . . • i,n i : s ′ i . W e deﬁne m i := do H T ( s i ) for 1 ≤ i ≤ n , then we hav e by Lem. 3.1 4: do H T ( t := f ( s 1 , . . . , s ar s ( f ) , • 1 , . . . , • ar d ( f ) )) = do H T ( t ′ := f ( • m 1 : σ , . . . , • m ar s ( f ) : σ , • 1 , . . . , • ar d ( f ) )) . W e choo se a deﬁning rule ρ of f as follows. I n case there exists a rule γ : f ( u 1 : σ 1 , . . . , u n : σ n , v 1 , . . . , v ar d ( f ) ) → t ∈ R such that m i < | u i | for some i ∈ { 1 , . . . , n } , then let ρ := γ . In ca s e ther e a re m ultiple p ossible choices for γ , we pick the minimal γ with r esp ect to > . Otherwise in exists a history-aw ar e data -exchange function G , and an outermo s t- fair rewrite sequence σ : t ′ → H T , G . . . in H T , G pro ducing only do H T ( t ′ ) element s . F rom exhaustivity o f T we get G ( t ′ ) is not a normal form, since all rules have 18 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks enough supply . Moreover, by orthogonality e xactly one deﬁning rule γ of f is ap- plicable, let ρ := γ . Ag a in, in ca se there a r e multiple p oss ible choices for γ , due to freedom in the choice of the data- exchange function G , we take the minimal γ with resp ect to > . W e deﬁne G T to insta n tiate: – for 1 ≤ i ≤ ar d ( f ) the o ccur rences of p ebbles • i in s , and – for 1 ≤ i ≤ ar s ( f ), 1 ≤ j ≤ n i the o ccurr ences of p ebbles • i,j in s with res pect to the rule ρ (Def. 3.7). Rewrite steps with resp ect to the low er bound data- e xch ange function G T do not change the d-o low er b ound of the pro duction o f a ter m. Lemma 3.1 8. F or al l s, s ∈ T er ( L Σ M ) S : s → A T , G T t implies do H T ( s ) = do H S ( t ) . Pr o of. W e use the notions intro duced in Def. 3.17. Note that the low er b ound data-exchange function G T is de ﬁned in a w ay that the p ebbles a re instantiated independent of the context ab ov e the stream function sym b ol. There fo re it is suﬃcient to conside r rew r ite steps s → A T , G T t a t the ro ot, closur e under con tex ts follows fro m Cor. 3 .16. ⊓ ⊔ Corollary 3.1 9. F or al l s ∈ T er ( L Σ M ) S we have: Π A T , G T ( s ) = do H T ( s ) . Pr o of. Direct conseq ue nc e of Lem. 3.18. ⊓ ⊔ Hence, as a co nsequence of Cor. 3.19 , the low er b ound o f the his tory-free d-o pro duction conincides with the lower b ound of the history-aw a re d-o pro duction. Lemma 3.2 0. F or al l s ∈ T er ( Σ ) S we have: do T ( s ) = do H T ( s ) . Pr o of. The direction do T ( s ) ≥ do H T ( s ) is trivial, and do T ( s ) ≤ do H T ( s ) follows from Cor . 3.19, that is, using the history-free data -exchange function G T no reduction pro duces more than do H T ( s ) elements. ⊓ ⊔ Lemma 3.2 1. F or al l s ∈ T er ( Σ ) S we have: do T ( s ) = do H T ( s ) . Pr o of. ⊆ Let n ∈ do T ( s ). Then ther e ex ists a data - exchange function G such that Π A T , G ( s ) = n , and a rewrite sequence σ : s 0 → A T , G . . . → A T , G s m with s m ≡ u 1 : . . . : u n : t . W e deﬁne for all h ∈ H T a data-exchange function G h by setting G h ( s ) := G ( s ) fo r all s ∈ T er ( L Σ M ) S . Mo reov er, we deﬁne the strategy B ⊆ H T , G to execute σ and con tin ue outermos t- fa ir afterwards. Since s m do es not produce mor e than n elements, ev ery maximal rewrite sequence with res pect to B will pro duce exactly n elements. Hence n ∈ do H T ( s ). ⊇ Let n ∈ do H T ( s ). Then there exist data- exchange functions G h for every h ∈ H T , and an o utermost-fair strateg y B ⊆ H T , G such that Π B ( s ) = n . There exists a (p ossibly inﬁnite) maxima l, outermost-fair r ewrite sequence σ : h s, ǫ i ≡ h s 0 , h 0 i → B . . . → B h s m , h m i → B . . . Data-Oblivious Stream Pro ductivity 19 such that s m ≡ u 1 : . . . : u n : t . Since the elements of H T , G are annotated with their histor y , we do not enco un ter rep etitions during the rewr ite sequence. Hence, w.l.o.g. we can a ssume that B is a deterministic stra tegy , a dmitting only the maximal r ewrite s e quence σ . Disregar ding the histo r y a nnotations there might exist re p etitions, that is, i < j ∈ N with s i ≡ s j . Nevertheless, from σ we ca n construct a history free data-exchange function G as follows. F or all s ∈ T er ( L Σ M ) S let S := { i ∈ N | s i ≡ s } and deﬁne: G ′ ( s ) :=      G h j ( s j ) if j := sup S < ∞ G h k ( s k ) if sup S = ∞ , k := inf S arbitrar y if ¬∃ i ∈ N . s i ≡ s ⊓ ⊔ 4 The Pro duction Calculus As a means to compute the d- o pro duction b ehaviour of strea m sp eciﬁcations, we introduce a ‘pro ductio n ca lculus’ with p erio dica lly increasing functions a s its central ingr edien t. This calculus is the term eq uiv a len t of the calculus of pebbleﬂow nets that was introduced in [2], see Se c . 4.3. 4.1 P erio d i cally Increasing F unctions W e use N := N ⊎ {∞} , the extende d natu r al numb ers , with the usual ≤ , +, a nd we deﬁne ∞ − ∞ := 0 . An inﬁnite sequence σ ∈ X ω is eventual ly p erio dic if σ = αβ β β . . . for s o me α ∈ X ∗ and β ∈ X + . A function f : N → N is eventual ly p erio dic if the sequence h f (0) , f (1 ) , f (2 ) , . . . i is even tually p erio dic. Deﬁnition 4.1 . A function g : N → N is called p erio dic al ly incr e asing if it is increasing , i.e. ∀ n. g ( n ) ≤ g ( n + 1), and the derivative of g , n 7→ g ( n + 1) − g ( n ), is even tua lly p erio dic. A function h : N → N is called p erio di- c al ly incr e asing if its restr ic tion to N is p erio dica lly increas ing and if h ( ∞ ) = lim n →∞ h ( n ). Finally , a k -ary function i : ( N ) k → N is called p erio dic al ly in- cr e asing if i ( n 1 , ..., n k ) = min( i 1 ( n 1 ) , . . . , i k ( n k )) for some unary p erio dically increasing functions i 1 , . . . , i k . Periodica lly increasing (p-i) functions can b e denoted by their v alue at 0 follow ed by a r epresentation o f their deriv ative. F or ex ample, 03 12 denotes the p-i function f : N → N with v a lues 0 , 3 , 4 , 6 , 7 , 9 , . . . . How e v e r , we use a ﬁner and more ﬂe x ible no ta tion ov er the alphab et { + , −} that will b e useful for computing several o per a tions on p-i functions. F or instance, we represent f as above by the ‘io-sequence’ − +++ − + − ++ = − +++ − + − ++ − + − ++ − + − + + . . . , in turn abbrevia ted by the ‘io- term’ h− +++ , − + − ++ i . 20 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks Deﬁnition 4.2 . Let ± := { + , − } . An io-se quenc e is a ﬁnite ( ∈ ± ∗ ) or inﬁnite sequence ( ∈ ± ω ) over ± , the set of io-sequences is denoted by ± ∞ := ± ∗ ∪ ± ω . W e let ⊥ denote the empty io- sequence, and use it as an ex plicit end marker. An io-term is a pair h α, β i of ﬁnite io-sequences α, β ∈ ± ∗ with β 6 = ⊥ . The set of io-ter ms is denoted b y I , a nd we use ι, κ to range ov er io-ter ms. F or a n io-sequence σ ∈ ± ω and an io-term ι = h α, β i ∈ I we say that ι denotes σ if σ = α β where β stands for the inﬁnite sequence β β β . . . . A sequence σ ∈ ± ∞ is r ational if it is ﬁnite or if it is denoted b y an io-term. T he se t of rational io-sequences is denoted by ± ∞ R . An inﬁnite sequence σ ∈ ± ω is pr o ductive if it contains inﬁnitely many +’s: σ ∈ ± ω is pr o ductive ⇐ ⇒ ∀ n. ∃ m ≥ n. σ ( m ) = + . W e let ± ω P denote the set of pro ductive sequences and deﬁne ± ∞ P := ± ∗ ∪ ± ω P . The reaso n for having b oth io-terms and io-s e quences is that, dep ending on the purp ose at hand, one o r the other is mor e conv enient to work with. Op erations are often easier to unders ta nd o n io -sequences than on io - terms, where as we need ﬁnite repr esentations to b e able to compute these op erations . Deﬁnition 4.3 . W e deﬁne [ [ σ ] ] : N → N , the interpr etation of σ ∈ ± ∞ , by: [ [ ⊥ ] ]( n ) = 0 (2) [ [+ σ ] ]( n ) = 1 + [ [ σ ] ]( n ) (3) [ [ − σ ] ](0) = 0 (4) [ [ − σ ] ]( n + 1) = [ [ σ ] ]( n ) (5) for all n ∈ N , and extend it to N → N by deﬁning [ [ σ ] ]( ∞ ) = lim n →∞ [ [ σ ] ]( n ). W e say that [ [ σ ] ] interpr ets σ , and, conversely , that σ r epr esents [ [ σ ] ]. W e overload notation and deﬁne [ [ ] ] : I → ( N → N ) b y [ [ h α, β i ] ] := [ [ αβ ] ]. It is easy to v e r ify tha t, for every σ ∈ ± ∞ , the function [ [ σ ] ] is increasing , and that, for every ι ∈ I , the function [ [ ι ] ] is p erio dica lly incr easing. F urthermor e, every increasing function is r epresented b y an io-sequence, and every p-i function is deno ted b y an io-term. Subsequently , for an e v e ntually co nstant function f : N → N , w e write f for the shortest ﬁnite io- sequence repr esent ing f (trailing − ’s can b e removed). F or an alwa ys even tually strictly increasing function f : N → N , i.e. ∀ n. ∃ m > n. f ( m ) > f ( n ), we write f for the unique io-s equence representing f ; note that then f ∈ ± ω P follows. F or a p erio dically incr easing function f : N → N , we w r ite f for the io-term ι = h α, β i such that [ [ ι ] ] = f and | α | + | β | is minimal; uniqueness follows fro m the following lemma: Lemma 4.4 . F or al l p-i functions f : N → N , the term f ∈ I is unique. Data-Oblivious Stream Pro ductivity 21 Pr o of. Co ns ider the ‘c o mpression’ TRS consisting o f the rules : h αx, β x i → h α, xβ i ( x ∈ ± ) (6) h α, β p i → h α, β i ( n > 1) (7) Clearly , ι → κ implies | ι | > | κ | wher e, for ι = h α, β i , | ι | := | α | + | β | . Hence , compressio n is terminating. There are tw o cr itical pair s due to an ov er lap of rule (7) with itself, a nd in rules (6 ) and (7): hh α, β n i , h α, β m ii orig ina ting from a term of the form h α, β n · m i , and hh α, ( xβ ) n i , h αx, β x ii originating fr om a term o f the form h αx, ( β x ) n i , r e- sp ectiv ely . Both pa irs are ea sily joinable in one step: the ﬁrst to h α, β i , a nd the second to h α, xβ i . Hence, the system is lo cally conﬂuent. Therefore, b y Newma n’s Lemma, it is a ls o co nﬂuen t, and no r mal forms a re unique. Finally , assume that t wo io - terms h α 1 , β 1 i , h α 2 , β 2 i ∈ I each of minimal length repr esent the sa me p-i function f . It is ea sy to see that then α 1 β 1 = α 2 β 2 . It follows that there exist h α, β i such that h α, β i → ∗ h α 1 , β 1 i and h α, β i → ∗ h α 2 , β 2 i . By conﬂuence and termination of compre s sion it follows that they hav e the same normal form h α 0 , β 0 i . Since compression reduces the length of io-terms it must follow tha t h α 1 , β 1 i = h α 0 , β 0 i = h α 2 , β 2 i . ⊓ ⊔ Note that we hav e that [ [ f ] ] = f fo r all increa s ing functions f , and [ [ f ] ] = f for all p-i functions f . As an example one can check tha t the in ter pretation of the afor emen tio ne d io -term f = h − +++ , − + − ++ i indeed ha s the sequence 0 , 3 , 4 , 6 , 7 , 9 , . . . as its g raph. R emark 4.5. Note that Def. 4.3 is well-deﬁned due to the pr o ductivit y requir e- men t on inﬁnite io-s e q uences. This ca n b e seen as follows: star ting o n [ [ σ ] ]( n ), after ﬁnitely many , say m ∈ N , − ’s, we either ar r ive at rule (3) (if m ≤ n ) o r a t rule (4) (if m > n ). Had we no t r e quired inﬁnite sequences σ ∈ ± ω to contain inﬁnitely many + ’s, then, e.g ., the computation o f [ [ −−− . . . ] ]( ∞ ) would co nsist of applications o f r ule (5) only , and hence would not lea d to an inﬁnite no rmal form. Prop osition 4.6. Unary p erio dic al ly incr e asing functions ar e close d under c om- p osition and p ointwise inﬁmum. W e wan t to co nstructively deﬁne the oper ations of comp osition, p oint wise in- ﬁm um, and least ﬁxed p oint calculation of increa sing functions (p-i functions) on their io- sequence (io - term) r epresentations. W e ﬁrst deﬁne these o per ations on io-s e quences by means of co inductiv e clauses. The op erations on io-terms are mo re in volved, beca use they require ‘lo op chec king’ on top. W e will pro ceed by showing that the operatio ns of c o mpo s ition and point wise inﬁm um on io- sequences preser v e r ationality . This will then give r ise to the needed algorithms for computing the op erations on io - terms. 22 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks Deﬁnition 4.7 . The oper ation c omp osition ◦ : ± ∞ P × ± ∞ P → ± ∞ P , h σ , τ i 7→ σ ◦ τ of (ﬁn ite or pr o ductive) io-se quenc es is deﬁned co inductiv e ly as follows: ⊥ ◦ τ = ⊥ (8) + σ ◦ τ = +( σ ◦ τ ) (9) − σ ◦ ⊥ = ⊥ (1 0) − σ ◦ + τ = σ ◦ τ (11) − σ ◦ − τ = − ( − σ ◦ τ ) (12) An a rgument simila r to Rem. 4.5 concer ning well-deﬁnedness applies for Def. 4.7 . R emark 4.8. Note that the deﬁning rules (8 )– (12) ar e orthogona l and exhaust all cases. As it stands, it is not immediately clea r that this deﬁnition is well-deﬁned (pro ductive!), the pr oblematic c a se b eing (11 ). Rule (11) is to b e thought of as an in ter nal communication b etw een comp onents σ and τ , as a silent step in the black b ox σ ◦ τ . How to guarantee tha t alwa ys , after ﬁnitely ma ny internal steps , either the pro cess ends or there will b e an ex ter nal step, in the for m o f o utput or a requir e men t for input? The recur sive call in (11) is not guarded. How ever, well-deﬁnedness o f comp osition can b e justiﬁed as follows: Consider arbitrar y sequences σ , τ ∈ ± ∞ P . B y deﬁnitio n o f ± ∞ P , ther e e x ists m ∈ N such that σ = − m + σ ′ for so me σ ′ ∈ ± ∞ P or σ = − m ⊥ ; likewise there exists n ∈ N such that τ = − n + τ ′ for some τ ′ ∈ ± ∞ P , or τ = − n ⊥ . Rules (11 ) and (12) a re decreasing with resp ect to the lexicographic order on ( m, n ). After ﬁnitely many applications of (11) and (12), one o f the r ules (8)–(10) must b e applied. Hence comp osition is well-deﬁned and the s equence pro duced is a n element of ± ∞ P , i.e. either it is a ﬁnite seq uence or it contains an inﬁnite nu mber o f +’s. Lemma 4.9 . Comp osition of io-se quenc es is asso ciative. Pr o of. Let R = { h σ ◦ ( τ ◦ υ ) , ( σ ◦ τ ) ◦ υ i | σ, τ , υ ∈ ± ∞ P } . T o show that R is a bisimulation, we prove that, for all σ, τ , υ , φ 1 , φ 2 ∈ ± ∞ P , if φ 1 = σ ◦ ( τ ◦ υ ) and φ 2 = ( σ ◦ τ ) ◦ υ , then either φ 1 = ⊥ = φ 2 or head ( φ 1 ) = head ( φ 2 ) and h tail ( φ 1 ) , tail ( φ 2 ) i ∈ R , by induction on the num b er n ∈ N of leading − ’s of σ 4 , and a sub-induction on the nu mber m ∈ N of leading − ’s of τ . If n = 0 a nd σ = ⊥ , then φ 1 = ⊥ = φ 2 . If σ = + σ ′ , we hav e φ 1 = +( σ ′ ◦ ( τ ◦ υ )), and φ 2 = +(( σ ′ ◦ τ ) ◦ υ ), a nd so h ta il ( φ 1 ) , tail ( φ 2 ) i ∈ R . If n > 0, then σ = − σ ′ , a nd we pr o c eed by s ub-induction o n m . If m = 0 and τ = ⊥ , then φ 1 = ⊥ = φ 2 . If τ = + τ ′ , we compute φ 1 = σ ′ ◦ ( τ ′ ◦ υ ), and φ 2 = ( σ ′ ◦ τ ′ ) ◦ υ , and conclude by IH. If m > 0 , then τ = − τ ′ , and we pro ceed by ca se distinction on υ . If υ = ⊥ , then φ 1 = ⊥ = φ 2 . If υ = + υ ′ , we compute φ 1 = σ ◦ ( τ ′ ◦ υ ′ ), a nd φ 2 = ( σ ◦ τ ′ ) ◦ υ ′ , and conclude by sub-IH. Finally , if υ = − υ ′ , then φ 1 = − ( σ ◦ ( τ ◦ υ ′ )), and φ 2 = − (( σ ◦ τ ) ◦ υ ′ ). Clear ly h tail ( φ 1 ) , tail ( φ 2 ) i ∈ R . ⊓ ⊔ 4 By deﬁnition of ± ∞ P the num b er of leading − ’s of any sequence in ± ∞ P is ﬁnite: for all σ ∈ ± ∞ P there exists n ∈ N such that either σ = − n ⊥ or σ = − n + σ ′ . Data-Oblivious Stream Pro ductivity 23 R emark 4.10. If we a llow to us e extended na tural num b ers a t pla c e s where an io-sequence is exp ected, using a co ercion n 7→ + n , with + 0 = ⊥ , o ne obser ves that the interpretation of a n io-se quence (Def. 4.3) is just a sp ecial case o f comp osition: [ [ σ ] ]( n ) = σ ◦ n . Prop osition 4.11 . F or al l incr e asing functions f , g : N → N : [ [ f ◦ g ] ] = [ [ f ] ] ◦ [ [ g ] ] . Pr o of. Immediate from Rem. 4.10 , and Lem. 4.9. ⊓ ⊔ Next, we show that c ompo sition of io-sequences pre s erves ratio na lit y . Lemma 4.1 2. If σ, τ ∈ ± ∞ R , then σ ◦ τ ∈ ± ∞ R . Pr o of. Let σ, τ ∈ ± ∞ R and set σ 0 := σ a nd τ 0 := τ . In the rewrite sequence starting with σ 0 ◦ τ 0 each of the steps is either of the form: σ n ◦ τ n → C n [ σ n +1 ◦ τ n +1 ] , (13) where σ n +1 ∈ { σ n , tail ( σ n ) } , τ n +1 ∈ { τ n , tail ( τ n ) } and C n ∈ {  , + :  , − :  } , or the rewr ite sequence ends with a step of the form: σ n ◦ τ n → ⊥ . In the latter ca se, σ ◦ τ results in a ﬁnite and he nc e rational io-sequence. Otherwise the rewrite se quence is inﬁnite, consisting o f s teps of form (13) only . F or θ ∈ ± ∗ we deﬁne the context C θ inductively: C θ ≡  , for θ ≡ ⊥ ; and C + θ ≡ + C θ as well as C − θ ≡ − C θ , for all θ ∈ ± ∗ . Becaus e the se quences σ , τ a re rationa l, the sets { σ n | n ∈ N } and { τ n | n ∈ N } are ﬁnite. Then, the pigeonhole principle implies the existence o f i, j ∈ N such that i < j , σ i = σ j and τ i = τ j . Now let α, γ ∈ ± ∗ with γ 6 = ⊥ s uc h tha t C α ≡ C 0 [ . . . C i − 1 [ C i ] . . . ], C γ ≡ C i +1 [ . . . C j − 1 [ C j ] . . . ], and: σ 0 ◦ τ 0 → ∗ C α [ σ i ◦ τ i ] , and σ i ◦ τ i → ∗ C γ [ σ j ◦ τ j ] = C γ [ σ i ◦ τ i ] . Then we ﬁnd an even tually ‘spir alling’ rewr ite seq uenc e : σ 0 ◦ τ 0 → ∗ C α [ σ i ◦ τ i ] = α ( σ i ◦ τ i ) → ∗ C α [ C γ [ σ i ◦ τ i ]] = αγ ( σ i ◦ τ i ) → ∗ C α [ C γ [ C γ [ σ i ◦ τ i ]]] = αγ γ ( σ i ◦ τ i ) , and therefore σ 0 ◦ τ 0 ։ ։ α γ . This shows that σ ◦ τ is ratio nal. ⊓ ⊔ Next, we deﬁne the op era tion of p oint wise inﬁmum o f increa sing functions on io-se q uences. 