A process calculus with finitary comprehended terms

A Pro cess Calculus with Finitary Comprehended T erms J.A. Bergstra and C.A. Middelburg Informatics Institute, F acult y of Science, U niv ersit y of Amsterd a m, Science P ark 904, 1098 XH Amsterdam, the Netherlands J.A.Bergst ra@uva.nl , C.A.Mi ddelburg@ uva.nl Abstract. W e introduce the n oti on of an ACP process a lgebra and the notion of a meadow enric hed ACP p rocess algebra. The former notion originates from the mo dels of the ax i om sy stem A CP. The latter notio n is a simple generalization of the former notion to pro cesses in whic h data are inv olv ed, the mathematical stru cture of data b eing a meadow. More- o ver, for all associative op erators from t h e sig nature of meadow enric hed ACP pro cess a lgebras that are n ot of an auxiliary n a ture, w e i ntroduce v aria ble-binding op era tors as generalizations. These v ariable-binding op- erators, whic h give ris e to comprehended terms, hav e the prop ert y that they can alwa y s be eliminated. Thus, w e obtain a process calculus whose terms can b e interpreted in all meado w enriched ACP pro cess algebras. Use of t he v ariable-binding operators can hav e a ma jor impact on the size of terms. Keywo rds: ACP process algebra, m eadow en ri c hed ACP process algebra, v aria ble-binding op erator, comprehended term, pro cess calculus. 1998 ACM Computin g Classificatio n: D.1.3, F.1.2, F.4.1. 1 In tro duction In many for ma lisms pr o posed for the description and analysis of pro cesses in which data are in v olved, algebraic s p ecifications of the da ta types concer ned hav e to b e given ov er and ov er again. This is a lso the case with the principa l A CP-based formalisms prop osed for the descr iption a nd analysis of pro cesses in which data are in volv ed, to wit µ CRL [1 6 ,17] and P SF [23]. There is a mismatch betw een the pro cess specification pa rt and the data specifica tio n part o f these formalisms. Firstly , there is a c hoice of one built-in t ype o f pr ocesses, whereas there is a choice o f all t yp e s of da ta that ca n b e sp ecified algebr aically . Secondly , the semantics of the data sp ecification par t is its initial algebra in the c a se of PSF and its clas s o f minimal Boolea n pres erving algebr a s in the cas e of µ CRL, whereas the semantics of the pro cess spec ifica tion pa r t is a mo de l based on transition systems and bisim ulation equiv alence. Stic king to this mismatc h, no lasting axio matizations in the st yle of ACP has emerged for pro cess algebras that have to do with pro cesses in w hich da ta a re inv olved. Our first main ob jective is to obtain a lasting axio matization in the st yle of ACP for pro cess algebras that have to do with pro cesses in which data are inv olv ed. T o achiev e this ob jective, we first intro duce the notio n of an ACP pro cess alg ebra a nd then the notion o f a mea dow e nric hed ACP pro cess algebra . A CP pr ocess algebras are essentially mo dels of the axio m system A CP. Meadow e nr ic hed A CP pr ocess algebr as are data enr ic hed A CP pr ocess alge- bras in which the mathema tica l structure for da t a is a meadow. Mea do ws were defined for the first time in [13]. The prime e x ample o f a meadow is the r ational nu m ber field with the m ultiplicative inv erse operatio n made total b y imp osing that the multiplicativ e inv erse of zero is zer o. Although the notion of a meadow enriched ACP pro cess algebra is a simple genera lization o f the notion of an ACP pro cess alg ebra, it is an in teresting one: there is a m ultitude of finite and infinite meadows and meadows obviate the need for Bo olean v alues and o perations on data that yield Bo olean v alues to deal with co nditions o n data. In the work on A CP, the emphasis has always been on axiom systems. In this pap er, we put the empha s is on algebras . That is , ACP pro cess algebr as ar e lo ok ed upo n in the sa me wa y as groups , rings, fie lds , e tc. are lo oked up on in uni- versal algebra (see e.g. [14]). The s e t of equations that ar e taken to characterize A CP pro cess algebra s is a revision of the axiom system ACP. The revisio n is pr i- marily a matter of streamlining. How ev er, it also involv es a minor g eneralization that allows for the generalization to meadow enriched ACP pr ocess a lg ebras