24 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks Deﬁnition 4.1 3. The op eration p ointwise inﬁmum ∧ : ± ∞ × ± ∞ → ± ∞ , h σ , τ i 7→ σ ∧ τ of io-se quenc es is deﬁned c o inductiv e ly by the following eq uations: ⊥ ∧ τ = ⊥ del ( ⊥ ) = ⊥ σ ∧ ⊥ = ⊥ del (+ σ ) = + del ( σ ) + σ ∧ + τ = +( σ ∧ τ ) d e l ( − σ ) = σ − σ ∧ τ = − ( σ ∧ del ( τ )) ( τ 6 = ⊥ ) σ ∧ − τ = − ( del ( σ ) ∧ τ ) ( σ 6 = ⊥ ) where del ( σ ) remov es the ﬁrst r equirement of σ ∈ ± ∞ (if any). W e let ◦ bind stronger than ∧ . Requirement remov al dis tributes over p oint wise inﬁmum: Lemma 4.1 4. F or al l σ, τ ∈ ± ∞ P , it holds that: del ( σ ∧ τ ) = del ( σ ) ∧ del ( τ ) . Pr o of. Check that {h del ( σ ∧ τ ) , del ( σ ) ∧ del ( τ ) i | σ, τ ∈ ± ∞ } ∪ {h σ , σ i | σ ∈ ± ∞ } is a bisimulation. ⊓ ⊔ Lemma 4.1 5. Inﬁmum is idemp otent, c ommutative, and asso ciative. Pr o of. By coinduction, with the use of Lem. 4 .14 in ca se of ass o ciativity . ⊓ ⊔ In a comp osition req uirement s co me from the second co mponent: Lemma 4.1 6. F or al l σ, τ ∈ ± ∞ P , it holds that: del ( σ ◦ τ ) = σ ◦ del ( τ ) . Comp osition distributes bo th left and rig h t ov er p oint wise inﬁmum: Lemma 4.1 7. F or al l σ, τ , υ ∈ ± ∞ P , it holds that: σ ◦ ( τ ∧ υ ) = σ ◦ τ ∧ σ ◦ υ . Pr o of. By co induction; let L = {h σ ◦ ( τ ∧ υ ) , σ ◦ τ ∧ σ ◦ υ i | σ , τ , υ ∈ ± ∞ P } . T o show tha t L is a bisim ulation, we prov e that, for all σ , τ , υ , φ 1 , φ 2 ∈ ± ∞ P , if φ 1 = σ ◦ ( τ ∧ υ ) and φ 2 = σ ◦ τ ∧ σ ◦ υ , then e ither φ 1 = ⊥ = φ 2 , or head ( φ 1 ) = head ( φ 2 ) and h tail ( φ 1 ) , tail ( φ 2 ) i ∈ L , b y induction o n the n umber n ∈ N o f leading − ’s of σ . If n = 0 and σ = ⊥ , then φ 1 = ⊥ = φ 2 . If n = 0 and σ = + σ ′ , then: φ 1 = +( σ ′ ◦ ( τ ∧ υ )), and φ 2 = +( σ ′ ◦ τ ∧ σ ′ ◦ υ ), and so h tail ( φ 1 ) , tail ( φ 2 ) i ∈ L . If n > 0 and σ = − σ ′ , we pro ceed by case ana lysis of τ and υ . If one of τ , υ is empty , then φ 1 = ⊥ = φ 2 . If τ = + τ ′ and υ = + υ ′ , then φ 1 = σ ′ ◦ ( τ ′ ∧ υ ′ ) and φ 2 = σ ◦ τ ∧ σ ◦ υ , and we conclude by the induction h y pothes is. If τ = + τ ′ and υ = − υ ′ , then φ 1 = − ( σ ◦ ( del ( τ ) ∧ υ ′ )) and φ 2 = − ( del ( σ ◦ τ ) ∧ σ ◦ υ ′ ) = − ( σ ◦ del ( τ ) ∧ σ ◦ υ ′ ) by Lem. 4.1 6. Th us h tail ( φ 1 ) , tail ( φ 2 ) i ∈ L . The case τ = − τ ′ , υ = + υ ′ is proved similarly . Finally , if τ = − τ ′ and υ = − υ ′ , w e compute φ 1 = − ( σ ◦ ( τ ′ ∧ υ ′ )) and φ 2 = − ( σ ◦ τ ′ ∧ σ ◦ υ ′ ), and conclude h tai l ( φ 1 ) , tail ( φ 2 ) i ∈ L . ⊓ ⊔ Data-Oblivious Stream Pro ductivity 25 Lemma 4.1 8. F or al l σ, τ , υ ∈ ± ∞ P , it holds that: ( τ ∧ υ ) ◦ σ = τ ◦ σ ∧ υ ◦ σ . Pr o of. Analo gous to the pro of o f Lem. 4.1 7. ⊓ ⊔ The op eration of p oint wise inﬁm um of (p erio dically ) incr easing functions is deﬁned on their io -sequence (io-term) representations. Prop osition 4.19 . F or al l incr e asing functions f , g : N → N : [ [ f ∧ g ] ] = [ [ f ] ] ∧ [ [ g ] ] . Pr o of. Immediate from Rem. 4.10 a nd Lem. 4.18. ⊓ ⊔ W e give a coinductive deﬁnition o f the calcula tion o f the least ﬁxed p oint of an io-se q uence. The op eratio n del was deﬁned in Def. 4 .13. Deﬁnition 4.2 0. The op er ation ﬁx : ± ∞ → N c omputing the le ast ﬁxe d p oint of a s e qu enc e σ ∈ ± ∞ , is deﬁned, for all σ ∈ ± ∞ , by: ﬁx ( ⊥ ) = 0 ﬁx (+ σ ) = 1 + ﬁx ( del ( σ )) ﬁx ( − σ ) = 0 Remov al of a requir emen t and feeding an input hav e equal eﬀect: Lemma 4.2 1. F or al l σ, τ ∈ ± ∞ P , it holds that: del ( σ ) ◦ τ = σ ◦ (+ τ ) . Pr o of. W e show that R = {h del ( σ ) ◦ τ , σ ◦ (+ τ ) i } ∪ {h σ , σ i} is a bis im ulatio n, b y case a nalysis of σ . F or σ = ⊥ , and σ = − σ ′ , h del ( σ ) ◦ τ , σ ◦ (+ τ ) i ∈ R follows by reﬂexivity . If σ = + σ ′ , then del ( σ ) ◦ τ = +( del ( σ ′ ) ◦ τ ), and σ ◦ (+ τ ) = + ( σ ′ ◦ (+ τ )). Hence, h tail ( de l ( σ ) ◦ τ ) , tail ( σ ◦ (+ τ )) i ∈ R . ⊓ ⊔ The following prop osition states that ﬁx ( σ ) is a ﬁxed po in t of [ [ σ ] ]: Prop osition 4.22 . F or al l σ ∈ ± ∞ P , it holds that: [ [ σ ] ]( ﬁx ( σ )) = ﬁx ( σ ) . Pr o of. W e prov e ﬁ x ( σ ) = σ ◦ ﬁx ( σ ) by case analysis and coinduction. If σ = ⊥ , then ﬁ x ( σ ) = ⊥ = σ ◦ ﬁx ( σ ). If σ = − σ ′ , then ﬁx ( σ ) = ⊥ , and σ ◦ ﬁ x ( σ ) = ( − σ ′ ) ◦ ⊥ = ⊥ . If σ = + σ ′ , then ﬁx ( σ ) = + ﬁx ( del ( σ ′ )) and σ ◦ ﬁx ( σ ) = +( σ ′ ◦ (+ ﬁx ( del ( σ ′ )))), and we have to prov e that ﬁx ( del ( σ ′ )) = σ ′ ◦ (+ ﬁx ( del ( σ ′ ))) which follows by an instance of Lem. 4 .21: σ ′ ◦ (+ ﬁx ( del ( σ ′ ))) = del ( σ ′ ) ◦ ﬁx ( del ( σ ′ )). ⊓ ⊔ Lemma 4.2 3. F or al l σ ∈ ± ∞ , it holds that: lfp([ [ σ ] ]) = ﬁ x ( σ ) . Pr o of. 26 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks 4.2 Computing Pro duction W e intro duce a term syn tax fo r the productio n calculus and rewrite r ules for ev aluating clos e d terms. Deﬁnition 4.2 4. Let X be a set of recurs ion v aria bles . The s et of pr o duction terms P is gener ated by: p ::= k | x | • ( p ) | f ( p ) | µx.p | min ( p, p ) where x ∈ X , for k ∈ N , the symbol k is a numer al (a term representation) for k , and, for a unar y p-i function f : N → N , f ∈ I , the io -term re pr esenting f . F or every ﬁnite set P = { p 1 , . . . , p n } ⊆ P , we use min ( p 1 , . . . , p n ) and min P as shorthands fo r the pr o duction term min ( p 1 , min ( p 2 , . . . , min ( p n − 1 , p n ))). The ‘pr o duction’ [ [ p ] ] ∈ N o f a clos e d pro duction term p ∈ P is deﬁned by induction on the term structure, interpreting µ as the least ﬁxed po in t op erator, f as f , k as k , and min as min , as follows. Deﬁnition 4.2 5. The pr o duction [ [ p ] ] α ∈ N of a term p ∈ P w ith r esp e ct to an assignment α : X → N is deﬁned inductively by: [ [ k ] ] α = k [ [ ι ( p )] ] α = [ [ ι ] ]([ [ p ] ] α ) [ [ x ] ] α = α ( x ) [ [ µx.p ] ] α = lfp( λn. [ [ p ] ] α [ x 7→ n ] ) [ [ • ( p )] ] α = 1 + [ [ p ] ] α [ [ min ( p 1 , p 2 )] ] α = min([ [ p 1 ] ] α , [ [ p 2 ] ] α ) where α [ x 7→ n ] denotes an up date of α , deﬁned by α [ x 7→ n ]( y ) = n if y = x , and α [ x 7→ n ]( y ) = α ( y ) otherwise . Finally , we let [ [ p ] ] := [ [ p ] ] α 0 with α 0 deﬁned by α 0 ( x ) = 0 for all x ∈ X . As b ecomes clear from Def. 4.25 , we could hav e done without the cla use • ( p ) in the BNF grammar for pro duction terms in Def. 4.2 4, as • can be abbrevia ted to + − +, an io -term that denotes the successo r function. How ever, w e take it up as a primitiv e constructor here in order to match with p ebbleﬂow nets, see Sec. 4 .3. F or faithfully mo delling the d-o low er b ounds of stream functions with strea m arity r , we employ r -a ry p-i functions, w hich we represent by ‘ r -a ry gates ’. Deﬁnition 4.2 6. An r - ary gate gate k ( σ 1 , . . . , σ r ) is deﬁned as a pro duction term co n tex t of the for m: gate k ( σ 1 , . . . , σ r ) := min ( k , σ 1 ( 2 1 ) , . . . , σ r ( 2 r )) , where k ∈ N and σ 1 , . . . , σ r ∈ ± ∞ . W e use γ as a sy n ta c tic v ar iable for gates. The interpr etation o f a g ate γ = gate k ( σ 1 , . . . , σ r ) is de ﬁned by: [ [ γ ] ]( n 1 , . . . , n r ) := min( k , [ [ σ 1 ] ]( n 1 ) , . . . , [ [ σ r ] ]( n r )) . In cas e k = ∞ , we simplify g ate k ( σ 1 , . . . , σ r ) to gate ( σ 1 , . . . , σ r ) := min ( σ 1 ( 2 1 ) , . . . , σ r ( 2 r )) . Data-Oblivious Stream Pro ductivity 27 It is po ssible to choose unique gate repres en ta tions f of p- i functions f that are eﬃciently c o mputable from other gate r epresentations, see Section 4 .3. Owing to the res tr iction to (term repres e n tatio ns o f ) p erio dically increasing functions in Def. 4.2 5 it is p ossible to calculate the pro duction [ [ p ] ] of terms p ∈ P . F or that purp ose, we deﬁne a rewr ite system which reduces any closed term to a numeral k . This system makes use of the computable op erations ◦ and ﬁx on io-terms mentioned ab ov e. Deﬁnition 4.2 7. The r ewrite r elation → R on pr o duction terms is deﬁned as the compa tible closure o f the following rules: • ( p ) → + − +( p ) ( R 1) ι 1 ( ι 2 ( p )) → ι 1 ◦ ι 2 ( p ) ( R 2) ι ( min ( p 1 , p 2 )) → min ( ι ( p 1 ) , ι ( p 2 )) ( R 3) µx. min ( p 1 , p 2 ) → min ( µx.p 1 , µx.p 2 ) ( R 4) µx.p → p if x 6∈ FV ( p ) ( R 5) µx.ι ( x ) → ﬁx ( ι ) ( R 6) min ( k 1 , k 2 ) → min( k 1 , k 2 ) ( R 7) ι ( k ) → [ [ ι ] ]( k ) ( R 8) µx.x → 0 ( R 9) The following tw o lemmas establish the use fulness of the rewrite relation → R . In order to compute the pro duction [ [ p ] ] o f a pro duction term p it suﬃces to obtain a → R -normal form of p . Lemma 4.2 8. The r ewrite r elation → R is pr o duction pr eserving: p → R p ′ = ⇒ [ [ p ] ] = [ [ p ′ ] ] . Pr o of. It suﬃces to prov e: C [ ℓ σ ] → R C [ r σ ] = ⇒ ∀ α. [ [ C [ ℓ σ ]] ] α = [ [ C [ r σ ]] ] α , where ℓ → r is a rule o f the T RS given in Def. 4.27 , a nd C a unary context over P . W e pro ceed by induction o n C . F o r the base case, C = [ ], we g ive the es sent ia l pro of steps only: F o r r ule ( R 1), observe that [ [ − +] ] is the identit y funct ion on N . F o r r ule ( R 2 ), we apply Pr op. 4.11 . F or rule ( R 3) the desired equality follows from C 1 on page 31. F or rule ( R 4) we conclude by C 2 ibid. F or rule ( R 6) we use Lem. 4.23. F or the remaining rules the statemen t trivially holds. F or the induction step, the statement easily follows fro m the induction hypotheses. ⊓ ⊔ Lemma 4.2 9. The r ewrite re lation → R is terminating and c onﬂuent, and every close d p ∈ P has a numer al k as its unique → R -normal form. Pr o of. T o see that → R is ter minating, let w : P → N be deﬁned by: w( x ) = 1 w( • ( p )) = 2 · w( p ) + 1 w( µx.p ) = 2 · w( p ) w( k ) = 1 w( σ ( p )) = 2 · w( p ) w( min ( p 1 , p 2 )) = w( p 1 ) + w( p 2 ) + 1 , 28 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks and obser v e that p → R q implies w( p ) > w( q ). Some of the r ules of → R ov erla p; e.g . rule ( R 2) with itself. F or each of the ﬁve critical pairs we can ﬁnd a common reduct (the critical pair h σ ◦ ( τ ◦ υ ) , ( σ ◦ τ ) ◦ υ i due to a n ( R 2 )/( R 2)-ov erlap ca n b e joined b y Lem. 4.9), and hence → R is lo cally conﬂuent, by the Critical Pairs Lemma (cf. [10 ]). B y Newman’s Le mma, we obtain that → R is conﬂuent. Thus nor mal forms a re unique. T o s how that every closed net normalis e s to a so urce, let p b e an arbitra r y normal form. Note that the se t of free v ariables of a net is clos ed under → R , and hence p is a clo sed net. Clearly , p does not contain p ebbles, otherwis e ( R 1) would be a pplicable. T o see that p contains no subterms of the form µx.q , supp ose it do es and consider the innermost such subterm, viz. q contains no µ . If q ≡ k or q ≡ x , then ( R 5), resp. ( R 9) is a pplicable. If q ≡ σ ( q ′ ), we further distinguish four cases: if q ′ ≡ k or q ′ ≡ x , then ( R 8 ) r esp. ( R 6) is applica ble; if the r o o t symbol of q ′ is one of b ox , min , then q constitutes a redex w.r .t. ( R 2), ( R 3), resp ectively . If q ≡ min ( q 1 , q 2 ), we hav e a redex w.r.t. ( R 4). Thus, there are no s ubter ms µx.q in p , and therefore, beca use p is clo sed, also no v a riables x . T o see that p has no s ubterms of the for m σ ( q ), suppo se it do es and co ns ider the innermost such subterm. Then, if q ≡ k or q ≡ min ( q 1 , q 2 ) then ( R 8) resp. ( R 3) is applicable; other cases have bee n excluded a b ove. Finally , p do es not co n tain s ubterms of the form min ( p 1 , p 2 ). F or if it do es, conside r the inner most o ccurre nce a nd note that, since the other cases hav e been excluded already , p 1 and p 2 hav e to b e sources, and s o we hav e a r e de x w.r.t. ( R 7). W e conclude that p ≡ k for so me k ∈ N . ⊓ ⊔ 4.3 P ebbleﬂow Nets Pro duction terms can b e visua lized by ‘p ebbleﬂow nets’ in tr oduce d in [2], a nd serve as a means to mo del the ‘da ta-oblivious’ co nsumption/pro duction b e- haviour of s tream sp eciﬁcations. The idea is to a bstract from the actual stre am elements (data) in a stream ter m in fav o ur of occur rences of the sym b ol • , which we call ‘p ebble’. Thus, a stream term u : s is tr a nslated to [ u : s ] = • ([ s ]), see Section 5 . W e give an o per ational description of p ebbleﬂow nets, and deﬁne the pro - duction of a net a s the n umber of pe bbles a net is able to pro duce at its output po rt. Then w e prov e that this deﬁnitio n of pr oductio n coincides with Def. 4.2 5. Pebbleﬂo w nets a re net works built of p ebble pro ces sing units (fans, boxes, meets, so ur ces) connected by wire s . W e us e the term syntax given in Def. 4 .24 for nets and the rules governing the ﬂow of pebbles thr ough a net, and then give an op eratio na l meaning of the units a net is built of. Data-Oblivious Stream Pro ductivity 29 Deﬁnition 4.3 0. The p ebbleﬂow r ewrite r elation → P is deﬁned as the co mpat- ible clo sure of the union of the fo llowing rules : min ( • ( p 1 ) , • ( p 2 )) → • ( min ( p 1 , p 2 )) ( P 1) µx. • ( p ( x )) → • ( µx.p ( • ( x ))) ( P 2) + σ ( p ) → • ( σ ( p )) ( P 3) − σ ( • ( p )) → σ ( p ) ( P 4) 1 + k → • ( k ) ( P 5) Wires a re unidirectio nal FIF O communication channels. They a re idealis ed in the s ense that there is no upp er b ound on the num b er of pebbles they can store; arbitrar ily long queues ar e a llow ed. Wires hav e no coun ter part on the term level; in this sense they are akin to the edges of a term tree. Wires connect b ox es , me ets , fans , and sour c es , that we descr ibe next. A me et is waiting for a p ebble at each o f its input p orts a nd only then pro duces o ne pebble a t its output p ort, see Fig. 9. Put diﬀerently , the num b er of p ebbles a meet pro duces equals the minimum of the num b ers of p ebbles av ailable at each o f its input por ts. Meets enable ex plicit branching; they are used to mo del stream functions of ar it y > 1 , as will b e expla ined b elow. A meet with an arbitrar y n umber n ≥ 1 of input p orts is implemented by using a single wire in case n = 1, and if n = k + 1 with k ≥ 1, by connecting the output po rt of a ‘ k -ary meet’ to one of the input p orts of a (binary) meet. p 1 p 2 p 1 p 2 min min Fig. 9: Rule ( P 1). p p Fig. 10: Rule ( P 2). The b ehaviour o f a fan is dual to that of a meet: a p ebble at its input p or t is duplicated along its output p orts. A fan can be seen as a n explicit sharing device, and thus enables the co nstruction of cyclic nets. More speciﬁc a lly , w e use fans only to implement feedback when dr awing nets; there is no explicit term representation for the fan in our term calculus. In Fig . 10 a p ebble is sent ov er the output wire of the net and at the same time is fed back to the ‘recursio n wire(s)’. T urning a cyclic net into a term (tree) means to intro duce a no tion of binding; certa in no des need to be la belled b y a na me ( µx ) so that a wir e p ointing 30 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks to that no de is replac ed by a name ( x ) r eferring to the la belled no de. In rule ( P 2) feedback is acco mplis hed by substituting • ( x ) for all free o ccurr e nc e s x of p . A sour c e has an output po rt only , contains a num b er k ∈ N of p ebbles, and can ﬁre if k > 0, see Fig . 13. The re write r elation → R given in Def. 4.27 c ollapses any clos ed net (without input p orts that is) into a so ur ce. A b ox consumes p ebbles at its input p ort and pro duces p ebbles at its output po rt, controlled by an io-se q uence σ ∈ ± ∞ asso ciated with the b ox. F o r exa mple, consider the unary s tream function dup , deﬁned a s follows, and its corresp onding io-sequence: dup ( x : σ ) = x : x : dup ( σ ) − ++ which is to b e thought of as: for dup to pr o duc e two out puts, it ﬁrst has to c onsume one input, and this pr o c ess r ep e ats indeﬁnitely . Intuitiv ely , the sy m b ol − represents a requirement for an input pebble, and + r epresents a rea dy state for an output pebble. Pebbleﬂow throug h boxes is visualised in Figs. 1 1 and 1 2. p p + σ σ Fig. 11: Ru le ( P 3). − σ p p σ Fig. 12: Rule ( P 4). k 1 + k Fig. 13: Ru le ( P 5). min σ 1 σ r Fig. 14: gate ( σ 1 , . . . , σ r ) The data-oblivio us production b ehaviour of stream functions f with a stream arity ar s ( f ) = r are mo delled by r -ar y gates (deﬁned in Def. 4.2 6) that express the contribution of each individual stre a m a r - gument to the total pro duction o f f , se e Fig. 14. The precise transla tion of stream functions in to gates is given in Sec. 5, in par ticular in Def. 5.9. Lemma 4.3 1. The p ebbleﬂow r ewrite r elation → P is c onﬂuent . Pr o of. The rules of → P can b e viewed as a higher-o rder r ewriting system (HRS) that is o rthogonal. Applying Thm. 11 .6.9 in [10 ] then establishes the lemma. ⊓ ⊔ Deﬁnition 4.3 2. The pr o duction function Π : P → N of nets is deﬁned by: Π ( p ) := sup { n ∈ N | p ։ P • n ( p ′ ) } , for all p ∈ P . Π ( p ) is called the pr o duction of p . Moreover, for a net p and a n assignment α ∈ X → N , let Π ( p, α ) := Π ( p α ) where p α denotes the net o btained by replac ing each fr ee v ariable x of p with • α ( x ) ( x ). Data-Oblivious Stream Pro ductivity 31 Note that for closed nets p (where FV ( p ) = ∅ ), p α = p and therefore [ [ p ] ] α = [ [ p ] ], for all assignments α . An impor tant pr ope r t y used in the fo llowing lemma is that functions o f the form λn ∈ N . Π ( p , α [ x 7→ n ]) are monotonic functions over N . Every mo no tonic function f : N → N in the complete chain N ha s a lea st ﬁxed p oint lfp( f ) which can b e computed by lfp( f ) = lim n →∞ f n (0). In what follows w e employ , for monotonic f , g : N → N , tw o basic pro p erties : ∀ n, m.  f (min ( n, m )) = min( f ( n ) , f ( m ))  ( C 1 ) lfp( λn. min( f ( n ) , g ( n ))) = min(lfp ( f ) , lfp( g )) ( C 2 ) Lemma 4.3 3. F or al l nets q ∈ P and al l assignments α , we have that Π ( µx.q , α ) is the le ast ﬁxe d p oint of λn ∈ N . Π ( q , α [ x 7→ n ]) . Pr o of. Let α : X → N b e an arbitrar y ass ignmen t a nd q 0 := q α [ x 7→ 0] . Obser v e that Π ( µx.q , α ) = Π ( µx.q 0 ) and consider a rewrite s equence of the form µx.q 0 → ∗ . . . → ∗ • n i ( µx.q i ) → ∗ • n i ( µx. • ℓ i ( q ′ i )) → ∗ • n i + ℓ i ( µx.q i +1 ) → ∗ . . . where ℓ i = Π ( q i ), n 0 = 0, n i +1 = n i + ℓ i , and q i +1 := q ′ i ( • ℓ i ( x )). Note that lim m →∞ n m = Π ( µx.q 0 ); ‘ ≤ ’ follows from ∀ m. µx.q 0 → ∗ • n m ( µx.q m ), a nd ‘ ≥ ’ since if lim m →∞ n m < ∞ then ∃ m ∈ N such that ℓ m := Π ( q m ) = 0 and therefo r e Π ( µx.q 0 ) = Π ( • n m ( µx.q m )) = n m by conﬂuence. Let f i = λn. [ [ q i ( • n ( x ))] ], and f ′ i = λn. [ [ q ′ i ( • n ( x ))] ]. W e prov e ∀ k ∈ N . f 0 ( n m + k ) = n m + f m ( k ) ( ∗ ) by induction over m . The base ca se m = 0 is trivial, we c onsider the induction step. W e hav e q m → ∗ • ℓ m ( q ′ m ) and b y substituting • k ( x ) for x we get ∀ k ∈ N . f m ( k ) = ℓ m + f ′ m ( k ) ( ∗∗ ) Moreov e r, since f m +1 ( k ) = f ′ m ( ℓ m + k ), we get n m +1 + f m +1 ( k ) = n m +1 + f ′ m ( ℓ m + k ) = n m + ℓ m + f ′ m ( ℓ m + k ) ( ∗∗ ) = n m + f m ( ℓ m + k ) ( ∗ ) = f 0 ( n m + ℓ m + k ) = f 0 ( n m +1 + k ). Let f := f 0 . W e pro ceed with showing ∀ m. f m (0) = n m by induction ov er m ∈ N . F or the base case m = 0 we have f 0 (0) = 0 and n 0 = 0, and for the induction step we get f m +1 (0) = f ( f m (0)) IH = f ( n m ) ( ∗ ) = n m + f m (0) = n m + ℓ m = n m +1 . Hence lfp( f ) = lim m →∞ f m (0) = lim m →∞ n m = Π ( µx.q 0 ) = Π ( µx.q , α ). ⊓ ⊔ Lemma 4.3 4. F or p ∈ P , σ ∈ ± ∞ P , α : X → N : Π ( σ ( p ) , α ) = [ [ σ ] ]( Π ( p, α )) . Pr o of. W e show tha t the rela tion R ⊆ N × N deﬁned as follows is a bisimulation: R :=  h Π ( σ ( p ) , α ) , [ [ σ ] ]( Π ( p, α )) i | σ ∈ ± ∞ P , p ∈ P , α : X → N  , 32 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks that is , we prove that, for all k 1 , k 2 ∈ N , σ ∈ ± ∞ P , p ∈ P , and α : X → N , if k 1 = Π ( σ ( p ) , α ) and k 2 = [ [ σ ] ]( Π ( p, α )), then either k 1 = k 2 = 0 or k 1 = 1 + k ′ 1 , k 2 = 1 + k ′ 2 and h k ′ 1 , k ′ 2 i ∈ R . Let k 1 , k 2 ∈ N , σ ∈ ± ∞ P , p ∈ P , a nd α : X → N , b e suc h that k 1 = Π ( σ ( p ) , α ) and k 2 = [ [ σ ] ]( Π ( p, α )). By deﬁnition o f ± ∞ , we have that σ ≡ − n + τ for so me n ∈ N a nd τ ∈ ± ∞ . W e pro ceed by induction on n . If n = 0, then k 1 = 1 + k ′ 1 with k ′ 1 = Π ( τ ( p ) , α ) a nd k 2 = 1 + k ′ 2 with k ′ 2 = [ [ τ ] ]( Π ( p, α )), and h k ′ 1 , k ′ 2 i ∈ R . If n = n ′ + 1, we distinguish cases: If Π ( p, α ) = 0, then k 1 = k 2 = 0. If Π ( p, α ) = 1 + m , then p ։ P • ( q ) for some q ∈ P with Π ( q , α ) = m . Thus we get k 1 = Π ( − n ′ + τ ( q ) , α ) and k 2 = [ [ − n ′ + τ ] ]( Π ( q , α )), and h k 1 , k 2 i ∈ R b y induction hypothesis. ⊓ ⊔ Next w e s how that productio n of a term p ∈ P (Def. 4.25) coincides with the maximal num b er of p ebbles pro duced by the ne t p via the r ewrite relatio n → P (Def. 4 .32). Lemma 4.3 5. F or al l n et s p and assignment s α , it holds Π ( p, α ) = [ [ p ] ] α . Pr o of. The statement of the lemma can b e pr ov ed by a straig h tfor w a rd induction on the n um b er of µ -bindings of a net p , with a subinduction on the size of p . In the case s p ≡ σ ( p ′ ) and p ≡ µx.p ′ Lem. 4.34 and Lem. 4.3 3 are applied, resp ectively . ⊓ ⊔ Subsequently , we will use the interpretation of ter ms in P and the pr o ductio n of p ebbleﬂow nets, [ [ ] ] and Π , interchangeably . 5 T ran slation in to Pr o duction T erms In this section we deﬁne a tr anslation from s tream constants in ﬂat or friendly nesting sp eciﬁcations to pro duction ter ms. In pa r ticular, the ro ot M 0 of a sp ec- iﬁcation T is mapped b y the translation to a pro duction term [ M 0 ] with the prop erty that if T is ﬂa t (friendly nesting), then the d-o lower b ound o n the pro duction of M 0 in T equals (is b ounded from b elow by) the pro duction o f [ M 0 ]. 5.1 T ranslation of Flat and F riendl y Nes ting Sym b ols As a ﬁrst step of the transla tion, we describ e how for a ﬂat (or friendly nesting) stream function symbol f in a stream sp eciﬁca tion T a p erio dically increa sing function h f i can b e c alculated that is (that b ounds from b elow) the d-o low er bo und on the pro duction of f in T . Let us ag ain consider the rules (i) f ( s ( x ) : y : σ ) → a ( s ( x ) , y ) : f ( y : σ ), and (ii) f ( 0 : σ ) → 0 : s ( 0 ) : f ( σ ) fro m Fig. 2. W e mo del the d-o lower b o und o n the pro duction o f f by a function from N to N deﬁned a s the unique solution for X f of the following system of equations. W e disr egard what the concrete s tream elements a re, and ther efore we ta ke the inﬁmum ov e r all p ossible tra ces: X f ( n ) = inf  X f , (i ) ( n ) , X f , (i i) ( n )  Data-Oblivious Stream Pro ductivity 33 where the solutions fo r X f , (i ) and X f , (i i) are the d-o lower b ounds of f assuming that the ﬁr st rule applied in the rewrite sequence is (i) or (ii), resp ectively . The rule (i) consumes t wo elemen ts, pro duces one elemen t and feeds one element back to the re cursive c a ll. F o r rule (ii) these num b ers ar e 1, 2 , 0 respectively . Therefore we get: X f , (i ) ( n ) = let n ′ := n − 2 , if n ′ < 0 then 0 else 1 + X f (1 + n ′ ) , X f , (i i) ( n ) = let n ′ := n − 1 , if n ′ < 0 then 0 else 2 + X f ( n ′ ) . The unique solution for X f is n 7→ n . − 1, represe n ted by the io -term −− +. In g eneral, functions may have m ultiple arguments, which during rewr iting may get p ermuted, deleted or duplicated. The idea is to trace sing le arg umen ts, and to take the inﬁmum over trace s in case an arg umen t is duplicated. In the deﬁnition of the trans lation of stre a m functions, we ne e d to distinguish the ca ses acc o rding to whether a symbol is weakly g uarded o r not: O n Σ s we deﬁne ; : = {h r oot ( ℓ ) , r oot ( r ) i | ℓ → r ∈ R S , ‘:’ 6 = r oot ( r ) ∈ Σ s } . the dep en- dency r elation betw een s y m b ols in Σ s . W e s ay tha t a symbol f ∈ Σ s is we akly guar de d if f is strongly normalising with respe c t to ; and ungu ar de d , other wise. Note that since Σ s is ﬁnite it is (easily) decidable whether a symbo l f ∈ Σ s is weakly guar ded or ungua rded. The tr anslation o f a str e am function symbol is deﬁned a s the unique so lutio n of a (usually inﬁnite) s y stem of deﬁning equations where the unknowns are functions. More precisely , for ea c h symbol f ∈ Σ fnest ⊇ Σ sfun of a ﬂat or friendly nesting stream sp eciﬁcation, this system ha s a p-i function h f i as the s olution for X f , which is unique among the co n tinuous functions. Later we will see (Pr op. 5.14 and Lem. 5.15) that the tra ns lation h f i of a ﬂat (friendly nesting) str eam function symbol f coincides with (is a lower b ound fo r ) the d-o low er b ound do T ( f ) o f f . Deﬁnition 5.1 . Let T = h Σ , R i b e a stream sp eciﬁcation. F or each ﬂat or friendly nes ting symbo l f ∈ Σ fnest ⊇ Σ ﬂat with arities k = ar s ( f ) and ℓ = ar d ( f ) we deﬁne h f i : N k → N , called the (p-i function) tr anslation of f in T , a s the unique solutio n s X f for X f of the following system of deﬁning equations , where the solution of an equation of the form X ( n 1 , . . . , n k ) = . . . is a function s X : N k → N (it is to b e understo o d that s X ∈ N if k = 0): F or all n 1 , . . . , n k ∈ N , i ∈ { ⋆, 1 , . . . , k } , a nd n ∈ N : X f ( n 1 , . . . , n k ) = min( X f ,⋆ , X f , 1 ( n 1 ) , . . . , X f ,k ( n k )) , X f ,⋆ = ( inf  X f ,⋆, ρ | ρ a de ﬁning rule of f  if f is weakly g ua rded, 0 if f is ung uarded, X f ,i ( n ) = ( inf  X f ,i,ρ ( n ) | ρ a deﬁning rule o f f  if f is weakly g ua rded, 0 if f is ung uarded. W e write u i : σ i for u i, 1 : . . . : u i,p : σ i , and | u i | for p . F or sp ecifying X f ,i,ρ we distinguish the p ossible forms the rule ρ can ha ve. If ρ is nes ting , then X f ,⋆, ρ = ∞ , 34 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks and X f ,i,ρ ( n ) = n for all n ∈ N . Other wise, ρ is non-nesting and of the for m: f (( u 1 : σ 1 ) , . . . , ( u k : σ k ) , v 1 , . . . , v ℓ ) → w 1 : . . . : w m : s , where either (a) s ≡ σ j , or (b) s ≡ g (( u ′ 1 : σ φ (1) ) , . . . , ( u ′ k ′ : σ φ ( k ′ ) ) , v ′ 1 , . . . , v ′ ℓ ′ ) with k ′ = ar s ( g ), ℓ ′ = ar d ( g ), and φ : { 1 , . . . , k ′ } → { 1 , . . . , k } . Let: X f ,⋆, ρ = ( ∞ case (a) m + X g ,⋆ case (b) X f ,i,ρ ( n ) = let n ′ := n − | u i | , if n ′ < 0 then 0 else m +      n ′ case (a), i = j ∞ case (a), i 6 = j inf  X g ,j ( n ′ + | u ′ j | ) | j ∈ φ − 1 ( i )  case (b) where we agree inf ∅ = ∞ . W e mention a useful intuition for understa nding the fabric of the for mal deﬁnition ab ov e of the translation h f i of a stream function f , the solution s X f for X f of the system of equa tions in Def. 5.1 . F or e a ch i ∈ { 1 , . . . , k } where k = ar s ( f ) the solution s X f ,i for X f ,i (a unary p-i function) in this system describ es to what extent consumption from the i - th co mp onent of f ‘delays’ the ov er a l pro duction. Since in a d-o rewr ite seq uence fro m f ( • n 1 : σ 1 , . . . , • n k : σ k ) it may happ en that all strea m v aria ble s σ 1 , . . . , σ k are eras e d at some po in t, and that the sequence subsequently contin ues rewriting stream co nstants, monitoring the delays caused by indiv idua l arguments is not suﬃcient alone to deﬁne the pro duction function of f . This is the reaso n for the use, in the deﬁnition of h f i , of the so lutio n s X f ,⋆ for the v ariable X f ,⋆ , which deﬁnes a ‘glas s ceiling’ for the not adequa te ‘overall delay’ function min( s X f , 1 ( n 1 ) , . . . , s X f ,k ( n k ) ), ta k ing account of s ituations in which in d-o rewr ite seq uences from terms f ( • n 1 : σ 1 , . . . , • n k : σ k ) all input comp onent s hav e b een er a sed. Concerning non-nesting r ules on which deﬁning rules for frie ndly nesting symbols dep end via , this translation uses the fact that their pro duction is bo unded below by ‘min ’. These b ounds are not necessar ily optimal, but can b e used to show pr o ductivit y of exa mples like X → 0 : f ( X ) with f ( x : σ ) → x : f ( f ( σ )). Def. 5 .1 can b e used to deﬁne, for every ﬂa t or friendly nesting symbol f ∈ Σ sfun in a str e am sp e ciﬁcation T , a ‘gate trans lation’ of f : b y deﬁning this translation by choos ing a gate that repre s en ts the p-i function h f i . How ever, it is desir able to deﬁne this tr anslation also in a n equiv alent wa y that lends itself b etter for co mputation, and wher e the result directly is a pro - duction term (gate) re presentation of a p-i function: b y sp ecifying the io-terms in a g ate tra nslation as denotatio ns of rationa l io-sequences that are the solutions of ‘io-seq uence speciﬁcations ’. In particular , the gate translatio n o f a symbol f ∈ Σ sfun will b e deﬁned in ter ms of the solutions, for each a rgument place of a stream function symbol f , of a ﬁnite io- sequence sp eciﬁcation that is extr acted from an inﬁnite one which precis ely determines the d-o lower bo und of f in that argument pla ce. Data-Oblivious Stream Pro ductivity 35 Deﬁnition 5.2 . Let X be a set of v ariables. The set of io-se qu enc e sp e ciﬁc ation expr essions over X is deﬁned b y the following gr ammar: E ::= ⊥ | X | − E | + E | E ∧ E where X ∈ X . Guardedness is deﬁned b y induction: An io -sequence sp eciﬁcation expression is called guar de d if it is of o ne of the for ms ⊥ , − E 0 , or + E 0 , or if it is o f the form E 1 ∧ E 2 for gua r ded E 1 and E 2 . Suppo se that X = { X α | α ∈ A } for some (countable) set A . Then by an io-se quenc e sp e ciﬁc ation over (the set of re cursion vari ables) X w e mean a family { X α = E α } α ∈ A of r e cu rsion e quat ions , where, fo r all α ∈ A , E α is a n io- sequence s peciﬁca tion expres sion ov er X . Let E b e an io-seque nce sp eciﬁcation. If E consists of ﬁnitely many recursion eq uations, then it is called ﬁnite . E is called guar de d ( we akly guar de d ) if the r ight -hand side of every recur sion equatio n in E is gua rded (or r esp e ctiv ely , can b e rewritten, using equa tional log ic a nd the equations o f E , to a gua r ded io -sequence sp eciﬁcation expressio n). Let E = { X α = E α } α ∈ A an io- sequence sp eciﬁcatio n. F ur ther more let α 0 ∈ A and τ ∈ ± ω . W e say that τ is a s olut ion of E for X α 0 if there exist io-sequences { σ α } α ∈ A such that σ α 0 = τ , and all of the recursion eq uations in E ar e true statements (under the interpretation o f ∧ as deﬁned in Def. 4.13), when, for all α ∈ A , σ α is subs tituted for X α , re s pectively . It turns out that weakly g uarded io- sequence sp eciﬁcations hav e unique so lu- tions, and tha t the solutions of ﬁnite, w e a kly gua r ded io -sequence sp eciﬁcatio ns are r ational io-sequences . Lemma 5.3 . F or every we akly guar de d io-se qu enc e sp e ciﬁc ation E and r e cursion variable X of E , t her e exists a unique solution σ ∈ ± ∞ of E for X . Mor e over, if E is ﬁn ite then the solut ion σ of E for X is a r ational io-se quenc e, and an io-term that denotes σ c an b e c ompute d on the input of E and X . Let T b e a stream sp eciﬁcation, and f ∈ Σ sfun . By exha ustiv eness of T for f , ther e is at least o ne deﬁning rule for f in T . Since T is a co nstructor stream TRS, it follows tha t every deﬁning rule ρ for f is of the form: f ( p 1 , . . . , p ar s ( f ) , q 1 , . . . , q ar d ( f ) ) → u 1 : . . . : u m : s ( ρ ) with p 1 , . . . , p ar s ( f ) ∈ T er ( C ( Σ )) S , q 1 , . . . , q ar d ( f ) ∈ T er ( C ( Σ )) D , u 1 , . . . , u m ∈ T er ( Σ ) D and s ∈ T er ( Σ ) S where r oot ( s ) 6 = ‘:’. If ρ is non-nesting then either s ≡ σ j or s ≡ g ( w 1 : σ φ (1) , . . . , w ar s ( g ) : σ φ ( ar s ( g )) , u ′ 1 , . . . , u ′ ar d ( g ) ) where w i : σ i is shorthand for w i, 1 : . . . : w i,m i : σ i , and φ : { 1 , . . . , ar s ( g ) } → { 1 , . . . , ar s ( f ) } is a function that descr ib es how the stream arg umen ts a re p ermuted and replicated. W e now deﬁne, for every g iven strea m sp eciﬁcation T , an inﬁnite io-s equence sp eciﬁcation E T that will b e instrumental for deﬁning gate translations of the stream function symbols in T . 36 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks Deﬁnition 5.4 . Let T = h Σ , R i b e a stream sp eciﬁcation. Bas e d on the set: A := {h ǫ, −i , h ǫ, + i , h ǫ , − + i} ∪ n h f , ⋆ i , h f , ⋆ , ρ i , h f , i, q i , h f , i, q , ρ i    f ∈ Σ fnest , q ∈ N , ρ deﬁning r ule for f , 1 ≤ i ≤ ar s ( f ) o of tuples we deﬁne the inﬁnite io-seq uence sp eciﬁcation E T = { X α = E α } α ∈ A by listing the eq uations of E T . W e start with the equations X h ǫ, −i = ⊥ , X h ǫ, + i = + X h ǫ, + i , X h ǫ, − + i = − + X h ǫ, − + i . Then we le t, for all friendly nesting (or ﬂat) f ∈ Σ sfun with arities k = ar s ( f ) and ℓ = ar d ( f ): X h f , ⋆ i = ( min  X h f , ⋆, ρ i | ρ a deﬁning rule o f f  if f is weakly g uarded, X h ǫ, −i if f is unguar ded, and for all 1 ≤ i ≤ ar s ( f ), and q ∈ N : X h f , i, q i = ( V  X h f , i, q , ρ i | ρ a de ﬁning rule of f  if f is weakly g ua rded, X h ǫ, −i if f is ung uarded. F or sp ecifying X h f , ⋆, ρ i and X h f , i, q, ρ i we dis tinguish the p ossible forms the rule ρ can hav e. In doing so , we abbreviate ter ms u i, 1 : . . . : u i,p : σ i with σ i a v aria ble of so rt s tream by u i : σ i and let | u i | := p . If ρ is nesting, then we let X h f , ⋆, ρ i = X h ǫ, + i , X h f , i, q , ρ i = X h ǫ, − + i . Otherwise, ρ is no n-nesting and o f the form: f (( u 1 : σ 1 ) , . . . , ( u k : σ k ) , v 1 , . . . , v ℓ ) → w 1 : . . . : w m : s , where either (a) s ≡ σ j , or (b) s ≡ g (( u ′ 1 : σ φ (1) ) , . . . , ( u ′ k ′ : σ φ ( k ′ ) ) , v ′ 1 , . . . , v ′ ℓ ′ ) with k ′ = ar s ( g ), ℓ ′ = ar d ( g ), and φ : { 1 , . . . , k ′ } → { 1 , . . . , k } . Let: X h f , ⋆, ρ i = ( X h ǫ, + i case (a) + m X h g , ⋆ i case (b) X h f , i, q, ρ i = let p := | u i | . − q , q ′ := q . − | u i | in − p + m      + q ′ X h ǫ, − + i case (a), i = j X h ǫ, + i case (a), i 6 = j V  X h g , j , q ′ + | u ′ j |i | j ∈ φ − 1 ( i )  case (b) where we agree V ∅ := X h ǫ, + i . W e formally state an ea sy o bserv ation about the sys tem E T deﬁned in Def. 5.9, and an immediate c o nsequence due to Lem. 5 .3. Data-Oblivious Stream Pro ductivity 37 Prop osition 5.5. L et T b e a str e am sp e ciﬁc ation. The io-se quenc e sp e ciﬁc a- tion E T (deﬁne d in Def. 5.9) is we akly gu ar de d. As a c onse quenc e, E T has a unique solution in ± ∞ for every variable X in E T . Another eas y obser v a tion is that the solutions for the v ariables X h f , ⋆ i in a system E T corres p ond very directly to num b ers in N . Prop osition 5.6. L et T b e a st r e am sp e ciﬁc ation and f ∈ Σ sfun . The unique solution of E T (deﬁne d in D ef. 5.9) for X h f , ⋆ i is of the form + ω or + n ⊥ . F urthermore observe that E T is inﬁnite in case