to pro ceed smo othly . In µ CRL and PSF, we find v ar iable-binding op erators gener alizing asso ciative op erators o f ACP . Our seco nd main ob jective is to deter mine to what extent such v ar iable-binding oper ators fit in with mea do w enric hed A CP pro cess algebras. T o ac hiev e this ob jective, we in troduce, for all asso ciativ e op erators from th e signature o f meadow enr ic hed ACP pr ocess a lgebras that a re not o f an auxiliary nature, v ariable-binding op erators a s g eneralizations. These v a riable-binding op erators, which give rise to co mprehended terms, hav e the prop ert y that they ca n a lw a ys be e limina ted. That is , for each co mpre- hended term, we c an derive from axioms c o ncerning the v a riable-binding op era- tors tha t the comprehended term is equal to a term ov er the sig nature of meadow enriched ACP pro cess algebr as. T ho se a xioms ar e axioms of a ca lculus b ecause the distinction betw een free and b ound v ar iables is essential in der iv ations. The terms of this process calculus are interpreted in meado w enr ic hed ACP pro cess algebras . F ull elimination of all v ariable-binding o p erators o c c urring in a compre - hended ter m can lead to a combinatorial explosio n. W e show that a combinatorial explosion can b e preven ted if v ariable-binding op erators tha t bind v ariables with a tw o-v alued rang e ar e still permitted in the res ultin g term. W e also show that in the latter ca s e the siz e of the resulting term ca n b e further reduced if we a dd an identit y element fo r sequential comp osition to meadow enriched ACP pro cess algebras . Moreover, we demonstra t e that there is an alternative to in tro ducing v ar iable-binding operato rs for several asso ciativ e o perators on pro cesses if w e add a sort of pro cess sequence s and suitable o perator s on proce s s seq ue nc e s to meadow enriched ACP pro cess algebr as. 2 F or rea da bilit y , it is impre c isely s aid a bov e that the mathematical str uc tur e for data in meadow enr ic hed ACP pr o cess a lgebras is a meadow. It is actually a signed meadow, i.e. a meadow expanded with a signum op eration. In the presence of a sig n um op eration, the o r dering o n the elements of a meadow that corres p onds with the usual order ing o n the elements of a field b ecomes definable. This pap er is organized as follows. Fir st, we give a brie f summary o f signed meadows (Section 2). Next, w e intro duce the notion of an ACP pr ocess alge- bra (Section 3 ) and the notion of a meadow enriched A CP pro cess algebra (Section 4). After that, we asso ciate a c a lculus with meadow e nr ic hed ACP pro cess algebras (Section 5) and define the interpretation of terms of this cal- culus in meadow enriched ACP pro cess a lg ebras (Sectio n 6). F ollowing this, we inv estigate the consequences o f elimination of v ariable-binding operato r s from comprehended terms on the size of the r esulting ter ms (Section 7). Then, we inv estigate the effects of adding a n identit y elemen t for sequent ial composition to ACP pro cess a lgebras (Section 8) and the effects of adding pro cess sequences to ACP pro cess alg ebras (Section 9). Fina lly , we make so me concluding r emarks (Section 10). This pap er conso lidates materia l fro m [9,1 0 ]. 2 Signed Meadows In this pap er, the mathematical structure for data is a signed mea dow. In this section, we g iv e a brief summary of sig ned meadows. A mea do w is a field with the multiplicative inv erse op eration made total by impo sing that the multiplicativ e in verse o f zero is zero. A sig ned mea do w is a meadow expanded with a sig n um op eration. Mea do ws were defined for the first time in [13] and w ere inv estigated in e.g. [5,6,1 1 ]. The expansion of mea do ws with a signum op eration or iginates fro m [5]. The signature of meadows is the same as the signature of fields. It is a one-sor t ed signa t ure. W e make the s ingle sort explicit b ecause w e will extend this signa t ure to a tw o-sorted sig na ture in Sectio n 4. The signatur e of meadows consists of the sor t Q of quantities and the following constants and op erators: – the co nstan ts 0 : → Q and 1 : → Q ; – the binar y addition op erator + : Q × Q → Q ; – the binar y mu ltip lic ation op erator · : Q × Q → Q ; – the unar y additive inverse op erator − : Q → Q ; – the unar y mu ltip lic ative inverse o perator − 1 : Q → Q . W e a ssume that there is a coun tably infinit e set U o f v ariables o f sor t Q , which contains u , v and w , with and without subscripts. T erms are built as usual. W e use infix no ta tion for the binary op erators + and · , pr efix notation for the unary op erator − , and p ostfix notation for the unary op erator − 1 . W e use the usua l precede nc e conv en tion to reduce the need for pa ren theses. W e int ro duce s ubt raction and division as a bbreviations: p − q abbreviates p + ( − q ) and p/q abbr e viates p · q − 1 . F or each non-nega tiv e natura l num ber n , we wr ite n 3 T able 1. A xioms for meadow s ( u + v ) + w = u + ( v + w ) u + v = v + u u + 0 = u u + ( − u ) = 0 ( u · v ) · w = u · ( v · w ) u · v = v · u u · 1 = u u · ( v + w ) = u · v + u · w ( u − 1 ) − 1 = u u · ( u · u − 1 ) = u for the numeral fo r n . That is, the term n is defined by induction on n as follows: 0 = 0 and n + 1 = n + 1 . W e a lso use the no tation p n for exponentiation with a natural num ber as exp onen t. F or ea c h term p ov er the signa ture o f meadows, the ter m p n is defined b y induction on n as follows: p 0 = 1 and p n +1 = p n · p . The consta nts a nd op erators from the signature o f meadows are ado pted fro m rational ar ithmetic, which gives an appropriate intuition ab out these constant s and op erators. A me ado w is an algebra with the signa tu re o f meado ws that satis fies the equations given in T able 1. Thus, a meadow is a commutative ring with identit y equipp e d w ith a multiplicative inv erse op e ration − 1 satisfying the reflexivity equation ( u − 1 ) − 1 = u and the r estricted inv erse equation u · ( u · u − 1 ) = u . F rom the eq uations given in T able 1, the equa tion 0 − 1 = 0 is deriv able (see [13]). A non- trivi al me ado w is a meadow that satisfies the sep ar ation axiom 0 6 = 1 ; and a c anc el lation me adow is a meadow that satisfies the c anc el lation axiom u 6 = 0 ∧ u · v = u · w ⇒ v = w , or e quiv alent ly , the gener al inverse law u 6 = 0 ⇒ u · u − 1 = 1 . Impo rtan t pr operties of non-trivial cancellation meado ws are u/ u = 0 ⇔ u = 0 and u/ u = 1 ⇔ u 6 = 0. Hencefor th, we will write p ⊳ r ⊲ q for (1 − r /r ) · p + ( r/ r ) · q . F or non-triv ial ca ncellation meadows, p ⊳ r ⊲ q can be r ead as follows: if r equals 0 then p else q . Each field with the multiplicativ e inv erse op eration made tota l by imp o sing that the mult iplicative inv erse of zero is zero is a non-trivial meadow. The prime example of a non-trivial ca ncellation meadow is the r ational num ber field with the multiplicativ e inverse op eration ma de total by imp osing that the multiplica- tive inv erse o f zero is zero. A signe d me ado w is a meado w expanded with a unary signum ope ration s satisfying the equations giv en in T able 2. In com bination with the cance lla tion axiom, the last equation in this table is equiv alent to the conditio na l equatio n 4 T able 2. A dditional axioms for signum op eration s ( u/u ) = u/u s (1 − u/u ) = 1 − u/u s ( − 1) = − 1 s ( u − 1 ) = s ( u ) s ( u · v ) = s ( u ) · s ( v ) (1 − s ( u ) − s ( v ) s ( u ) − s ( v ) ) · ( s ( u + v ) − s ( u )) = 0 s ( u ) = s ( v ) ⇒ s ( u + v ) = s ( u ). In signed meadows, the predicates < and > a re defined a s follows: u < v ⇔ 1 + s ( u − v ) = 0 , u > v ⇔ 1 − s ( u − v ) = 0 . In [5], it is shown that the equational theories of signed meadows and signed cancellation meadows a re ide ntical. 3 A CP Pro cess Algebras In this s ection, we introduce the notion of an ACP pro cess algebra . This notion originates f rom the mode ls of ACP, an axiom system that w as first presen ted in [7]. A comprehens iv e introduction to ACP can b e found in [3,15]. It is a ssumed that a fix ed but a r bitrary set A o f atomic action names , with δ / ∈ A , has b een given. The signature of ACP pro cess algebra s is a o ne-sorted signatur e. W e make the single sort explic it b ecause we will extend this signa ture to a tw o-sorted signa ture in Sectio n 4. The signatur e of ACP pro cess alg ebras consists o f the sort P of pr o c esses and the following co ns t ants, o perators , a nd pr edicate s ym bols: – the de ad lo ck constant δ : → P ; – for ea ch e ∈ A , the atomic action co nstan t e : → P ; – the binar y