that Σ sfun 6 = ∅ , and hence t y pically is inﬁnite. Wherea s Lem. 5 .3 implies unique solv ability of E T for indi- vidual v ar iables in view o f (the ﬁr s t statement in) Pr op. 5.5, it will usually not guarantee that these unique s olutions are r ational io-sequences . Nevertheless it can be shown that all solutions of E T are r ational io -sequences. F or our purp oses it will suﬃce to show this only for the solutions of E T for certain of its v ariables . W e will only b e int e rested in the solutions of E T for v ariables X h f , i, 0 i and X h f , ⋆ i . It turns o ut that from E T a ﬁnite weakly g uarded speciﬁca tion E ′ T can b e extracted that, fo r each of the v ar iables X h f , i, 0 i and X h f , ⋆ i , ha s the same solution as E T , r espe ctiv ely . Lem. 5.7 b elow states that, for a stream sp eciﬁcation T , the ﬁnite io-sequence speciﬁca tio n E ′ T can alw ays b e obtained algor ithmically , which together with L e m. 5.3 implies that the unique solutio ns o f E T for the v ariables X h f , i, 0 i and X h f , ⋆ i are r ational io -sequences for which representing io-terms ca n be computed. As a conseq uence these solutions, for a stream sp eciﬁcation T , of the rec ur sion system E T , ca n b e v iew e d to r epresent p-i functions: the p-i func- tions that are r epresented b y the io-term de no ting the resp ective solution, a rational io-s equence. As an e xample let us c onsider an io -sequence sp eciﬁcation that corres ponds to the deﬁning rule s for f in Fig. 2 : X = − ++ X ∧ −− + Y Y = ++ X ∧ − + Y . The unique s olution for X of this system is the r a tional io- sequence (and res pec- tively , io- term) − − +, which is the transla tion o f f (as mentioned ea rlier). Lemma 5.7 . L et T b e a str e am sp e ciﬁc ation. Ther e exists an io-se quenc e sp e c- iﬁc ation E ′ T = { X α = E ′ α } α ∈ A ′ such that: (i) E ′ T is ﬁnite and we akly guar de d; (ii) {h f , i, 0 i , h f , ⋆ i | f ∈ Σ sfun , i, q ∈ N , i ∈ { 1 , . . . , ar s ( f ) } } ⊆ A ′ ; for e ach f ∈ Σ sfun ∩ Σ fnest and i ∈ { 1 , . . . , ar s ( f ) } , E ′ T has t he same solution for X h f , i, 0 i , and r esp e ctively for X h f , ⋆ i , as the io-se quenc e sp e ciﬁc ations E T (se e Def. 5.4); (iii) on the input of T , E ′ T c an b e c omput e d. Pr o of (S ketch). An algor ithm for obtaining E ′ T from E T can b e obtained as follows. On the input of E T , set E := E T and rep eat the following step on E a s long as it is applicable: 38 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks (RPC) Detect and remov e a r e achable non-c onsuming pseudo-cycle from the io- sequence sp eciﬁcatio n E : Supp ose that, for a function sy m b ol h , for j, k , l ∈ N , and for a recurs ion v ariable X h h , j , k i that is r eachable from X h f , i, 0 i , we hav e X h h , j , k i w − → X h h , j , l i (from the recursion v aria ble X h h , j , k i the v aria ble X h h , j , l i is reachable via a path in the sp eciﬁcation on which the ﬁnite io- sequence w is enco un ter e d as the word formed by consecutive labels ), where k < l a nd w only contains sym b ols ‘+’. Then modify E by setting X h h , j , k i = X h ǫ, + i . It is not diﬃcult to show that a step (RPC) preser v e s weakly g ua rdedness and the unique solution of E T , a nd that, on the input o f E T , the algorithm terminates in ﬁnitely many steps, having pro duced an io-seq ue nce s peciﬁca tion E ′ T with [ f ] i as the solution for X h f , i, 0 i and the pro per t y that only ﬁnitely many recursion v ariables are reachable in E ′ T from X h f , i, 0 i . ⊓ ⊔ As an immediate consequence of Le m. 5.7, and of Lem. 5.3 w e obtain the following lemma. Lemma 5.8 . L et T b e a st r e am sp e ciﬁc ation, and let f ∈ Σ sfun . (i) F or e ach i ∈ { 1 , . . . , ar s ( f ) } ther e is pr e cisely one io-se quenc e σ t hat solves E T for X h f , i, 0 i ; furthermor e, σ is r ational. Mor e over t her e ex ists an algorithm that, on the input of T , f , and i , c omputes the shortest io-term that denotes the solution of E T for X h f , i, 0 i . (ii) Ther e is pr e cisely one io-se quenc e σ that s olves E T for X h f , ⋆ i ; this io-se quenc e is r ational. And ther e is an algorithm that, on the input of T and f , c omputes the shortest io-term that denotes the solut ion of E T for X h f , ⋆ i . Lem. 5.8 g uarantees the well-deﬁnedness in the deﬁnition below of the ‘g ate translation’ for ﬂat or friendly nesting s tream functions in a stream sp eciﬁcatio n. Deﬁnition 5.9 . Let T = h Σ , R i b e a str e am deﬁnitio n. F or each ﬂat or friendly nesting symbol f ∈ Σ sfun ∩ Σ fnest with stream arity k = ar s ( f ) we deﬁne the gate tr anslation [ f ] of f by: [ f ] := g ate [ f ] ⋆ ([ f ] 1 , . . . , [ f ] ar s ( f ) ) , where [ f ] ⋆ ∈ N is deﬁned by: [ f ] ⋆ := ( n the unique solution of E T for X h f , ⋆ i is + n ⊥ ∞ the unique s o lution of E T for X h f , ⋆ i is + ω and, for 1 ≤ i ≤ ar s ( f ), [ f ] i is the shortest io- term that deno tes the unique solu- tion for X h f , i, 0 i of the weakly g ua rded io-sequence spe c iﬁc a tion E T in Def. 5.4. F rom Lem. 5.8 w e also obtain that, for ev er y stream sp eciﬁca tio n, the function which ma ps stream function symbols to their gate tr a nslations is computable. Data-Oblivious Stream Pro ductivity 39 Lemma 5.1 0. Ther e is an algorithm t hat, on the input of a stre am sp e ciﬁc a- tion T , and a ﬂat or friend ly n esting symb ol f ∈ Σ sfun ∩ Σ fnest , c omputes the gate tr anslation [ f ] of f . Example 5.11. Consider a ﬂat s tream sp eciﬁcation co nsisting of the rules: f ( x : σ ) → x : g ( σ, σ, σ ) , g ( x : y : σ , τ , υ ) → x : g ( y : τ , y : υ , y : σ ) . The translation of f is [ f ] = gate ([ f ] 1 ), where [ f ] 1 is the unique so lution for X h f , 1 , 0 i of the io-seq ue nc e sp eciﬁcatio n E T : X h f , 1 , 0 i = − +( X h g , 1 , 0 i ∧ X h g , 2 , 0 i ∧ X h g , 3 , 0 i ) X h g , 1 , 0 i = −− + X h g , 3 , 1 i X h g , 1 , 1 i = − + X h g , 3 , 1 i X h g , 1 , q i = + X h g , 3 , q − 1 i ( q ≥ 2) X h g , 2 , q i = + X h g , 1 , q +1 i ( q ∈ N ) X h g , 3 , q i = + X h g , 2 , q +1 i ( q ∈ N ) By the algor ithm referre d to in Lem. 5.10 this inﬁnite sp eciﬁcation can be turned int o a ﬁnite one. T he ‘no n-consuming pseudo cycle’ X h g , 3 , 1 i +++ − → X h g , 3 , 2 i justiﬁes the mo diﬁcation o f E T by s etting X h g , 3 , 1 i = + X h g , 3 , 1 i ; likewise we set X h g , 3 , 0 i = + X h g , 3 , 0 i . F ur thermore a ll equations not reachable from X h f , 1 , 0 i are remov ed (garbag e collection), and we o bta in a ﬁnite sp eciﬁcation E ′ T , which, by Lem. 5.3 , has a p erio dically incr easing solution, with io -term-denotation [ f ] 1 = h− + −− , + i . The reader may try to ca lculate the gate corresp onding to g , it is: [ g ] = gate ( −− + , + − + , +). Second, cons ide r the ﬂat strea m sp eciﬁcation with data constructor symbols 0 and 1 : f ( 0 : σ ) → g ( σ ) , f ( 1 : x : σ ) → x : g ( σ ) , g ( x : y : σ ) → x : y : g ( σ ) , denoted ρ f 0 , ρ f 1 , and ρ g , resp ectively . Then, [ f ] 1 is the solution for X h f , 1 , 0 i of X h f , 1 , 0 i = X h f , 1 , 0 , ρ f 0 i ∧ X h f , 1 , 0 , ρ f 1 i X h f , 1 , 0 , ρ f 0 i = − X h g , 1 , 0 i X h f , 1 , 0 , ρ f 1 i = −− + X h g , 1 , 0 i X h g , 1 , 0 i = −− ++ X h g , 1 , 0 i . Example 5.12. Consider a pure stream sp eciﬁcation with the function layer: f ( x : σ ) → x : g ( σ, σ, σ ) , g ( x : y : σ , τ , υ ) → x : g ( y : τ , y : υ , y : σ ) . 40 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks The tra nslation of f is h f i , the unique solution for X f of the system: X f ( n ) = min( X f ,⋆ (0) , X f , 1 ( n )) X f , 1 ( n ) = let n ′ := n − 1 if n ′ < 0 then 0 else 1 + inf  X g , 1 ( n ′ ) , X g , 2 ( n ′ ) , X g , 3 ( n ′ )  X f ,⋆ ( n ) = 1 + X g ,⋆ (0) X g , 1 ( n ) = let n ′ := n − 2 , if n ′ < 0 then 0 else 1 + X g , 3 (1 + n ′ ) X g , 2 ( n ) = 1 + X g , 1 (1 + n ) X g , 3 ( n ) = 1 + X g , 2 (1 + n ) X g ,⋆ ( n ) = 1 + X f ,⋆ (0) An algor ithm for solving such s ystems of e quations is describ ed in (the pro of of ) Lemma 5.7 ; here w e solve the system dire ctly . Note that X g , 3 ( n ) = 1 + X g , 2 ( n + 1) = 2 + X g , 1 ( n + 2) = 3 + X g , 3 ( n ), hence ∀ n ∈ N . X g , 3 ( n ) = ∞ . Likewise we obtain X g , 2 ( n ) = ∞ if n ≥ 1 a nd 1 for n = 0 , and X g , 1 ( n ) = ∞ if n ≥ 2 a nd 0 for n ≤ 1. Then we get h f i (0) = 0, h f i (1) = h f i (2) = 1, a nd h f i ( n ) = ∞ for all n ≥ 2, represented by the ga te h f i = gate ( − + −− +). The gate cor resp onding to g is h g i = gate ( −− + , + − + , +). Example 5.13. Consider a ﬂat strea m function sp eciﬁcation with the following rules which use pa tter n matching on the data constr uctors 0 and 1 : f ( 0 : σ ) → g ( σ ) f ( 1 : x : σ ) → x : g ( σ ) g ( x : y : σ ) → x : y : g ( σ ) denoted ρ f 0 , ρ f 1 , and ρ g , re s pectively . Then, h f i is the so lution for X f , 1 of: X f ( n ) = min( X f ,⋆ (0) , X f , 1 ( n )) X f , 1 ( n ) = inf  X f , 1 ,ρ f 0 ( n ) , X f , 1 ,ρ f 1 ( n )  X f , 1 ,ρ f 0 ( n ) = let n ′ := n − 1 , if n ′ < 0 then 0 else X g , 1 ( n ′ ) X f , 1 ,ρ f 1 ( n ) = let n ′ := n − 2 , if n ′ < 0 then 0 else 1 + X g , 1 ( n ′ ) X f ,⋆ ( n ) = min( X g ,⋆ (0) , 1 + X g ,⋆ (0)) X g , 1 ( n ) = let n ′ := n − 2 , if n ′ < 0 then 0 else 2 + X g , 1 ( n ′ ) X g ,⋆ ( n ) = 2 + X g ,⋆ (0) . As solution we obtain a n ov erla pping of bo th traces h f i 1 ,ρ f 0 and h f i 1 ,ρ f 1 , that is , h f i 1 ( n ) = n . − 2 re presented by the gate h f i = gate ( −−− +). The following prop osition explains the c orresp ondence b etw een Def. 5.1 a nd Def. 5.9. Prop osition 5.14 . L et T b e a str e am sp e ciﬁc ation. F or e ach f ∈ Σ sfun it holds: h f i ( n 1 , . . . , n k ) = [ [[ f ]] ]( n 1 , . . . , n k ) (for al l n 1 , . . . , n k ∈ N ), wher e k = ar s ( f ) . That is, the p-i fun ction tr anslation h f i of f c oincides with the p-i function t hat is r epr esente d by the gate tr anslation [ f ] . Data-Oblivious Stream Pro ductivity 41 The following lemma sta tes that the tra nslation h f i of a ﬂat stream function symbol f (as deﬁned in Def. 5.1) is the d-o lower b ound on the pro duction function of f . F or friendly nesting stre am symbols f it states that h f i p oint wisely bo unds from below the d-o low e r b ound on the pro duction function of f . Lemma 5.1 5. L et T b e a str e am sp e ciﬁc ation, and let f ∈ Σ fnest ⊇ Σ ﬂat . (i) If f is ﬂat, then: h f i = do T ( f ) . H enc e, do T ( f ) is p erio dic al ly incr e asing. (ii) If f is friend ly nesting, then it holds: h f i ≤ do T ( f ) (p ointwise ine quality). 5.2 T ranslation of Stream Constants In the seco nd step, we now deﬁne a translatio n of strea m cons ta n ts in a ﬂa t o r friendly nesting stream sp eciﬁca tio n into pro duction terms under the a ssump- tion that gate tr a nslations for the stream functions are g iven. Her e the idea is that the recursive deﬁnition of a strea m co nstant M is unfolded step by step; the terms thus aris ing are tra nslated accor ding to their structure using gate tra ns - lations o f the s tream function symbols from a given family of g ates; whenever a stream constant is met that has b een unfolded b efore, the trans lation stops after establishing a binding to a µ -binder created ea rlier. Deﬁnition 5.1 6. Let T = h Σ , R i be a stream sp eciﬁcation, and F = { γ f } f ∈ Σ sfun a family of gates that a re asso cia ted with the sy mbo ls in Σ sfun . Let T er ( Σ ) 0 S be the set of terms in T of sor t strea m that do not co ntain v ariables o f sort stream. The tra ns lation function [ · ] F : T er ( Σ ) 0 S → P is deﬁned by s 7→ [ s ] F := [ s ] F ∅ based on the following deﬁnition of expressions [ s ] F α , where α ⊆ Σ sc on , by induction o n the structur e of s ∈ T er ( Σ ) 0 S , using the cla uses: [ M ( u )] F α := ( µM . mi n { [ r ] F α ∪{ M } | M ( v ) → r ∈ R } if M 6∈ α M if M ∈ α [ u : s ] F α := • ([ s ] F α ) [ f ( s 1 , . . . , s ar s ( f ) , u 1 , . . . , u ar d ( f ) )] F α := γ f ([ s 1 ] F α , . . . , [ s ar s ( f ) ] F α ) where M ∈ Σ sc on , u = h u 1 , . . . , u ar d ( M ) i a vector o f terms in T er ( Σ ) D , f ∈ Σ sfun , s, s 1 , . . . s ar s ( f ) ∈ T er ( Σ ) 0 S , and u, u 1 , . . . , u ar d ( f ) ∈ T er ( Σ ) D . Note that the deﬁ- nition of [ M ( u )] F α do es not dep end o n the vector u of terms in T er ( Σ ) D . There- fore we deﬁne, for all M ∈ Σ sc on , the tr anslation of M with r esp e ct to F b y [ M ] F := [ M ( x )] F ∅ ∈ P where x = h x 1 , . . . , x ar d ( M ) i is a vector of data v ariables. The fo llowing lemma is the basis of our result in Sec. 6, Thm. 6.1 , concerning the decidability of d-o pr o ductivit y for ﬂat s tream sp eciﬁcations. In pa rticular the le mma states that if we us e gates that repr e sent d-o optimal lower b ounds on the pro duction of the strea m functions, then the tr anslation of a strea m constant M yie lds a pro duction term that rewrites to the d-o low er b ound of the pro duction of M . 42 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks Lemma 5.1 7. L et T = h Σ , R i b e a stre am sp e ciﬁc ation, and F = { γ f } f ∈ Σ sfun b e a family of gates such t hat, for al l f ∈ Σ sfun , the arity of γ f e quals the stre am arity of f . Then t he fol lowing st atemen ts hold: (i) S upp ose that [ [ γ f ] ] = do T ( f ) holds for al l f ∈ Σ sfun . Then for al l M ∈ Σ sc on and ve ctors u = h u 1 , . . . , u ar d ( M ) i of data t erms [ [[ M ] F ] ] = do T ( M ( u )) holds. And c onse qu en t ly, T is d-o pr o ductive if and only if [ [[ M 0 ] F ] ] = ∞ . (ii) Su pp ose that [ [ γ f ] ] = do T ( f ) holds for al l f ∈ Σ sfun . Then for al l M ∈ Σ sc on and ve ctors u = h u 1 , . . . , u ar d ( M ) i of data t erms [ [[ M ] F ] ] = do T ( M ( u )) holds. And c onse qu en t ly, T is d-o non-pr o ductive if and only if [ [[ M 0 ] F ] ] < ∞ . Lem. 5 .17 is an immediate consequence of the following lemma, whic h also is the basis o f our result in Sec. 6 conc e rning the recogniza bilit y o f pro ductivity for ﬂat and friendly nesting stream speciﬁc a tions. In particular the lemma b elow as s erts that if we us e gates that r epresent p-i functions which are low er b ounds o n the pro duction of the str e am functions, then the tra nslation of a stream constant M yields a pro duction term that rewrites to a num b er in N s ma ller or equa l to the d-o low e r b ound of the pro duction of M . Lemma 5.1 8. L et T b e a str e am s p e ciﬁc ation, and let F = { γ f } f ∈ Σ sfun b e a family of gates such that, for al l f ∈ Σ sfun , the arity of γ f e quals the str e am arity of f . Supp ose t hat one of t he fol lowing st atements holds: (a) [ [ γ f ] ] ≤ do T ( f ) for al l f ∈ Σ sfun ; (b) [ [ γ f ] ] ≥ do T ( f ) for al l f ∈ Σ sfun ; (c) do T ( f ) ≤ [ [ γ f ] ] for al l f ∈ Σ sfun ; (d) do T ( f ) ≥ [ [ γ f ] ] for al l f ∈ Σ sfun . Then, for al l M ∈ Σ sc on and ve ctors u = h u 1 , . . . , u ar d ( M ) i of data terms in T , the c orr esp onding one of t he fol lowing statements holds: (a) [ [[ M ] F ] ] ≤ do T ( M ( u )) ; (b) [ [[ M ] F ] ] ≥ do T ( M ( u )) ; (c) do T ( M ( u )) ≤ [ [[ M ] F ] ] ; (d) do T ( M ( u )) ≥ [ [[ M ] F ] ] . 6 Deciding Data-Oblivious Pro ductivit y In this section we assemble our res ults concerning decis ion of d-o pr o ductivit y , and automata ble recognitio n of pro ductivity . W e deﬁne metho ds: ( DOP ) for deciding d-o pro ductivity o f ﬂat stre am sp eciﬁcations, ( DP ) for deciding pro ductivity of pure stre a m sp eciﬁcations, a nd ( RP ) for rec o gnising pr o ductivity of friendly nes ting stream s peciﬁca tions, that pro cee d in the following steps: Data-Oblivious Stream Pro ductivity 43 (i) T a ke a s input a ( DOP ) ﬂat, ( DP ) pure, or ( RP ) fr iendly nesting s tream sp eci- ﬁcation T = h Σ , R i . (ii) T r anslate the stream function symbols int o ga tes F := {h f i } f ∈ Σ sfun (Def. 5.1 ). (iii) Co nstruct the pr oductio n term [ M 0 ] F with res pect to F (Def. 5.16). (iv) Co mpute the pro duction k of [ M 0 ] F using → R (Def. 4.27 ). (v) Give the fo llowing output: ( DOP ) “ T is d-o pr o ductive” if k = ∞ , else “ T is not d-o pr o ductive”. ( DP ) “ T is pro ductive” if k = ∞ , else “ T is not pro ductive”. ( RP ) “ T is pro ductive” if k = ∞ , else “don’t know”. Note that all o f thes e steps are automa table (cf. o ur pro ductivity to ol, Sec. 8 ). Our main r esult states that d-o pr o ductivity is de c ida ble for ﬂat stream sp ec- iﬁcations. It rests o n the fact tha t the a lgorithm DOP o btains the low er b ound do T ([ M 0 ]) on the pro duction of M 0 in T . Since d-o productivity implies pro- ductivit y (Pr op. 3.4), we obtain a computable, d-o o ptimal, suﬃcient condition for pro ductivity o f ﬂat stream s peciﬁca tions, which canno t b e impr ov ed by any other d-o a na lysis. Seco nd, since for pure stream sp eciﬁcations d-o pro ductivity and pro ductivity are the same, we get that pro ductivity is decida ble for them. Theorem 6.1. (i) D OP de cides d-o pr o ductivity of ﬂat str e am sp e ciﬁc ations, (ii) DP de cides pr o ductivity of pur e stre am sp e ciﬁc ations. Pr o of. Let k b e the pro ductio n of the term [ M 0 ] F ∈ P in step (iv) of DOP / DP . (i) By Lem. 5.15 (i), L e m. 5.17 (i), Lem. 4.28, Lem. 4.2 9 w e ﬁnd: k = do T ( M 0 ). (ii) F o r pure sp eciﬁca tio ns we additionally no te: Π T ( M 0 ) = do T ( M 0 ). ⊓ ⊔ Third, w e obtain a computable, suﬃcient conditio n for pro ductivity o f friendly nesting strea m sp eciﬁcations. Theorem 6.2. A friend ly nesting (ﬂat) str e am sp e ciﬁc ation T is pr o ductive if the algorithm R P ( DO P ) r e c o gnizes T as pr o ductive. Pr o of. Let k b e the pro ductio n of the term [ M 0 ] F ∈ P in step (iv) of RP / DO P . B y Lem. 5.1 5 (ii), Lem. 5.1 8 (a), Lem. 4.28 a nd Lem. 4.29 : k ≤ do T ( M 0 ) ≤ Π T ( M 0 ). ⊓ ⊔ Example 6.3. W e illustra te the translation and decision o f d-o pro ductivit y by means o f Pascal’s triangle, Fig. 2. The trans lation of the stream function sym- bo ls is F = {h f i } with h f i = gate ( −− +), see page 32. W e calculate [ P ] F , the translation of P , a nd reduce it to normal form with re spect to → R : [ P ] F = µP . • ( • ( − − +( P ))) ։ R µP. + + −− +( P ) → R ∞ Hence do T ( P ) = ∞ , and P is d-o pro ductive a nd therefore pr o ductive. 44 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks 7 Examples Pro ductivity o f all of the follo wing exa mples is recognized fully automatica lly by a Haskell implementation of our decision algo rithm for data-o blivious pro duc- tivit y . The to ol and a num b er o f examples can b e fo und at: http://inf inity.few.vu.nl/pr o ductivity . In Subsections 7.1–7.6 b elow, we give p ossible input-r epresentations a s well as the to ol-output for the exa mples of stream sp eciﬁcatio ns in Section 2 a nd for an example of a str e am function sp eciﬁca tio n in Section 3. 7.1 Example in Fig. 2 on P age 3 F or applying the automated pro ductivity prover to the ﬂat s tream spe ciﬁcation in Fig. 2 on page 2 of the stream o f rows in Pascal’s triangle, one can use the input: Signature( P : stream(nat ), 0 : nat, f : stream(nat ) -> stream(nat), a : nat -> nat -> nat, s : nat -> nat ) P = 0:s(0):f(P ) f(s(x):y:s igma) = a(s(x),y):f(y :sigma) f(0:sigma) = 0:s(0):f(s igma) a(s(x),y) = s(a(x,y)) a(0,y) = y On this input the automated pro ductivit y prov e r pro duces the following output: The a uto mated pro ductivity prover has been applied to : Signature( -- stream symbols -- P : stream(nat ), f : stream(nat ) -> stream(nat), -- data symbol s -- 0 : nat, a : nat -> nat -> nat, s : nat -> nat ) -- stream layer -- P = 0:s(0):f(P ) f(s(x):y:s igma) = a(s(x),y):f(y :sigma) Data-Oblivious Stream Pro ductivity 45 f(0:sigma) = 0:s(0):f(s igma) -- data layer -- a(s(x),y) = s(a(x,y)) a(0,y) = y T ermination o f the data lay er ha s b een proven automatically . The function s ym b ol f is ﬂat, we can compute the precis e data-o blivious low er bo und: [ f ] = g ate ([ f ] ⋆, 0 ( 0 ) , [ f ] 1 , 0 ) [ f ] 1 , 0 = µ f 1 , 0 . ∧          −− + ∧    µ f 1 , 1 . ∧    − + ∧ n f 1 , 1 ++ ∧ n f 1 , 0 − ++ ∧ n f 1 , 0 = − − + [ f ] ⋆, 0 = µ f ⋆, 0 . ∧ ( + f ⋆, 0 ++ f ⋆, 0 = + P dep ends o nly on ﬂat stream functions, we ca n decide data-obli vious pro- ductivit y . W e tra nslate P into a p ebbleﬂow net a nd colla ps e it to a sour c e : 5 [ P ] = µP. • ( • ([ f ]( P ))) = µP . • ( • ( −− +( P ))) ։ R µP. + − +( • ( −− +( P ))) ։ R µP. + − +(+ − +( −− +( P ))) ։ R µP. + + − ( −− +( P )) ։ R µP. + + −− +( P ) ։ R ∞ The sp eciﬁcatio n of P is pr o ductive. 7.2 Example in Fig. 3 on P age 6 F or applying the automated pro ductivity prover to the ﬂat s tream spe ciﬁcation in Fig. 3 o n page 6 of the ternary T hue-Morse sequence , one can use the input: Signature( Q, Qprime : stream (char), 46 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks f : stream(cha r) -> stream(char ), a, b, c: char ) Q = a:Qprime Qprime = b:c:f(Qpr ime) f(a:sigma) = a:b:c:f(si gma) f(b:sigma) = a:c:f(sigm a) f(c:sigma) = b:f(sigma) On this input the automated pro ductivit y prov e r pro duces the following output: The a uto mated pro ductivity prover has been applied to : Signature( -- stream symbols -- Q : stream(cha r), Qprime : stream(ch ar), f : stream(cha r) -> stream(char ), -- data symbol s -- a : char, b : char, c : char ) -- stream layer -- Q = a:Qprime Qprime = b:c:f(Qpr ime) f(a:sigma) = a:b:c:f(si gma) f(b:sigma) = a:c:f(sigm a) f(c:sigma) = b:f(sigma) -- data layer -- Data-Oblivious Stream Pro ductivity 47 The function s ym b ol f is ﬂat, we can compute the precis e data-o blivious low er bo und: [ f ] = g ate ([ f ] ⋆, 0 ( 0 ) , [ f ] 1 , 0 ) [ f ] 1 , 0 = µ f 1 , 0 . ∧          − +++ ∧ n f 1 , 0 − ++ ∧ n f 1 , 0 − + ∧ n f 1 , 0 = − + [ f ] ⋆, 0 = µ f ⋆, 0 . ∧      +++ f ⋆, 0 ++ f ⋆, 0 + f ⋆, 0 = + Q depe nds only on ﬂat str eam functions, we can decide data-oblivio us pro- ductivit y . W e tra nslate Q into a p ebbleﬂow net and colla ps e it to a sourc e: 6 [ Q ] = µQ. • ( µQpr ime. • ( • ([ f ]( Qpr ime ))) ) = 1 µQ. • ( µQpr ime. • ( • ( − +( Qpri me )))) ։ R µQ. + − +( µQpri me. • ( • ( − +( Qpr ime )))) ։ R µQ. + − +( µQpri me. + − +( • ( − +( Qpr ime )))) ։ R µQ. + − +( µQpri me. + − +(+ − +( − +( Qpr ime )))) ։ R µQ. + − +( µQpri me. ++ − ( − +( Qpr ime ))) ։ R µQ. + − +( µQpri me. ++ − ( Qpr ime )) ։ R µQ. + − +( ∞ ) ։ R µQ. ∞ ։ R ∞ The sp eciﬁcatio n of Q is pro ductive. Qpr i me dep ends only on ﬂat str eam functions, we ca n decide data-oblivious pro ductivit y . 48 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks W e tra nslate Qprime into a pe bbleﬂow net and co llapse it to a so urce: 7 [ Qpr i me ] = µQpr i me. • ( • ([ f ]( Qpr ime ))) = µQpr ime. • ( • ( − +( Qpr ime ))) ։ R µQpr ime. + − +( • ( − +( Qpri me ))) ։ R µQpr ime. + − +(+ − +( − +( Qpr ime ))) ։ R µQpr ime. ++ − ( − +( Qpr ime )) ։ R µQpr ime. + + − ( Qpri me ) ։ R ∞ The sp eciﬁcatio n of Qprime is pro ductive. 7.3 Example in Fig. 4 on P age 7 F or a pplying the automated pro ductivit y prov er to the pure stream speciﬁca tion in Fig. 4 o n page 7 of the ternary T hue-Morse sequence , one can use the input: Signature( Q : stream(cha r), M : stream(bit ), zip : stream(x ) -> stream(x) -> stream(x), inv : stream(b it) -> stream(bit ), tail : stream( x) -> stream(x), diff : stream( bit) -> stream(char), i : bit -> bit, X : bit -> bit -> char, 0, 1 : bit, a,b,c : char ) Q = diff(M) M = 0:zip(inv( M),tail(M)) zip(x:s,t) = x:zip(t,s) inv(x:s) = i(x):inv(s) tail(x:s) = s diff(x:y:s ) = X(x,y):diff(y :s) i(0) = 1 i(1) = 0 X(0,0) = b X(0,1) = a X(1,0) = c X(1,1) = b The a uto mated pro ductivity prover then gives the following o utput: Data-Oblivious Stream Pro ductivity 49 The a uto mated pro ductivity prover has been applied to : Signature( -- stream symbols -- Q : stream(cha r), M : stream(bit ), zip : stream(x ) -> stream(x) -> stream(x), inv : stream(b it) -> stream(bit ), tail : stream( x) -> stream(x), diff : stream( bit) -> stream(char), -- data symbol s -- i : bit -> bit, X : bit -> bit -> char, 0 : bit, 1 : bit, a : char, b : char, c : char ) -- stream layer -- Q = diff(M) M = 0:zip(inv( M),tail(M)) zip(x:s,t) = x:zip(t,s) inv(x:s) = i(x):inv(s) tail(x:s) = s diff(x:y:s ) = X(x,y):diff(y :s) -- data layer -- i(0) = 1 i(1) = 0 X(0,0) = b X(0,1) = a X(1,0) = c X(1,1) = b T ermination o f the data lay er ha s b een proven automatically . 50 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks The function sy m b ol zip is pure, we can compute its precise pr o ductio n mo dulus: [ zip ] = gate ([ zip ] ⋆, 0 ( 0 ) , [ zip ] 1 , 0 , [ zip ] 2 , 0 ) [ zip ] 1 , 0 = µ zip 1 , 0 . ∧ n − + ∧ n µ zip 2 , 0 . ∧ n + ∧ n zip 1 , 0 = − + + [ zip ] 2 , 0 = µ zip 2 , 0 . ∧ n + ∧ n µ zip 1 , 0 . ∧ n − + ∧ n zip 2 , 0 = + − + [ zip ] ⋆, 0 = µ zip ⋆, 0 . ∧ n + zip ⋆, 0 = + The function symbol inv is pure, we can compute its precis e pr o duction mo dulus: [ inv ] = gate ([ inv ] ⋆, 0 ( 0 ) , [ inv ] 1 , 0 ) [ inv ] 1 , 0 = µ inv 1 , 0 . ∧ n − + ∧ n inv 1 , 0 = − + [ inv ] ⋆, 0 = µ inv ⋆, 0 . ∧ n + inv ⋆, 0 = + The function s y m b ol tail is pure, we can compute its precise pro ductio n mo dulus: [ tail ] = gate ([ tail ] ⋆, 0 ( 0 ) , [ tail ] 1 , 0 ) [ tail ] 1 , 0 = µ tail 1 , 0 . ∧ n − µx. − + x = −− + [ tail ] ⋆, 0 = µ tail ⋆, 0 . ∧ n µx. + x = + The function symbol di ﬀ is pure, we can co mpute its precise pro duction mo dulus: [ diﬀ ] = g ate ([ diﬀ ] ⋆, 0 ( 0 ) , [ diﬀ ] 1 , 0 ) [ diﬀ ] 1 , 0 = µ diﬀ 1 , 0 . ∧ n −− + ∧ n µ diﬀ 1 , 1 . ∧ n − + ∧ n diﬀ 1 , 1 = − − + [ diﬀ ] ⋆, 0 = µ diﬀ ⋆, 0 . ∧ n + diﬀ ⋆, 0 = + Data-Oblivious Stream Pro ductivity 51 Q dep ends only on pure s tr eam functions , we can de cide pro ductivit y . W e tra nslate Q into a p ebbleﬂow net and colla ps e it to a sourc e: 8 [ Q ] = µQ. [ diﬀ ]( µM . • ([ zip ]([ inv ]( M ) , [ tail ]( M )))) = µQ. −− +( µM . • ( min ( − ++( − +( M )) , + − +( −− + ( M ))))) ։ R µQ. − − +( µM . • ( mi n ( − ++( M ) , + −− ++ ( M )))) ։ R µQ. −− +( µM . + − +( min ( − ++( M ) , + −− ++( M )))) ։ R µQ. − − +( µM . min (+ − +( − ++( M )) , + − +(+ −− ++( M )))) ։ R µQ. −− +( µM . min (+ − +( M ) , ++ −− ++( M ))) ։ R µQ. −− +( min ( µM . + − +( M ) , µM . + + −− ++( M ))) ։ R µQ. min ( − − +( µM . + − +( M )) , −− +( µM . ++ −− ++( M ))) ։ R min ( µQ. −− +( µM . + − +( M )) , µQ. −− +( µM . ++ −− ++( M ))) ։ R min ( µQ. − − +( ∞ ) , µQ. −− +( ∞ )) ։ R min ( µQ. ∞ , µQ. ∞ ) ։ R min ( ∞ , ∞ ) ։ R ∞ The sp eciﬁcatio n of Q is pro ductive. M dep ends o nly on pure strea m functions, we can decide pro ductivit y . W e tra nslate M into a p ebbleﬂow net a nd collapse it to a so urce: 8 [ M ] = µM . • ([ zip ]([ inv ]( M ) , [ tail ]( M ))) = µM . • ( min ( − ++( − +( M )) , + − +( −− +( M )))) ։ R µM . • ( min ( − ++( M ) , + −− ++( M ))) ։ R µM . + − +( min ( − ++( M ) , + −− ++( M ))) ։ R µM . mi n (+ − +( − ++( M )) , + − +(+ −− ++( M ))) ։ R µM . mi n ( + − +( M ) , ++ −− ++( M )) ։ R min ( µM . + − +( M ) , µM . + + −− ++( M )) ։ R min ( ∞ , ∞ ) ։ R ∞ The sp eciﬁcatio n of M is pro ductive. 7.4 Example in Fig. 5 on P age 7 F or a pplying the automated pro ductivit y prov er to the pure stream speciﬁca tion in Fig. 5 on page 7 of the Thue–Morse sequence using the D0L-system 0 7→ 01, 1 7→ 10, one can use the input: Signature( 52 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks M : stream(bit ), Mprime : stream(bi t), h : stream(bit ) -> stream(bit), 0, 1 : bit ) M = 0:Mprime Mprime = 1:h(Mprim e) h(0:sigma) = 0:1:h(sigm a) h(1:sigma) = 1:0:h(sigm a) The a uto mated pro ductivity prover then gives the following o utput: The a uto mated pro ductivity prover has been applied to : Signature( -- stream symbols -- M : stream(bit ), Mprime : stream(bi t), h : stream(bit ) -> stream(bit), -- data symbol s -- 0 : bit, 1 : bit ) -- stream layer -- M = 0:Mprime Mprime = 1:h(Mprim e) h(0:sigma) = 0:1:h(sigm a) h(1:sigma) = 1:0:h(sigm a) -- data layer -- The function symbol h is pur e, we ca n compute its precise pro duction mo dulus: [ h ] = gate ([ h ] ⋆, 0 ( 0 ) , [ h ] 1 , 0 ) [ h ] 1 , 0 = µ h 1 , 0 . ∧    − ++ ∧ n h 1 , 0 − ++ ∧ n h 1 , 0 = − ++ [ h ] ⋆, 0 = µ h ⋆, 0 . ∧ ( ++ h ⋆, 0 ++ h ⋆, 0 = + M dep ends o nly on pure strea m functions, we can decide pro ductivit y . Data-Oblivious Stream Pro ductivity 53 W e tra nslate M into a p ebbleﬂow net a nd collapse it to a so urce: 9 [ M ] = µM . • ( µM p rime . • ([ h ]( M pr ime ))) = µM . • ( µM pri me. • ( − ++( M prime ))) ։ R µM . + − +( µM pri me. • ( − ++( M prime ))) ։ R µM . + − +( µM pr ime. + − +( − ++( M pr ime ))) ։ R µM . + − +( µM pri me. + − +( M prime )) ։ R µM . + − +( ∞ ) ։ R µM . ∞ ։ R ∞ The sp eciﬁcatio n of M is pro ductive. Mprime dep ends only on pure s tream functions, we can decide pro ductivit y . W e tra nslate Mprime in to a p ebbleﬂow net and collapse it to a sour c e : 9 [ Mprime ] = µM pr ime. • ([ h ]( M pri me )) = µM pr i me. • ( − ++( M pr ime )) ։ R µM p r ime. + − +( − ++( M p r ime )) ։ R µM p r ime. + − +( M pri me ) ։ R ∞ The sp eciﬁcatio n of Mprime is pr o ductive. 7.5 Example in Fig. 6 on P age 8 F or a pplying the automated pro ductivit y prov er to the pure stream speciﬁca tion in Fig. 6 on page 8 of the Thue–Morse sequence using the D0L-system 0 7→ 01, 1 7→ 10, one can use the input: Signature( nats, ones : stream(nat ), conv, add : stream (nat) -> stream(nat) -> stream(nat), times : stream (nat) -> nat -> stream(nat ), 0 : nat, s : nat -> nat, a, m : nat -> nat -> nat ) nats = 0:conv(ones ,ones) ones = s(0):ones conv(x:sig ma,y:tau) = m(x,y):add(tim es(tau,x),conv(sigma,y:tau)) times(x:si gma,y) = m(x,y):times (sigma,y) add(x:sigm a,y:tau) = a(x,y):add(sigm a,tau) 54 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks a(0,y) = y a(s(x),y) = s(a(x,y)) m(0,y) = 0 m(s(x),y) = a(y,m(x,y)) The a uto mated pro ductivity gives as output: The a uto mated pro ductivity prover has been applied to : Signature( -- stream symbols -- nats : stream( nat), ones : stream( nat), conv : stream( nat) -> stream(nat) -> stream(nat), add : stream(n at) -> stream(nat ) -> stream(nat), times : stream (nat) -> nat -> stream(nat ), -- data symbol s -- 0 : nat, s : nat -> nat, a : nat -> nat -> nat, m : nat -> nat -> nat ) -- stream layer -- nats = 0:conv(ones ,ones) ones = s(0):ones conv(x:sig ma,y:tau) = m(x,y):add(tim es(tau,x),conv(sigma,y:tau)) times(x:si gma,y) = m(x,y):times (sigma,y) add(x:sigm a,y:tau) = a(x,y):add(sigm a,tau) -- data layer -- a(0,y) = y a(s(x),y) = s(a(x,y)) m(0,y) = 0 m(s(x),y) = a(y,m(x,y)) T ermination o f the data lay er ha s b een proven automatically . The function symbol co nv is friendly nesting , we can compute a data -oblivious Data-Oblivious Stream Pro ductivity 55 low er b ound: [ conv ] = gate ([ conv ] ⋆, 0 ( 0 ) , [ conv ] 1 , 0 , [ conv ] 2 , 0 ) [ conv ] 1 , 0 = µ c onv 1 , 0 . ∧ n µx. − + x = − + [ conv ] 2 , 0 = µ c onv 2 , 0 . ∧ n µx. − + x = − + [ conv ] ⋆, 0 = µ c onv ⋆, 0 . ∧ n µx. − + x = − + The function symbol ad d is pure, we can compute its precise pro duction mo dulus: [ add ] = gate ([ add ] ⋆, 0 ( 0 ) , [ add ] 1 , 0 , [ add ] 2 , 0 ) [ add ] 1 , 0 = µ add 1 , 0 . ∧ n − + ∧ n add 1 , 0 = − + [ add ] 2 , 0 = µ add 2 , 0 . ∧ n − + ∧ n add 2 , 0 = − + [ add ] ⋆, 0 = µ add ⋆, 0 . ∧ n + add ⋆, 0 = + The function symbol times is pure, we can compute its prec is e pro duction mo d- ulus: [ times ] = gate ([ times ] ⋆, 0 ( 0 ) , [ times ] 1 , 0 ) [ times ] 1 , 0 = µ times 1 , 0 . ∧ n − + ∧ n times 1 , 0 = − + [ times ] ⋆, 0 = µ times ⋆, 0 . ∧ n + times ⋆, 0 = + nats dep ends only o n ﬂat strea m functions, we try to pro ve pro ductivit y . 56 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks W e tra nslate nats into a p ebbleﬂow net and collapse it to a sour c e: 10 [ nats ] = µnats. • ([ co n v ]( µones. • ( ones ) , µones. • ( ones ))) = µnats. • ( min ( − +(0) , min ( − +( µones. • ( ones )) , − +( µones. • ( ones ))))) ։ R µnats. + − +( min ( − +(0) , min ( − +( µone s. • ( ones )) , − +( µones. • ( ones ))))) ։ R µnats. min (+ − +( − +(0)) , + − +( min ( − +( µones. • ( ones )) , − +( µones. • ( ones ))))) ։ R µnats. min (+ − (0) , + − +( min ( − +( µone s. • ( ones )) , − +( µones. • ( ones ))))) ։ R min ( µnats. + − (0) , µnats. + − +( min ( − +( µones. • ( ones )) , − +( µones. • ( ones ))))) ։ R min ( µnats. 1 , µnats. min (+ − +( − +( µone s. • ( ones ))) , + − +( − +( µones. • ( ones ))))) ։ R min ( µnats. 1 , µnats. min (+ − ( µones. • ( ones )) , + − ( µones. • ( ones )))) ։ R min (1 , min ( µnats. + − ( µones. • ( one s )) , µnats. + − ( µones. • ( ones )))) ։ R min (1 , min ( µnats. + − ( µones. + − +( ones )) , µnats. + − ( µones. + − +( ones )))) ։ R min (1 , min ( µnats. + − ( ∞ ) , µnats. + − ( ∞ ))) ։ R min (1 , min ( µnats. ∞ , µna ts. ∞ )) ։ R min (1 , min ( ∞ , ∞ )) ։ R min (1 , ∞ ) ։ R 1 F ailed to prov e pro ductivity of n a ts . ones dep ends only on pure s tream functions, we can decide pro ductivity . W e tra nslate ones into a p ebbleﬂow net a nd colla ps e it to a sourc e: 10 [ ones ] = µones. • ( ones ) = µones. • ( ones ) ։ R µones. + − +( ones ) ։ R ∞ The sp eciﬁcatio n of ones is pro ductive. 7.6 Example on P age 12 Now we consider the strea m function s p eciﬁca tio n b elow Def. 3.5 on page 12, which is not a stream sp eciﬁcation b ecaus e it do es not contain a stream co nstant as its ro ot. Still, the automated pr o ductivit y prov er can b e used to co mpute the precise data -oblivious low er b ound of the stream function f . On the input: Signature( f : stream(bit ) -> stream(bit), g : stream(bit ) -> stream(bit) -> stream(bit ), 0, 1 : bit ) Data-Oblivious Stream Pro ductivity 57 f(s) = g(s,s) g(0:y:s,x: t) = 0:0:g(s,t) g(1:s,x1:x 2:x3:x4:t) = 0:0:0:0:0:g(s ,t) the pro ductivity prov er g enerates the following output: The a uto mated pro ductivity prover has been applied to : Signature( -- stream symbols -- f : stream(bit ) -> stream(bit), g : stream(bit ) -> stream(bit) -> stream(bit ), -- data symbol s -- 0 : bit, 1 : bit ) -- stream layer -- f(s) = g(s,s) g(0:y:s,x: t) = 0:0:g(s,t) g(1:s,x1:x 2:x3:x4:t) = 0:0:0:0:0:g(s ,t) -- data layer -- The function s ym b ol f is ﬂat, we can compute the precis e data-o blivious low er bo und: [ f ] = gate ([ f ] ⋆, 0 ( 0 ) , [ f ] 1 , 0 ) [ f ] 1 , 0 = µ f 1 , 0 . ∧                ∧                µ g 1 , 0 . ∧    −− ++ ∧ n g 1 , 0 − +++++ ∧ n g 1 , 0 µ g 2 , 0 . ∧    − ++ ∧ n g 2 , 0 −−−− +++++ ∧ n g 2 , 0 = −− −− ++ − ++ − + −− ++ − + − ++ − [ f ] ⋆, 0 = µ f ⋆, 0 . ∧ ( µ g ⋆, 0 . ∧ ( ++ g ⋆, 0 +++++ g ⋆, 0 = + 58 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks The function symbol g is ﬂa t, we can compute the precis e data-oblivio us low e r bo und: [ g ] = g a t e ([ g ] ⋆, 0 ( 0 ) , [ g ] 1 , 0 , [ g ] 2 , 0 ) [ g ] 1 , 0 = µ g 1 , 0 . ∧    −− ++ ∧ n g 1 , 0 − +++++ ∧ n g 1 , 0 = −− ++ [ g ] 2 , 0 = µ g 2 , 0 . ∧    − ++ ∧ n g 2 , 0 −−−− +++++ ∧ n g 2 , 0 = −− −− ++ − ++ − + [ g ] ⋆, 0 = µ g ⋆, 0 . ∧ ( ++ g ⋆, 0 +++++ g ⋆, 0 = + 8 Conclusion and F urther W ork In order to for malize quantitativ e appro aches fo r r ecognizing pro ductivity of stream s peciﬁca tions, we deﬁned the notion of d-o r ewriting and in vestigated d-o pro ductivity . F or the sy n ta c tic clas s o f ﬂat s tr eam s peciﬁca tions (that employ pattern matching on data), we devised a decision a lg orithm for d-o pro ductiv- it y . In this wa y we settled the pro ductivity r ecognition problem for ﬂat stream sp eciﬁcations from a d-o p ersp ective. F or the e v e n larg er class including friendly nesting stream function rules , we obtained a co mputable suﬃcient co nditio n for pro ductivity . F or the sub class of pure str eam s peciﬁca tions (a substantial exten- sion of the cla ss given in [2]) we s how ed that pro ductivity and d- o pro ductivity coincide, and ther e b y obtained a decision algor ithm for pr o ductivit y of pure sp eciﬁcations. W e hav e implemented in Haskell the decision algorithm for d- o pro ductivity . This tool, together with more information including a man ual, examples, our related papers, and a compa rison of our criteria with those o f [3 , 9, 1] can b e found at our web page http://infinity.few .vu.nl/productivity . The rea de r is invited to exp eriment with our to ol. It is no t p ossible to obtain a d-o optimal criterio n for non-pro ductivit y of ﬂat sp eciﬁcations in an analogo us wa y to how we es tablished such a criterion for pro ductivity . This is b ecause d-o upp er b ounds on the pro duction of str eam functions in ﬂat stre am sp eciﬁcations are not in ge ne r al p erio dica lly increasing Data-Oblivious Stream Pro ductivity 59 functions. F o r example, fo r the fo llowing strea m function sp eciﬁcatio n: f ( x : σ , τ ) → x : f ( σ , τ ) , f ( σ, y : τ ) → y : f ( σ, τ ) , it holds that do ( f )( n 1 , n 2 ) = n 1 + n 2 , which is not p-i. While this example is not orthogo nal, do ( f ) is als o no t p-i for the following simila r orthog onal exa mple: f ( 0 : x : σ , y : τ ) → x : f ( σ , τ ) , f ( 1 : σ , x : y : τ ) → y : f ( σ , τ ) . Currently we are developing a metho d tha t g o es b eyond a d-o ana lysis, one that would, e.g., pr ov e pro ductivity of the example B given in the intro duction. Moreov e r, we study a reﬁned pro duction calculus that accounts for the delay of ev alua tio n of str eam elements, in o rder to obtain a faithful mo delling of la zy ev aluation, needed for example for S on pag e 4, where the ﬁrst element dep ends on a ‘future’ e x pansion of S . A cknow le dgement. W e thank J an Willem K lop, Car los Lombardi, Vincent v an Oostro m, and Ro el de V rijer for useful dis c us sions. References 1. W. Buchholz. A T erm Calculus for (Co-)Recursive Deﬁn itions on Streamlike Data Structures. Anna l s of Pur e and Applie d Lo gic , 136(1-2):75–90, 2005. 2. J. En d rullis, C. Grabmay er, D. H endriks, A. Isihara, and J.W. Klop. Pro ductivity of Stream D eﬁnitions. I n Pr o c. of F CT 2007 , LNCS, pages 274–2 87. Sp ringer, 2007. 3. J. Hughes, L. Pareto, and A . Sabry . Proving the Correctness of R eactiv e Systems Using S ized T y p es. In POPL ’96 , p ages 410–423, 1996. 4. D. Kapu r, P . Narendran, D.J. Rosenkrantz, and H. Zhang. Suﬃcient-Completeness, Ground-Redu cibilit y and their Complexity. A cta Inf ormatic a , 28(4):311– 350, 1991. 5. J.W. Klop and R . de V rijer. Inﬁ nitary Normalization. In S. Arte- mo v , H. Barringer, A.S. d’Avila Garcez, L.C. Lamb, and J. W o od s, editors, We W il l Show Them: Essa ys in Honour of Dov Gabb ay , vol u me 2, pages 169–192. College Publications, 2005. Item 95 at http://web .mac.com/janwillem klo p/iWeb/Site/Bibliography.html . 6. J.J.M. M. Rutten . Beha vioural Diﬀerential Equ ations: a Coinductive Calculus of Streams, Automata, and Po wer Series. The or etic al Computer Scienc e , 308(1-3):1– 53, 2003. 7. A. Salomaa. Jewels of F ormal L anguage The ory . Pitman Publishing, 1981. 8. B.A. Sijtsma. On the Prod uctivity of R ecursive List Deﬁnitions. ACM T r ansactions on Pr o gr amming L anguages and Systems , 11(4):633– 649, 1989. 9. A. T elford and D. T urn er. Ensuring Streams Flo w. In AMAST , pages 509– 523, 1997. 10. T erese. T erm R ewriting Systems , volume 55 of Cambridge T r acts i n The or etic al Computer Scienc e . Cambridge Universit y Press, 2003. 60 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks App endix A Solving W eakly Guarded IO-T erm Sp eciﬁcations (Lemma 5.3) Let E = { X α = E α } α ∈ A be a weakly guarded ﬁnite io-sequence s peciﬁca tion, r o ot ∈ A , and σ r oot the unique solution o f E for X r oot . Deﬁnition A.1. W e deﬁne a ﬁnite, ro oted gra ph G E = hV , L , v r oot i that repre- sents traces through the expressions in E . The set of no des V co nsists o f all pa ir s of α ∈ A tog ether with po s itions in E α , the right-hand side o f the corr e spo nding equation in E : V = {h α, p i | α ∈ A, p ∈ Pos ( E α ) } . Note that the set of no des V is ﬁnite. The ro ot no de v r oot is h r o ot , hii . The set of lab elled edg es L ⊆ V × {− , + , ǫ } × V is the smallest set such that: for all α ∈ A , and every p osition p ∈ Pos ( E α ) the following conditions hold: (i) if E α | p = X α ′ then h α, p i ǫ → h α ′ , hii ∈ L ; (ii) if E α | p = − E ′ then h α, p i − → h α, p · 1 i ∈ L ; (iii) if E α | p = + E ′ then h α, p i + → h α, p · 1 i ∈ L ; (iv) if E α | p = E ′ ∧ E ′′ then h α, p i ǫ → h α, p · 1 i and h α, p i ǫ → h α, p · 2 i ∈ L . where we hav e written v ℓ → w to denote h v , ℓ, w i ∈ L . Mo reov er, for ℓ ∈ {− , + , ǫ } , we let ℓ → denote the rela tion {h v , w i | v ℓ → w } . Note that by weak guardedness o f E the relation ǫ → is terminating. W e pro ceed by deﬁning the set of traces T ( G E , v ) thro ugh the gr a ph G E starting a t no de v ∈ V . Deﬁnition A.2. F or a path ρ : v 1 l 1 → v 2 l 2 → . . . in G E , where l 1 , l 2 , . . . ∈ {− , + , ǫ } , we use t r ac e ( ρ ) to denote l 1 · l 2 · · · ∈ ± ∞ , the word obtained by concatenating the symbo ls ‘ − ’ and ‘+’ along the path ρ , thus treating the lab el ‘ ǫ ’ as the empt y word. Note that tra c e ( ρ ) ∈ ± ω whenever the path ρ is inﬁnite by weak guardedness of E . The set o f tra ces T ( G E , v ) ⊆ ± ω through G E from v ∈ V is deﬁned as follows: T ( G E , v ) := { tr ac e ( ρ ) | ρ is an inﬁnite path starting from no de v in G E } . F or Z ≡ { τ 1 , . . . , τ n } ⊆ ± ω , let V τ ∈ Z τ denote τ 1 ∧ . . . ∧ τ n , where ∧ is the binary o p era tion deﬁned in Def. 4 .13. If Z = ∅ , w e set V τ ∈ Z τ = +. The following lemma establishes a cor resp ondence b etw een the sp eciﬁcation E and the g r aph G E , in particular it implies that the unique solution σ r oot of E for X r oot equals the inﬁmum over all tr aces through G E from the ro ot no de. Lemma A.3. F or α ∈ A , p ∈ Pos ( E α ) , let σ ( α, p ) ∈ ± ω denote the u nique solu- tion for the ex pr ession E α | p in E , and let τ ( α, p ) b e shorthand for V τ ∈ T ( G E , h α, p i ) τ . Then, for al l α ∈ A , p ∈ Pos ( E α ) , we have σ ( α, p ) = τ ( α, p ) . Data-Oblivious Stream Pro ductivity 61 Pr o of. By coinduction, i.e., we show that the relatio n R ∈ ± ω × ± ω deﬁned by: R := {h σ ( α, p ) , τ ( α, p ) i | α ∈ A, p ∈ Pos ( E α ) } is a bisimulation. W e pro ceed b y well-founded induction on pairs h α, p i with resp ect to ǫ → . (Base) If v := h α, p i is in norma l for m with r esp ect to ǫ → , then v has pre c is ely one outg o ing edge: v − → o r v + → . In case v − → , we hav e that E α | p = − E ′ , and therefore σ ( α, p ) = − σ ( α, p · 1). F urther more, w e hav e τ ( α, p ) = − τ ( α, p · 1), and clearly h σ ( α, p · 1) , τ ( α, p · 1 ) i ∈ R . The pro of for case v + → pro ceeds ana logously . (Step) If v := h α, p i is not a normal form with r e spect to ǫ → , then either v has one o utgoing edge v ǫ → v ′ or tw o outgo ing edges v ǫ → v ′ and v ǫ → v ′′ . In the ﬁrst case we have E α | p = X α ′ , and then v ′ = h α ′ , hii , σ ( α, p ) = σ ( α ′ , hi ) and τ ( α, p ) = τ ( α ′ , hi ), and we conclude by a n application of the induction hypothesis for h α ′ , hii . In the second ca se we hav e tha t E α | p = E ′ ∧ E ′′ , a nd then v ′ = h α, p · 1 i and v ′′ = h α, p · 2 i . Thus we ge t σ ( α, p ) = σ ( α, p · 1) ∧ σ ( α, p · 2 ), and, by ass o ciativity of ∧ , τ ( α, p ) = τ ( α, p · 1) ∧ τ ( α, p · 2). W e conclude b y a double a pplication of the induction h y p othesis . ⊓ ⊔ The set T ( G E , v ) consists of inﬁnite, p ossibly non-pro ductive io-sequences. Therefore w e adopt the deﬁnition of the pro duction function [ [ σ ] ] : N → N to b e applicable to non-pro ductive io -sequences σ by dropping ∞ from the domain. Deﬁnition A.4. The pr o duction funct ion ˜ π σ of a sequence σ ∈ ± ω is corecur - sively deﬁned by , for all n ∈ N , ˜ π σ ( n ) := ˜ π ( σ, n ): ˜ π (+ σ, n ) = 1 + ˜ π ( σ, n ) , ˜ π ( − σ , 0 ) = 0 , ˜ π ( − σ, n + 1) = ˜ π ( σ, n ) . A pr o per t y o f the o p era tion ∧ given in Def. 4.13 is tha t the mapping [ [ ] ] : ± ∞ → N → N preser ves inﬁm um, see Prop. 4.19 . Clearly , a simila r pr op erty holds for ˜ π : for all σ 1 , σ 2 ∈ ± ω , w e hav e: ˜ π σ 1 ∧ σ 2 = ˜ π σ 1 ∧ ˜ π σ 2 . Then, the fo llowing coro llary dire ctly follows fr om Lemma A.3. Corollary A.5. F or al l n ∈ N we have ˜ π σ ro ot ( n ) = inf { ˜ π τ ( n ) | τ ∈ T ( G E , v r oot ) } . Mor e over, given n ∈ N , we c an always pick a tr ac e τ ∈ T ( G E , v r oot ) such that ˜ π τ ( n ) = ˜ π σ ro ot ( n ) . Using G E we deﬁne a ‘tw o-dimensiona l dia g ram’ D E ⊆ V × N × N re pr esenting the traces o f E star ting fro m X r oot as step functions. F or v ∈ V , x, y ∈ N let v x,y be shorthand for h v , x, y i . The ho rizontal axis ( x -axis) of T co rresp onds to input, and the vertical a xis ( y -axis) to output; that is, a ‘ − ’ in the trace yields a step to the right in the dia gram and ‘+’ a s tep upw ards . 62 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks Deﬁnition A.6. F or U ⊆ V × N × N w e deﬁne G E ( U ) ⊆ V × N × N , the one-step closur e of U under G E , as follows: G E ( U ) := U ∪ G E ,ǫ ( U ) ∪ G E , + ( U ) ∪ G E , − ( U ) G E ,ǫ ( U ) := { w x,y | v x,y ∈ U, v ǫ → w ∈ L} G E , + ( U ) := { w x,y +1 | v x,y ∈ U, v + → w ∈ L} G E , − ( U ) := { w x +1 ,y | v x,y ∈ U, v − → w ∈ L} F urthermore we deﬁne G ∗ E ( U ) := [ n ∈ N G n E ( U ) , the many-step closur e of U under G E . Deﬁnition A.7. W e deﬁne D E ⊆ V × N × N as D E := G ∗ E ( { v r oot 0 , 0 } ). The dia g ram D E contains the traces throug h G E starting from v r oot as ‘step functions’. Lemma A.8. F or al l v ∈ V , x, y ∈ N we have v x,y ∈ D E if and only if ther e exists a p ath ρ : v r oot → ∗ v in G E such that x = # − ( tr ac e ( ρ )) , y = # + ( tr ac e ( ρ )) . Pr o of. F ollows immediately from the deﬁnition of D E . ⊓ ⊔ W e pro ceed by showing that the so lutio n σ r oot of E for X r oot coincides with the ‘low er b ound’ o f the traces in D E . Deﬁnition A.9. F or U ⊆ V × N × N a nd G E = hV , L , v r oot i as a bove, w e deﬁne low U : N → N , the lower b ound of U , as follows: low U ( x ) := inf { y ∈ N | v x,y ∈ U, ∃ w. v − → w ∈ L} , where inf ∅ = ∞ . Lemma A.10 . The lower b ound of D E is the solut ion σ r oot of E for X r oot : ∀ n ∈ N . low D E ( n ) = ˜ π σ ro ot ( n ) Pr o of. Let n ∈ N , we show ˜ π σ ro ot ( n ) ≥ low D E ( n ) and ˜ π σ ro ot ( n ) ≤ low D E ( n ). ≥ F or ˜ π σ ro ot ( n ) = ∞ the pr o o f oblig a tion is trivial, thus as sume ˜ π σ ro ot ( n ) < ∞ . F rom Coro llary A.5 it follows that there exists a seq uenc e τ ∈ T ( G E , v r oot ) such that ˜ π τ ( n ) = ˜ π σ ro ot ( n ). Since ˜ π τ ( n ) < ∞ and τ ∈ ± ω , there exis ts m ∈ N with # − ( τ 0 ...m − 1 ) = n < n + 1 = # − ( τ 0 ...m ), and # + ( τ 0 ...m − 1 ) = ˜ π τ ( n ). Then ther e is a path ρ : v r oot → ∗ v − → v ′ in G E such that tr ac e ( ρ ) = τ 0 ...m . B y Lemma A.8 we hav e v n, ˜ π σ ro ot ( n ) ∈ D E , and together with v − → v ′ we conclude ˜ π σ ro ot ( n ) ≥ low D E ( n ) acco rding to Deﬁnition A.9. Data-Oblivious Stream Pro ductivity 63 ≤ Assume low D E ( n ) < ∞ . F r om Deﬁnition A.9 it follows that there exist no des v , v ′ ∈ V with v n, low D E ( n ) ∈ D E and v − → v ′ . Co nsequently by Lemma A.8 there exists a pa th ρ : v r oot → ∗ v − → v ′ in G E such that # − ( tr ac e ( ρ )) = n + 1 and # + ( tr ac e ( ρ )) = low D E ( n ). W e c ho os e a τ ∈ ± ω that has tr ac e ( ρ ) as a pre ﬁx, so that ˜ π τ ( n ) = low D E ( n ). Then, by Corolla ry A.5, we know that ˜ π σ ro ot ( n ) ≤ low D E ( n ). W e co nc lude ˜ π σ ro ot ( n ) = low D E ( n ). ⊓ ⊔ Although in principle D E could be used to calculate σ r oot for every n ∈ N , it remains to be shown how a rationa l representation of the solution σ r oot of E for X r oot can b e obtained. F or this purp ose w e construct a ‘simpler’ dia gram D 0 E ⊆ D E that has the sa me low er b ound. W e construct D 0 E such tha t there a re only ﬁnitely many no des o n each vertical line; ther e by we employ the idea that whenever ther e are t wo no des w x,y , w x,y ′ with y < y ′ , then we can omit the upper occur rence w x,y ′ (including all traces r ising from w x,y ′ ) without having an inﬂuence on the low er b ound low D E ( x ). Deﬁnition A.11 . W e deﬁne functions G 0 E , omit : ( V × N × N ) → ( V × N × N ): G 0 E ( U ) := U ∪ G E ,ǫ ( U ) ∪ G E , + ( U ) omit ( U ) := { v x,y | v x,y ∈ U, ¬∃ y ′ < y . v x,y ′ ∈ U } Let D 0 E ( n ) ⊆ V × N × N for all n ∈ N be deﬁned as follows: D 0 E (0) := vclosur e ( { v r oot 0 , 0 } ) D 0 E ( n + 1) := D 0 E ( n ) ∪ D 0 E | n +1 D 0 E | n +1 := vclosur e ( step right ( n )) step righ t ( n ) := { w n +1 ,y | v n,y ∈ D 0 E ( n ) , v − → w ∈ L} , where vclosur e ( U ) ⊆ V × N × N is deﬁned to b e vc o m ( U ) for the s mallest m ∈ N such that vc o m +1 ( U ) = vc o m ( U ) where vc o := omit ◦ G 0 E , the vertical one-step closure follow ed by an application of omit . F urthermore w e deﬁne D 0 E := S n ∈ N D 0 E ( n ) . Note that D 0 E ( n ) is ﬁnite for all n ∈ N . The impo rtant step in the constr uction is the termination of vclosur e ( ) which is gua ranteed by the following lemma. Lemma A.12 . The c omputation of vclosur e ( U ) is terminating for ﬁnite sets U ⊆ V × N × N , that is, t her e exists m ∈ N su ch that vc o m +1 ( U ) = vc o m ( U ) . Pr o of. F or U ⊆ V × N × N we intro duce auxiliar y deﬁnitions : X ( U ) := { x ∈ N | ∃ v ∈ V , y ∈ N . v x,y ∈ U } ⊆ N V ( U, x ) := { v ∈ V | ∃ y ∈ N . v x,y ∈ U } ⊆ V Y ( U, x, v ) := { y ∈ N | v x,y ∈ U } ⊆ N y ( U, x, v ) := min( Y ( U, x, v )) ∈ N . 64 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks Note that Y ( W , x, v ) = { y ( W, x, v ) } is a singleton se t for every W := omit ( U ), x ∈ X ( W ) a nd v ∈ V ( W, x ) with U ⊆ V × N × N arbitra ry . Let U ⊆ V × N × N . W e deﬁne U 0 := U a nd for n = 1 , 2 , . . . let U n := vc o ( U n − 1 ) . Note that U n = vc o n ( U ). F rom the deﬁnition of G 0 E and omit it follows that: (i) ∀ n ∈ N . X ( U n ) = X ( U ), (ii) ∀ n ∈ N . U n ⊆ G 0 E ( U n ), (iii) ∀ n ∈ N , x ∈ X ( U ) . V ( U n , x ) ⊆ V ( U n +1 , x ), since V ( U n , x ) ⊆ V ( G 0 E ( U n ) , x ) = V ( vc o ( U n ) , x ) = V ( U n +1 , x ) (iv) ∀ n ≥ 1 , x ∈ X ( U ) . |V ( U n , x ) | ≤ |V | . By (iii) and (iv) we conclude that there exists n 0 ∈ N : ∀ n ≥ n 0 , x ∈ X ( U ) . V ( U n , x ) = V ( U n 0 , x ) , (1) that is, the s e t of no des V ( U n , x ) on ea c h vertical str ip x b ecomes ﬁxed after n 0 steps. F ur thermore by (ii) and the deﬁnition of omit we get ∀ n ∈ N , x ∈ X ( U ) , v ∈ V ( U n , x ) . y ( U n +1 , x, v ) ≤ y ( U n , x, v ) , (2) that is , the heigh ts of the no des are non- inc r easing. Therefor e for ea ch x ∈ X ( U ) and no de v ∈ V ( U n 0 , v ) there ex is ts m x,v ≥ n 0 such that: ∀ n ≥ m x,v . y ( U n , x, v ) = y ( U m x,v , x, v ) Let m := max { m x,v | x ∈ X ( U ) , v ∈ V ( U n 0 , v ) } , then the heights of all no des are ﬁxed from m onw ards a nd consequently we hav e ∀ n ≥ m. U n = U m . ⊓ ⊔ Lemma A.13 . D E and D 0 E have the same lower b ound, that is, low D E = low D 0 E . Pr o of. Now w e constr uct a rational r epresentation of the low e r bo und of D 0 E , which by Le mma A.13 and A.10 is the solution σ r oot of E for X r oot . The co nstruction is based on a ‘rep etition search’ o n D 0 E , that is, w e search tw o v er tical strips x 1 and x 2 that contain similar constellatio ns of no des allowing to conclude that from the vertical po sition x 1 onw ards the low er b ound is ‘quasi per io dic’. Deﬁnition A.14 . F or x ∈ N , v ∈ V let V ( x ) := { v ∈ V | ∃ y ∈ N . v x,y ∈ D 0 E ( x ) } ⊆ V y ( x, v ) := ( y if there exists y ∈ N with v x,y ∈ D 0 E ( x ) ∞ otherw is e . Note that y ( x, v ) is well-deﬁned since there ex ists at most one s uch y ∈ N . Data-Oblivious Stream Pro ductivity 65 Deﬁnition A.15 . F or x ∈ N we deﬁne h ( x ) := min { y ( x, v ) | v ∈ V } , the height of the low est no de on the vertical strip x . Deﬁnition A.16 . F or x ∈ N , v ∈ V let y r ( x, v ) := y ( x, v ) − h ( x ) , the height of the no de v relative to the low est no de on str ip x . Deﬁnition A.17 . F or x 1 < x 2 ∈ N with V ( x 1 ) 6 = ∅ a nd V ( x 2 ) 6 = ∅ let d ( x 1 , x 2 ) := h ( x 2 ) − h ( x 1 ) , the height diﬀerence betw ee n the vertical strips x 1 and x 2 . Deﬁnition A.18 . F or x 1 < x 2 ∈ N , v ∈ V with v ∈ V ( x 1 ) and v ∈ V ( x 2 ) let d ( x 1 , x 2 , v ) := y ( x 2 , v ) − y ( x 1 , v ) , the height diﬀerence of the no de v b et ween the vertical str ips x 1 and x 2 , and d r ( x 1 , x 2 , v ) := h ( x 1 , x 2 , v ) − d ( x 1 , x 2 ) , the relative heig h t diﬀerence of the no de v b etw een the vertical strips x 1 and x 2 . Lemma A.19 . F or x 1 , x 2 ∈ N , v ∈ V ( x 1 ) it holds: y r ( x 2 , v ) = y r ( x 1 , v ) + d r ( x 1 , x 2 , v ) . Pr o of. Immediately from the deﬁnitions. ⊓ ⊔ W e pro ceed with the deﬁnition of ‘pseudo rep etitions’ h x 1 , x 2 i ∈ N 2 . That is, we require that the vertical strips x 1 and x 2 po ssess a s imila r constella tion of no des. A pse udo repetition do es not yet guar antee ‘qua si per io dicit y ’ of the lower bo und (having found the cyclic pa rt of the io-se quence), as will b e explained a fter the deﬁnition. Deﬁnition A.20 . A pseudo r ep etition is a pair h x 1 , x 2 i ∈ N 2 such that: (i) The vertical s trips x 1 and x 2 contain the s a me set of no des v ∈ V : V ( x 1 ) = V ( x 2 ) (ii) F r o m strip x 1 to x 2 all no des have increased their height by at least d ( x 1 , x 2 ): ∀ v ∈ V . d r ( x 1 , x 2 , v ) ≥ 0 66 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks F or a pseudo r epe titio n h x 1 , x 2 i we c an dis tinguish b etw een no des v for which d r ( x 1 , x 2 , v ) = 0 a nd those with d r ( x 1 , x 2 , v ) > 0. If for a ll no des v ∈ V ( x 1 ) it holds that d r ( x 1 , x 2 , v ) = 0 then we ha ve an exact r epetitio n, yielding ‘quasi per io dicit y ’ of the low er b ound. On the other hand assume that there is a no de v ∈ V ( x 1 ) which increases its relative height and c o n tr ibutes to the low er b ound betw een the vertical s tr ips x 1 and x 2 . Then we do not yet have a ‘quasi per io dic’ low er b o und, b ecause due to the increas ing relative height of v , the contribution v to the low er bo und will change. F or this rea s on we will later strengthen the conditions on ps eudo rep etitions, yielding ‘rep etitions’ which then will guara n tee ‘quasi p erio dicity’, see Deﬁnition A.27. Lemma A.21 . Ther e exist pseudo r ep etitions in D 0 E . Pr o of. F or every x ∈ N we hav e V ( x ) ∈ P ( V ) and P ( V ) is a ﬁnite s et. Hence it follows fro m the Pigeonho le Principle tha t ther e exist indices i 1 < i 2 < i 3 < . . . with V ( i 1 ) = V ( i 2 ) = V ( i 3 ) = . . . , let V ′ := V ( i 1 ) = { v 1 , . . . , v k } with k = |V ( i 1 ) | . Let ≤ on N k be deﬁned as follows: ( a 1 , . . . , a l ) ≤ ( b 1 , . . . , b l ) ⇔ a 1 ≤ b 1 ∧ . . . ∧ a l ≤ b l . F or x ∈ { i 1 , i 2 , . . . } we deﬁne the tuple t x := ( y r ( x, v 1 ) , . . . , y r ( x, v k )) ∈ N k . It is a well-known fact that ≤ is an almost full relation on N k , see [10]. Hence there exist x 1 , x 2 ∈ { i 1 , i 2 , . . . } with t x 1 ≤ t x 2 . Then d r ( x 1 , x 2 , v ) ≥ 0 for all v ∈ V ′ which es tablishes that h x 1 , x 2 i is a pseudo rep etition. ⊓ ⊔ The following deﬁnitions and lemmas a re auxilia ry for the purp ose of proving Lemma A.26. Deﬁnition A.22 . F or U , W ⊆ V × N × N and d x , d y ∈ N we deﬁne U ≤ d x ,d y W ⇐ ⇒ ∀ v x,y ∈ W. ∃ y ′ < y − d. v x − d x ,y ′ ∈ U . Lemma A.23 . F or x 1 < x 2 ∈ N and d y ∈ N we have: D 0 E | x 1 ≤ x 2 − x 1 ,d y D 0 E | x 2 ⇐ ⇒ ∀ v ∈ V ( x 1 ) . y ( x 2 , v ) ≥ d y + y ( x 1 , v ) Pr o of. F ollows from the deﬁnitions. ⊓ ⊔ Lemma A.24 . If U 1 , U 2 ⊆ V × N × N , d x , d y ∈ N with U 1 ≤ d x ,d y U 2 then: (i) vc o ( U 1 ) ≤ d x ,d y vc o ( U 2 ) , and (ii) vclosur e ( U 1 ) ≤ d x ,d y vclosur e ( U 2 ) . Pr o of. F r o m U 1 ≤ d x ,d y U 2 it follows that G 0 E ( U 1 ) ≤ d x ,d y G 0 E ( U 2 ) holds, and then we get omit ◦ G 0 E ( U 1 ) ≤ d x ,d y omit ◦ G 0 E ( U 2 ) by deﬁnition of omit . By induction w e get ∀ n ∈ N . vc o n ( U 1 ) ≤ d x ,d y vc o n ( U 2 ). F urthermor e ther e exist m 1 , m 2 ∈ N such that vclosur e ( U i ) = vc o m i ( U i ) for i ∈ { 1 , 2 } b y deﬁnition. F rom ∀ m ′ i ≥ m i . vc o m ′ i ( U i ) = vc o m i ( U i ) it follows that for i ∈ { 1 , 2 } we hav e vclosur e ( U i ) = vc o max( m 1 ,m 2 ) ( U i ). Hence vclosur e ( U 1 ) ≤ d x ,d y vclosur e ( U 2 ). ⊓ ⊔ Data-Oblivious Stream Pro ductivity 67 Lemma A.25 . If x 1 < x 2 ∈ N with V ( x 1 ) = V ( x 2 ) , then V ( x 1 + 1 ) = V ( x 2 + 1 ) . If mor e over D 0 E | x 1 ≤ x 2 − x 1 ,d y D 0 E | x 2 for d y ∈ N , then D 0 E | x 1 +1 ≤ x 2 − x 1 ,d y D 0 E | x 2 +1 . Pr o of. F r o m V ( x 1 ) = V ( x 2 ) and the deﬁnitio ns of D 0 E ( x 1 + 1) and D 0 E ( x 2 + 1) it follows that V ( x 1 + 1) = V ( x 2 + 1). Assume D 0 E | x 1 ≤ x 2 − x 1 ,d y D 0 E | x 2 and consider Deﬁnition A.1 1. Then it follows that step right ( x 1 ) ≤ x 2 − x 1 ,d y step right ( x 2 ). W e conclude with a n application of Lemma A.24 yielding D 0 E | x 1 +1 ≤ x 2 − x 1 ,d y D 0 E | x 2 +1 . ⊓ ⊔ Lemma A.26 . L et h x 1 , x 2 i b e a pseudo r ep etition. Then t he p air h x 1 + m, x 2 + m i is a pseudo r ep etition for al l m ∈ N . Pr o of. It suﬃces to show that h x 1 + 1 , x 2 + 1 i is a ps e udo rep etition, for m > 1 the claim follows by induction. W e g et V ( x 1 + 1) = V ( x 2 + 1) by Lemma A.25. F urthermore b y the deﬁnition of pseudo rep etition we hav e ∀ v ∈ V ( x 1 ) . d r ( x 1 , x 2 , v ) ≥ 0, which is equiv alen t to D 0 E | x 1 ≤ x 2 − x 1 ,d ( x 1 ,x 2 ) D 0 E | x 2 since V ( x 1 ) = V ( x 2 ). Emloying Le mma A.25 we get D 0 E | x 1 +1 ≤ x 2 − x 1 ,d ( x 1 ,x 2 ) D 0 E | x 2 +1 , that is ∀ v ∈ V ( x 1 + 1) . d r ( x 1 + 1 , x 2 + 1 , v ) ≥ 0. Hence h x 1 + 1 , x 2 + 1 i is a pseudo rep etition. ⊓ ⊔ W e pro ceed with the deﬁnition of ‘rep etitions’. Then we show that rep etitions exist a nd that these indeed gua rantee quasi p erio dicity of the low er b ound. Deﬁnition A.27 . A r ep etition is a pseudo re p etition h x 1 , x 2 i if: (i) h x 1 , x 2 i is a pseudo rep etition (ii) Let I ( x 1 , x 2 ) := { v x,y | v x,y ∈ T 0 , x = x 1 , i ( x 1 , x 2 , p ) = 0 } . W e req uire that only no des from I ( x 1 , x 2 ) contribute to the lower b ound b etw een x 1 and x 2 and these no des r econstruct themselfs o n the str ip x 2 : ∀ x ∈ N , x 1 ≤ x ≤ x 2 . low G E ( I ) ( x ) = low T 0 ( x ) (3) ∀ v x,y ∈ I ( x 1 , x 2 ) . v x 2 ,y + d ( x 1 ,x 2 ) ∈ G E ( I ) (4) Both conditions can b e eﬀectively chec ked since G E ( I ) | ≤ n can b e computed for every n ∈ N . Having suc h a re p etition is is easy to see that the low er b ounds behaves ‘quas i p erio dic’ fro m this po in t onw ards, that is the io-sequence is in- deed ra tional. This is b ecause the the nodes with i ( x 1 , x 2 , v ) = 0 consitute a exactly r epea ting pattern. F urthermore the no des with increa sing rela tiv e heig h t i ( x 1 , x 2 , v ) > 0 do not contribute to the low er b ound betw een x 1 and x 2 , a nd since their height ca nno t decrea se they will a lso not co n tribute in the future. It remains to show that we indeed a lw ays encounter a rep etition h x 1 , x 2 i . W e know that there exists a pseudo rep etition h x 1 , x 2 i . Then we have for a ll m ∈ N we hav e h x 1 + m · ( x 2 − x 1 ) , x 2 + m · ( x 2 − x 1 ) i is a pseudo r e petition. F or all m ∈ N and v ∈ V we have i ( x 1 + m · ( x 2 − x 1 ) , x 2 + m · ( x 2 − x 1 ) , v ) ≥ 0, therefore it follows that there ar e some no des for which the relative height incr ease will even tually stay a t 0 while o ther no des contin ue increasing their rela tiv e height. The la tter will from some p oint on no longer contribute to the lower bo und and the pattern. Hence there e x ists m 0 for which h x 1 + m 0 · ( x 2 − x 1 ) , x 2 + m 0 · ( x 2 − x 1 ) i is a rep etition. 68 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks App endix B T ermination of the F unction La yer T ranslation (Lemma 5.10) It suﬃces to show that, for every stream function symbol f and ar gument p osition i of f , the inﬁnite io-sequence sp eciﬁcatio n E T deﬁned in Def. 5.4 that has [ f ] i as unique solution for the recurs io n v ariable X h f , i, 0 i can be transformed in ﬁnitely many steps into a n io-sequence speciﬁc a tion E ′ with the same so lution for the recursion v aria ble X h f , i, 0 i and the pro p erty that only ﬁnitely man y r e cursion v ariables are re a chable in E ′ from X h f , i, 0 i . (This is b ecaus e from such a n io - sequence sp eciﬁcation E ′ even tually a ﬁnite io-sequence sp eciﬁcation E ′ 0 with [ f ] i as unique s olution for X h f , i, 0 i can b e e x tracted.) An a lgorithm for this purp ose is o btained as follows. On the input of E T , set E := E T and rep eat the following step o n E a s long a s it is applicable: (RPC) Detect and remov e a r e achable non-c onsuming pseudo-cycle from the io- sequence sp eciﬁcatio n E : Supp ose that, for a function sy m b ol h , for j, k , l ∈ N , and for a recurs ion v ariable X h h , j , k i that is r eachable from X h f , i, 0 i , we hav e X h h , j , k i w − → X h h , j , l i (from the recursion v aria ble X h h , j , k i the v aria ble X h h , j , l i is reachable via a path in the sp eciﬁcation on which the ﬁnite io- sequence w is enco un ter e d as the word formed by consecutive labels ), where l < k a nd w only contains sym b ols ‘+’. Then modify E by setting X h h , j , k i = + X h h , j , k i . It is not diﬃcult to show that a step (RPC) preser v e s weakly g ua rdedness and the unique solution of E , and tha t, o n input E T , the algo rithm terminates in ﬁnitely man y steps, pr oducing an io-sequence sp eciﬁcation E ′ with [ f ] i as the solution for X h f , i, 0 i and the prop erty that only ﬁnitely many r ecursion v ar iables are r eachable in E ′ from X h f , i, 0 i . Data-Oblivious Stream Pro ductivity 69 App endix C Soundness of the F unct ion La yer T ranslation Throughout, let T = h Σ , R i b e a ﬁxed strea m sp eciﬁcation. Fir st w e intro duce a few axillary lemmas and deﬁnitio ns. Deﬁnition C.1 . F or all stream ter ms t ∈ T er ( L Σ M ) we deﬁne by # • ( t ) : = sup { n ∈ N | t = • k ( t ′ ) } the nu mb er of le ading s t r e am p ebbles of t . Deﬁnition C.2 . Let h ∈ Σ sfun be a a stream function symbol and j, k ∈ N . W e deﬁne h j,k := h ( . . . • ∞ . . . , • k ( σ ) , . . . • ∞ . . . ) ∈ T er ( L Σ M ) wher e the j -th ar g umen t po sition is • k ( σ ) while all other ar gument s a re • ∞ . Deﬁnition C.3 . F or terms s ≡ g ( • n 1 ( σ ) , . . . , • n k ( σ )) ∈ T er ( L Σ M ) t ≡ h ( • m 1 ( σ ) , . . . , • m k ( σ )) ∈ T er ( L Σ M ) with n 1 , . . . , n k , m 1 , . . . , m k ∈ N , we write s ⊳ t if n i ≤ m i for all i . Lemma C.4 . If s, t ∈ T er ( L Σ M ) as in Def. C.3 with s ⊳ t then do T ( s ) ≤ do T ( t ) . Pr o of. Immediately from the deﬁnition. ⊓ ⊔ W e pro o f the soundness o f the function layer translatio n, that is: Lemma C.5 . L et T b e a st r e am deﬁnition, and let f ∈ Σ sfun . (i) If T is ﬂat, t hen it holds: [ [[ f ]] ] = do T ( f ) . Henc e, do T ( f ) is a p erio dic al ly incr e asing function. (ii) If T is friend ly nest ing, then it holds: [ [[ f ]] ] ≤ do T ( f ) . Pr o of. W e start w ith (i). Le t E T be the weakly gua rded io-sequence s p eciﬁca tio n (ov er v ar iables X ) obta ined from the tr a nslation. Let V : X → ( N → N ) be the unique solutio n o f E T , which exists b y Lemma 5 .3. Then by de ﬁnitio n we hav e [ [[ f ]] ] = min( V ( X h f , 1 , 0 i ) , . . . , V ( X h f , n, 0 i )). Assume f is not weakly g uarded. There e x ists g ∈ Σ sfun and γ : f ; ∗ g ; + g without pro duction; let Σ γ be the set of sym b ols o ccurr ing in γ and ρ 1 ,. . . , ρ n ∈ R the rules applied in γ (all having zer o pro duction). W e deﬁne T γ as the set of terms t ∈ T er ( L Σ M ) with r oot ( t ) ∈ Σ γ . Accor ing to Lemma 3.9 we can contruct a data- exchange function G such that fo r e very term t ∈ T γ either G ( t ) is not a r e dex or a redex with res pect to a ρ i . Then every r educt of a term t ∈ T γ in A T ( G ) as aga in in T γ and has ther efore no pro duction. Hence do T ( f ) = 0 and by deﬁnition [ [[ f ]] ] = 0 since X h f , i, 0 i = X h ǫ, −i for all 1 ≤ i ≤ ar s ( f ). The r emaining pro o f obligation is [ [[ f ]] ] = do T ( f ) for w ea kly gua rded f . W e start with [ [[ f ]] ] ≥ do T ( f ). Let p 1 , . . . , p ar s ( f ) ∈ N a nd deﬁne o := [ [[ f ]] ]( p 1 , . . . , p ar s ( f ) ). W e have V ( X h f , i, 0 i )( p i ) = o for some 1 ≤ i ≤ ar s ( f ). Note tha t do T ( f )( p 1 , . . . , p i , . . . , p ar s ( f ) ) ≤ do T ( f )( . . . ∞ . . . , p i , . . . ∞ . . . ) = do T ( f i,p i ) . 70 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks Hence it suﬃces to show do T ( g i,q + p ) ≤ V ( X h g , i, q i )( p ) for a ll g ∈ Σ sfun , argument po sitions 1 ≤ i ≤ ar s ( g ) ∈ N and q , p ∈ N . F or the case V ( X h g , i, q i )( p ) = ∞ there is no thing to b e shown, hence let V ( X h g , i, q i )( p ) < ∞ . E mploying well-founded induction with r e s pect to the ordering > ov er tuples h g , i , q , p i with V ( X h g , i, q i )( p ) < ∞ we constr uct a data-exchange function G together with an outermost-fair rewrite sequence in A T ( G ) sta rting from g i,p + q as witness of do T ( g i,q + p ) ≤ V ( X h g , i, q i )( p ). The o r dering > is deﬁned as follows: h g , i , q , p i > h g ′ , i ′ , q ′ , p ′ i ⇐ ⇒ g is weakly-guarded ∧ ( o > o ′ ∨ ( o = o ′ ∧ g ; + g ′ )) where o := V ( X h g , i, q i )( p ) , o ′ := V ( X h g ′ , i ′ , q ′ i )( p ′ ) Let G be an ar bitrary data-exchange function. The induction basis consists of the tuples h g , i, q , p i wher e – g is not weakly guar de d, or – V ( X h g , i, q i )( p ) = 0 a nd g is a norma l for m with resp ect to ; + . The not weakly guarded symbo ls hav e be en discussed a bove, therefore let g b e a normal for m with resp ect to ; + . By deﬁnition X h g , i, q i = V ρ ∈ R g X h g , i, q, ρ i , and hence V ( X h g , i, q, ρ i )( p ) = 0 for so me ρ ∈ R sfun . Then X h g , i, q, ρ i is either of s hape (shap e 1) − in ( ρ,i ) . − q + . . . or (shape 2 ) − ( in ( ρ,i ) . − q ) +1 + . . . T o see this cons ider the c a se distinction in the deﬁnitio n of X h g , i, q, ρ i : X h g , i, q, ρ i = − in ( ρ,i ) . − q + out ( ρ )      V j ∈ φ − 1 ( i ) X h h , j , ( q . − in ( ρ,i ))+ su ( ρ,j ) i for (a) ( C 1) + q . − in ( ρ,i ) X h ǫ, − + i for (b) if j = i ( C 2) X h ǫ, + i for (b) if j 6 = i ( C 3) W e have (shap e 1) in cas e of: – (C1) since g being a ; normal form and therefore out ( ρ ) > 0 , – (C2) if out ( ρ ) + ( q . − in ( ρ, i )) > 0, and – (C3) The (shap e 2) o ccurs in c ase (C2) if out ( ρ ) + ( q . − in ( ρ, i )) = 0. Let t ≡ g i,q + p and G ′ = G ( t, ρ ), then: Data-Oblivious Stream Pro ductivity 71 – Assu me X h g , i, q, ρ i is of (shap e 1). F rom V ( X h g , i, q , ρ i )( p ) = 0 it fo llows in ( ρ, i ) . − q > p and hence in ( ρ, i ) > q + p . G ′ ( t ) cannot b e a ρ -redex since ρ has at a rgument p o sition i a co nsumption greater than q + p . Hence g i,q + p is a normal form in A T ( G ′ ) by Lem. 3.9 and it follows that do T ( g i,q + p ) = 0. – Assu me X h g , i, q, ρ i is of (shap e 2). Then we have out ( ρ ) + ( q . − in ( ρ, i )) = 0 , hence out ( ρ ) = 0 and q ≤ in ( ρ, i ). F rom V ( X h g , i, q, ρ i )( p ) = 0 it follows that ( in ( ρ, i ) . − q ) + 1 = in ( ρ, i ) − q + 1 > p and therefore in ( ρ, i ) ≥ q + p . Then g i,q + p is a normal form if in ( ρ, i ) > q + p or g i,q + p → σ if in ( ρ, i ) = q + p in A T ( G ′ ); in b oth cas es do T ( g i,q + p ) = 0. This co ncludes the ba se case of the induction. Now we consider the induction step. By deﬁnition X h g , i, q i = V ρ ∈ R g X h g , i, q, ρ i , and hence V ( X h g , i, q i )( p ) = V ( X h g , i, q, ρ i )( p ) for some ρ ∈ R sfun . Let t ≡ g i,q + p and G ′ = G ( t, ρ ). Again we co nsider the case distinctio n (C1),(C2) and (C3) in the deﬁnition of X h g , i, q, ρ i . If in ( ρ, i ) . − q > p then in all ca s es V ( X h g , i, q, ρ i )( p ) = 0 and g i,q + p is a nor mal form in A T ( G ′ ), consequently do T ( g i,q + p ) = 0. Therefo re assume in ( ρ, i ) . − q ≤ p . (C1) X h g , i, q, ρ i = − in ( ρ,i ) . − q + out ( ρ ) V j ∈ φ − 1 ( i ) X h h , j , ( q . − in ( ρ,i ))+ su ( ρ,j ) i There exists j ∈ φ − 1 ( i ) with V ( X h g , i, q, ρ i )( p ) = e V ( p ) where e := − in ( ρ,i ) . − q + out ( ρ ) X h h , j , q ′ i q ′ := ( q . − in ( ρ, i )) + su ( ρ, j ) . Since in ( ρ, i ) . − q ≤ p , let p ′ := p − ( in ( ρ, i ) . − q ). W e get V ( X h g , i, q, ρ i )( p ) = out ( ρ ) + V ( X h h , j , q ′ i )( p ′ ) . (5) G ′ ( g i,q + p ) is a ρ -redex by Lem. 3.9, hence g i,q + p → A T ( G ′ ) • out ( ρ ) ( h ( • n 1 ( σ ) , . . . , • n ar s ( h ) ( σ ))) where by deﬁnition of ρ : n j = su ( ρ, j ) + ( q + p − in ( ρ, i )) = q ′ + p ′ . Therefore h ( • n 1 ( σ ) , . . . , • n ar s ( h ) ( σ )) ⊳ h j,q ′ + p ′ . W e hav e V ( X h g , i, q, ρ i )( p ) > V ( X h h , j , q ′ , ρ i )( p ′ ) or g ; h by (5 ), ther efore we get do T ( h j,q ′ + p ′ ) ≤ V ( X h h , j , q ′ i )( p ′ ) by induction hypo thes is and do T ( g i,q + p ) ≤ out ( ρ ) + do T ( h j,q ′ + p ′ ) ≤ out ( ρ ) + V ( X h h , j , q ′ i )( p ′ ) = V ( X h g , i, q, ρ i )( p ) = V ( X h g , i, q i )( p ) which pr ov es the claim. 72 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks (C2) X h g , i, q, ρ i = − in ( ρ,i ) . − q + out ( ρ ) + q . − in ( ρ,i ) X h ǫ, − + i Then V ( X h g , i, q, ρ i )( p ) = out ( ρ ) + ( q . − in ( ρ, i )) + V ( X h ǫ, − + i )( p − ( in ( ρ, i ) . − q )) = out ( ρ ) + ( q . − in ( ρ, i )) + ( p − ( in ( ρ, i ) . − q )) = out ( ρ ) + q + p − in ( ρ, i ) F urthermore in cas e (C2) ρ is collaps ing with i = j , therefore g i,q + p → A T ( G ′ ) • out ( ρ )+( q + p − in ( ρ,i )) ( σ ) . Hence do T ( g i,q + p ) ≤ out ( ρ ) + q + p − in ( ρ, i ) = V ( X h g , i, q, ρ i )( p ) . (C3) X h g , i, q, ρ i = − in ( ρ,i ) . − q + out ( ρ ) X h ǫ, + i Then V ( X h g , i, q, ρ i )( p ) = ∞ = do T ( g i,q + p ) since ρ is collaps ing with i 6 = j . This co ncludes the dir ection [ [[ f ]] ] ≥ do T ( f ). The rema ining pro of oblig ation is [ [[ f ]] ] ≤ do T ( f ). Let p 1 , . . . , p ar s ( f ) ∈ N and deﬁne o := do T ( f )( p 1 , . . . , p ar s ( f ) ) = do T ( f ( • p 1 , . . . , • p ar s ( f ) )). Assume o < ∞ , otherwis e there is nothing to b e shown. Then there exists a data-exchange function G and a ﬁnite r eduction sequence f ( • p 1 ( σ ) , . . . , • p ar s ( f ) ( σ )) → n A T ( G ′ ) • o ( t ) such that do T ( t ) = 0 and in particular – t is a normal for m in A T ( G ′ ), or – g is not weakly guar de d. W.l.o.g. choo se G and the reduction seq uence such that n is minimal. W e apply induction on the length of the re ductio n n to show – There exists 1 ≤ i ≤ ar s ( f ) such that do T ( f ( • p 1 ( σ ) , . . . , • p ar s ( f ) ( σ ))) = do T ( f ( • ∞ , . . . , • p i , . . . , • ∞ )) , – and [ [[ f ]] ] ≤ do T ( f ( • p 1 ( σ ) , . . . , • p ar s ( f ) ( σ ))). Data-Oblivious Stream Pro ductivity 73 App endix D Soundness of the Str eam La y er T ranslation (Lemma 5.17 and Lemma 5.18) D.1 Pro of Sk etch of Lemm a 5. 17 W e o nly s ketch a pro of of statement (i) o f Lem. 5.17 b ecause item (ii) o f this lemma ca n b e demonstr ated analog ously . Pr o of (of L em. 5.17, (i)). Let T b e an SCS, a nd let F = { γ f } f ∈ Σ sfun be a family of g ates such that, fo r all f ∈ Σ sfun , [ [ γ f ] ] = do T ( f ). W e give the ideas for the pro ofs of the inequations “ ≤ ” and “ ≥ ” that make part of the eq ua tion do T ( M 0 ) = [ [[ M 0 ] F ] ] . (6) W e assume, without loss of ge ner ality , that every f ∈ Σ sfun has prec isely one o ccurrence in the SCS- layer. Note that, by replacing for every f ∈ Σ sfun the individual o ccurrences of f in the SCS-lay er o f T b y individual o ccurrences of the s ym b ols f , f ′ , f ′′ , . . . , resp ectively , and intro ducing deﬁning rules for f ′ , f ′′ , . . . in the SFS-part analogous to the deﬁning rules for f that ar e a lready present there, an SCS T ′ is obtained that satisﬁes the restr iction, and for whic h it is easy to establish that it holds: do T ( M 0 ) = do T ′ ( M 0 ), [ M 0 ] F = [ M 0 ] F ′ , and henc e [ [[ M 0 ] F ] ] = [ [[ M 0 ] F ′ ] ], where F ′ = { γ f } f ∈ Σ ′ sfun is the family of gates ( Σ ′ sfun is the stream function signature of T ′ ) b y adding to F , for all symbo ls f ∈ Σ sfun , the corres p onding copies γ f ′ , γ f ′′ , . . . o f γ f . Note that we also hav e, for all f ′ ∈ Σ ′ sfun , [ [ γ f ′ ] ] = do T ( f ′ ). “ ≤ ”: W e only have to consider the ca se in which for k := [ [[ M 0 ] F ] ] it ho lds k < ∞ . Hence we assume that k < ∞ . Then ther e exists a ﬁnite rewrite s equence ρ : [ M 0 ] F ։ P • k ( p ′ ) . Then with p := [ M 0 ] F it follows that p ′ has the sa me ‘g ate structure’ as p , but po ssibly the io-sequences in the gates hav e changed, a nd there are likely to appea r many p ebbles queuing inside p ′ . By the deﬁnition of k it follows that [ [ p ′ ] ] = 0. Without loss of genera lit y , we ass ume that during ρ a ll extractable pe bbles have b een extracted from ‘inside’ the gates in p ′ , tha t is, there is no o ccurr e nce of a r edex in p ′ with re spect to the p ebbleﬂow rule min ( • ( p 1 ) , • ( p 2 )) → • ( min ( p 1 , p 2 )). Note that the o nly o ccurrence s of the symbol min in terms of ρ are pa r t of ga tes. (If this would not b e the case fo r ρ , then we would extend ρ by ﬁnitely many → P -steps to a r ewrite sequence ρ ′ which ha s this pr op e r t y , and base the further a rgumentation on ρ ′ .) Due to the ass umption that function sym b ols in Σ sfun only o ccur o nce in the SCS-lay er, to every f ∈ Σ sfun there cor resp onds in p (via the net translation in Def. 5.16) a unique gate γ f = [ f ]. W e now monitor, for every ga te γ f in p , the co nsumption/pro duction b ehaviour dur ing ρ . Let f ∈ Σ sfun . Suppo se that dur ing ρ , γ f consumes n 1 , . . . , n ar s ( f ) pebbles from its input comp onents, resp ectively , a nd that in total it pro duces n p ebbles. B y the assumption 74 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks that no extractable p ebbles ar e stuck inside γ f at the end of ρ , in par ticu- lar, [ [ γ f ] ]( n 1 , . . . , n ar s ( f ) ) = n follows. Mo reov er, by the a ssumption that the data-oblivio us lower bound do T ( f ) o f a sy m b ol f ∈ Σ sfun coincides with the pro duction function o f the gate γ f , we g et do T ( f )( n 1 , . . . , n ar s ( f ) ) = n . As a consequence, ther e is a rewrite sequence: ˜ ρ f : f ( • n 1 : σ 1 , . . . , • n ar s ( f ) : σ ar s ( f ) , • , . . . , • | {z } ar d ( f ) ) ։ A T , G • n : t in an ARS A T , G such that ˜ ρ f can b e extended to an outermost-fair re write sequence that do es not pro duce further pebbles. Then fro m ˜ ρ f we extra ct a rational gate ˜ γ f with the prop erties: (a) ˜ γ f precisely describ es the con- sumption/pro duction behaviour that happ ens during ˜ ρ f , (b) therea fter ˜ γ f contin ues like the gate corresp onding to f in p ′ . W e gather the new gates in the new fa mily of g ates: ˜ F := { ˜ γ f } f ∈ Σ sfun . Note that, for all f ∈ Σ sfun , [ [ γ f ] ] ≤ [ [ ˜ γ f ] ] holds . Nevertheless, it can be shown that there is actually also a rewrite se q uence: ˜ ρ : [ M 0 ] ˜ F ։ P • k ( ˜ p ′ ) . such that [ [ ˜ p ′ ] ] = 0. Using this p ebbleﬂow r e w r ite sequence ˜ ρ as well as the rewrite sequences ˜ ρ f on the abs tracted versions o f T , it is p ossible to obtain a rewr ite sequence in an ARS A T , G of the form: ¯ ρ : T T ( M 0 ) ։ ։ mω A T , G • k : s , where m is the le ng th of the r eduction seq ue nc e ˜ ρ , that can b e extended in A T , G to a rewrite seq uence ¯ ¯ ρ which consists of outer most-fair sequences of rewrite sequence s of length ω in which r e s pectively , for some g ∈ Σ sfun , g -r edexes ar e reduced uniformously in all unfoldings, and s uc h that further- more ¯ ¯ ρ do es not contain terms with a preﬁx of more than k p ebbles. The existence o f the rewrite sequence ¯ ¯ ρ on M 0 in an ARS A T , G demonstrates that do T ( M 0 ) ≤ k . “ ≥ ”: This dir e ction o f the pro o f is a c o nsequence o f three s tatemen ts tha t a re men tioned b elow. First, it holds that: F or all p ∈ P : [ [ T ( p )] ] = [ [ p ] ] , (7) where T ( p ) denotes the inﬁnite unfolding of p into a p ossibly inﬁnite ‘p eb- bleﬂow tree’. Second: F or all stream ter ms s in T : Π T ( s ) = Π T ( T T ( s )) , (8) where by s ∈ T er ( Σ ) we mean the result of inﬁnitely unfolding all stream constants in s . Finally , third, it holds that: F or all strea m terms s in T : do T ( T T ( s )) ≥ [ [ T ([ s ] F )] ] . (9) Putting (7), (8), and (9) tog e ther , we obtain: do T ( M 0 ) = do T ( T T ( M 0 )) ≥ [ [ T ([ M 0 ] F )] ] = [ [[ M 0 ] F ] ] , which s hows “ ≥ ”. Data-Oblivious Stream Pro ductivity 75 This co ncludes our sketc h of the pro of of Lem. 5.1 7 (i). D.2 Pro of of Lemma 5 .18 Let T be a stream sp eciﬁcatio n, and let F = { γ f } f ∈ Σ sfun be a fa mily of gates such that, for all f ∈ Σ sfun , the arity o f γ f equals the strea m ar it y o f f . Supp ose that one of the following sta temen ts holds: (a) [ [ γ f ] ] ≤ do T ( f ) for all f ∈ Σ sfun ; (b) [ [ γ f ] ] ≥ do T ( f ) for all f ∈ Σ sfun ; (c) do T ( f ) ≤ [ [ γ f ] ] for all f ∈ Σ sfun ; (d) do T ( f ) ≥ [ [ γ f ] ] for all f ∈ Σ sfun . In this s ection we show that then, for a ll M ∈ Σ sc on , the co rresp onding one of the following statements holds: (a) [ [[ M ] F ] ] ≤ do T ( M ) ; (b) [ [[ M ] F ] ] ≥ do T ( M ) ; (c) do T ( M ) ≤ [ [[ M ] F ] ] ; (d) do T ( M ) ≥ [ [[ M ] F ] ] . Multiple Numbe red Co n texts Deﬁnition D.1. L et T = h Σ , R i b e a str e am sp e ciﬁc ation, and k , l ∈ N . A (multiple numb er e d) str eam co n tex t C [ ] over str e am holes 2 1 , . . . , 2 k , and data holes 2 · 1 , . . . , 2 · ℓ is a (str e am) term C [ ] ∈ T er ∞ ( Σ ⊎ { 2 1 , . . . , 2 k } ⊎ { 2 · 1 , . . . , 2 · ℓ } , X ) S wher e the 2 i :: S and 2 · i :: D ar e distinct c onstant symb ols not o c curring in Σ . We denote by C [ t 1 , . . . , t k ; u 1 , . . . , u ℓ ] the r esu lt of r eplacing the o c curre n c es of 2 i by s i and 2 · j by t j in C [ ] T , r esp e ctively. R emark D.2. Each of the hole symbols 2 1 , . . . , 2 k , and 2 · 1 , . . . , 2 · ℓ may have m ultiple o ccurrences in a s tream co n text C [ ] o ver 2 1 , . . . , 2 k and 2 · 1 , . . . , 2 · ℓ , and this includes the case that it do es not o ccur in C [ ] at a ll. Example D.3. Let T b e a stream sp eciﬁcation with data constants b and c , a unary strea m function symbol f , and a bina ry stream function symbol g . Then the expression b : f ( c : g ( 2 · 1 : 2 1 , g ( 2 2 , g ( 2 1 , 2 3 )))) is a stream context ov er 2 1 , 2 2 , 2 3 and 2 · 1 . Also f ( b : 2 3 ) is a str eam co n text ov er 2 1 , 2 2 , 2 3 and 2 · 1 . But b : 2 1 : σ is no t a str eam context over 2 1 , b ecaus e the symbol 2 1 o ccurs at a ‘data po sition’ (this is excluded by ma n y- sortedness, : :: D → S → S and 2 1 :: S ). W e extend the d-o pr o duction function of stream function symbols f ∈ Σ sfun , Def. 3.5, to multiple num b ered contexts. 76 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks Deﬁnition D.4. Let T = h Σ , R i b e a str eam sp eciﬁcation, and C [ ] a mu l- tiple num b ered stream cont ext over stream holes 2 1 , . . . , 2 k , a nd data holes 2 · 1 , . . . , 2 · ℓ . The d-o pr o duction ra n ge do T ( C [ ] ) : N k → N of t he c ontext C [ ] is: do T ( C [ ] )( n 1 , . . . , n k ) := do T ( L C [ ] M (( • n 1 : σ ) , . . . , ( • n k : σ )) ) , where • m : σ := m time s z }| { • : . . . : • : σ . The d-o lower and u pp er b ounds on the pr o duction of g are deﬁned by do T ( C [ ] ) : = inf ( do T ( C [ ] )) and do T ( C [ ] ) := sup( do T ( C [ ] )), resp ectively . Note that it is indeed a n extension o f Def. 3 .5, for every g ∈ Σ sfun we hav e do T ( g ) = do T ( g ( 2 1 , . . . , 2 ar s ( g ) , 2 · 1 , . . . , 2 · ar d ( g ) )) . The following lemma states that data- oblivious lower and upp e r b ounds a re reachable. Lemma D.5. L et T = h Σ , R i b e a str e am sp e ciﬁc ation, and s ∈ T er ∞ ( Σ ) S . Colouring T erms, T racing Data Fl ow F or the purp ose of trac ing the ﬂow of data-elements during rewrite sequences , we introduce auxiliary ‘tracker symbols’ and partitio n ter ms in to ‘co loured’ multip le num b ered co n texts. Usua l rewriting will then b e allowed o nly within contexts. Rewrite s teps cross ing the b order betw een contexts are prohibited. In order to cir c umven t the loss of po ssible reduction steps, we introduce r ules for exchanging p ebbles b etw een neighboring contexts (rec olouring of the p ebbles). W e int r o duce aux ilia ry , unary function symbols ⋄ c and ⋄ · c in the pe bbleﬂow rewrite s ystem as w ell as in the str eam sp eciﬁcation T . The symbols ⋄ c mark the top of a context o f colo ur c a nd at the sa me time terminate the context ab ov e it. The symbols ⋄ · c only close, but do not o p en a context. W e call them t ra cker symb ols in the sequel, since they are also used to track p ebble/data exchange betw een neighbo r ing cont ex ts. Therefore we enrich the p ebbleﬂow rewrite system with the following rules ⋄ c ( • ( p )) → • ( ⋄ c ( p )) ⋄ · c ( • ( p )) → • ( ⋄ · c ( p )) (10) and the stream sp eciﬁcation T with rules ⋄ c ( t : σ ) → t : ⋄ c ( σ ) ⋄ · p ( t : σ ) → t : ⋄ · c ( σ ) (11) for every function symbol ⋄ c and ⋄ · c . W e deﬁne ⋄ c , ⋄ · c :: stream → stream , that is, the tra cker symbols ca nno t b e pla ced at data po sitions. Deﬁnition D.6. F or terms s, t we say that t is an enrichment of s with tr acker symb ols , denoted s ∠ t , if s ca n b e obtained from t b y removing tracker s ym b ols, that is, t → ∗ s via r ules ⋄ c ( u ) → u a nd ⋄ · c ( u ) → u . Data-Oblivious Stream Pro ductivity 77 Lemma D.7. Every p ebbleﬂow r e duction c an b e lifte d to en richte d terms: ∀ s, s ′ ∈ P , s ։ P s ′ , s ∠ t. ∃ t ′ . t ։ P ⋄ t ′ ∧ s ′ ∠ t ′ and str e am sp e ciﬁc ations T Every p ebbleﬂow rewr ite s equence a nd every stream sp eciﬁca tion rewrite sequence c an be lifted to enriched terms, i.e. ∀ s ′ և s ∠ t. ∃ t ′ . t ; ∗ t ′ ∧ s ′ ∠ t ′ where ; = → ∪ → (10) (or ; = → ∪ → (11) ). Sometimes the tra c ker symbols are in the wa y of rule application, but, bec a use of the spec ia l structure of p ebbleﬂow (resp. SCS) rules, it is alwa ys p ossible to move them awa y using rules (10 ) (resp. (11)). Now we s ket ch the pr o of for the inequalities “ ≤ ” (i) and “ ≥ ” (ii). “ ≤ ”: W e need to show [ [[ M ] F ] ] ≤ Π T ( M ). F or the pro of of (i) we map a given ﬁ- nite p ebbleﬂow r ewrite seq ue nce on [ M ] F pro ducing n p ebbles to a T rewrite sequence o n the inﬁnite unfolding of M that pro duces a t least n data ele- men ts. (F or (ii) the pro of pro ce e ds the other wa y a rround.) Thereby no t every rewrite step will be mapp ed, but we trace ‘essential’ steps and map them to a man y- step (possibly inﬁnite) reductio n on ‘the other side’. F or this pur pos e we partition the p ebbleﬂow term [ M ] F as well a s inﬁnite unfolding of M into co loured co n texts (using the same c olours for the net and the unfolding of M ). The pa rtitioning is arr anged in such a wa y tha t contexts o f the s ame colour ha ve the same pro duction function. In the sequel we will refer to this pro per t y as the c ontext pr o duction pr op erty (CPP ). Usual r ewriting will then b e allow ed only within contexts. Rewrite s teps crossing the bo rder b et ween co n tex ts are prohibited. In order to circumv ent the los s of p ossible reduction s teps , we in tro duce additional rules for e x - changing p ebbles b et ween neighboring contexts (reco louring o f the p ebbles). The construction o f the mapp ed reduction sequence will b e driven b y the g o al to uphold CPP . It can b e shown p ebbleﬂow rewr iting and SCS r ewriting do es not change the pro duction function of a context. Hence in order to maintain CPP it is not necessa ry to map internal steps o cur ring within co n tex ts . The ‘essential’ s teps are the exchange steps b etw een the contexts. W e us e C P c to refer to the context of co lour c in the p ebbleﬂow net [ M ] F . Likewise we use C T c for the co n tex t of colour c in the inﬁnite unfolding of the stream co ns tan t M . Although there may be multiple, typically inﬁnitely many , o ccur rences of a context with c o lour c o n the ter m level, contexts with the same colour are syntactically equal. Initially all contexts are of one of the following (simple) shap es: Clearly [ [ ] ] C P c = [ [ ] ] C T c for case 1. An impo rtant step in the pro of is to show that the tra nslation of functions s ym b ols into rational gates pre serves the quantitativ e pro ductio n be haviour, yielding [ [ ] ] C P c = [ [ ] ] C T c for cas e 2 . The pr o of now co n tinues as follows. Every p ebble exchange step in the p eb- bleﬂow net is mapp ed to a n inﬁnite rewrite sequence on the co rresp onding coloured SCS term. F or the deﬁnition of these corr esp onding re w r ite s teps we apply the assumption [ [ ] ] C P c = [ [ ] ] C T c guaranteeing that a data element 78 J¨ org En drullis, Clemens Grabmay er, and D imitri Hendriks case C P c C T c 1 • ( 2 1 ) d : 2 1 2 min ( σ f , 1 ( 2 1 ) , . . . , σ f ,r ( 2 r )) f ( 2 1 , . . . , 2 r ) Fig. 15: Initial context conﬁgurations can also b e extracted from the cont e x t on SCS term le vel, and that after the step we hav e again [ [ ] ] C P c = [ [ ] ] C T c for all colours c . In this way , we deﬁne a rewrite sequence ρ T in T by induction on the num b er of re w r ite steps of the given p ebbleﬂow reduction ρ such that ρ T and ρ pro duce the s ame amount of data elements and p ebbles, resp ectively . “ ≥ ”: Here w e need to show [ [[ M ] F ] ] ≥ Π T ( M ). T he pr oo f of this inequalit y pro ceeds similarly to that of “ ≤ ” . Let ρ T in T b e a r eduction se quence in T . In o rder to ma p ρ T back to a p ebbleﬂow rewrite sequence ρ it is crucial that the co n texts of one colour are c hanged synchronously and in the sa me w ay . It cannot b e assumed that ρ T po sesses this pr ope r t y . F or this purp ose ρ T is transformed into a sequence ρ ′ T of complete dev elopments. This is always p ossible since c o mplete dev e lo pmen ts in orthogo nal TRSs a re a coﬁnal rewrite str ategy . F or ma pping ρ ′ T to a rewrite s equence ρ with the same pr o ductio n o n the p ebbleﬂow net, we can then pro ceed by induction on the length o f ρ ′ T , in co n verse direction to the a rgument for “ ≤ ”, ab ov e.

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