alternative c omp osition op erator + : P × P → P ; – the binar y se quential c omp osition oper ator · : P × P → P ; – the binar y p ar al lel c omp osition op erator k : P × P → P ; – the binar y left mer ge oper ator ⌊ ⌊ : P × P → P ; – the binar y c ommunic ation mer ge oper ator | : P × P → P ; – for ea ch H ⊆ A , the unary enc apsulation o p erator ∂ H : P → P ; – the unar y atomic action predica te symbo l A : P . W e a ssume that there is a countably infinite set X of v ariables of sort P , which contains x , y and z , with and without subscr ipts. T erms are built a s usua l. W e use infix notation for the bina ry op erators. W e use the fo llowing precedence conv en tions to reduce the need for par en theses: the oper ator + binds weak er than all other binary op erators and the op erator · binds stro nger than all o ther binary op erators. Let P a nd Q b e clo sed terms o f so r t P . Intuitiv ely , the consta n ts, op erators and predica te symbo ls intro duced ab o v e ca n b e expla ined as follows: 5 – δ is not capable of doing anything; – e is only c a pable o f performing atomic a ction e and next terminating suc- cessfully; – P + Q b eha v es either as P or as Q , but not b oth; – P · Q fir st b eha ves a s P and on succes sful terminatio n of P it next b eha v es as Q ; – P k Q b eha v es as the pr o cess that pro ceeds with P and Q in pa rallel; – P ⌊ ⌊ Q b eha v es the same as P k Q , except that it star ts with p erforming an atomic a ction o f P ; – P | Q behaves the same a s P k Q , except that it starts with performing an atomic a ction o f P and a n atomic action of Q s ync hronously; – ∂ H ( P ) b ehav es the same a s P , e xcept that atomic actions fr o m H are blo c k ed; – A ( P ) ho lds if P is an atomic action. The oper a tors ⌊ ⌊ and | a re o f an auxiliary nature. They are needed for the a x- iomatization of ACP pro cess alg ebras. The predicate symbo l A is used to distinguish a tomic actions from other pro cesses. This predicate sy m bol, whic h do es no t o ccur in the axiom sy s tem A CP, obviates the nee d to have a consta n t for each atomic actio n. An alterna tive wa y to distinguish ato mic a ctions fro m other pr ocesses is to hav e a subs o rt A of the sort P . W e hav e not c hosen this alternativ e w a y because it co mplicates matters co nsiderably . Mor e over, we prefer to keep close to e lemen tary alg ebraic sp ecification (see e.g. [12]). By the notational co n v en tion intro duced b elo w, we seldom have to use the predicate symbol A e xplicitly . In eq uations between terms of sor t P , we will use a notational conv ent ion which requir es the following assumption: ther e is a countably infinite set X ′ ⊆ X that contains a , b and c , with a nd without subscripts, but do es not contain x , y and z , with and witho ut subscr ipts. Let φ b e a n equation b et w een terms of so rt P , a nd let { a 1 , . . . , a n } b e the set of all v a riables fr om X ′ that o ccur in φ . Then we write φ for A ( x 1 ) ∧ . . . ∧ A ( x n ) ⇒ φ ′ , where φ ′ is φ with, for all i ∈ [1 , n ], all o ccurrences of a i replaced by x i , and x 1 , . . . , x n are v ariables from X that do not o ccur in φ . An ACP pr o c ess algebr a is an algebr a with the signatur e of A CP pro cess algebras tha t s atisfies the formulas given in T able 3. Three formulas in this table are actually schemas of form ulas: e is a syntactic v ariable whic h stands for an arbitrar y constant of so rt P (i.e. an atomic a ction co ns tan t or the deadlo c k constant). A side condition is added to tw o schemas to r estrict the co ns tan ts for which the s yn tactic v ariable stands. Because the notational con ven tion introduced above is used, the four equa- tions in T able 3 that are a c t ually conditional equations lo ok the same as their counterpart in the axiom system ACP. It ha ppens that thes e c o nditional equa- tions allow for the g eneralization to meadow enriched ACP pro cess alg ebras to pro ceed smo othly . Apart from this, the set of for m ulas given in T able 3 differs from the ax io m system ACP o n thr ee p oin ts. Firstly , the equa tions x | y = y | x , ( x | y ) | z = x | ( y | z ), and δ | x = δ ha v e b een a dded. In the axiom sys tem ACP, 6 T able 3. A xioms for ACP pro cess algebras x + y = y + x ( x + y ) + z = x + ( y + z ) x + x = x ( x + y ) · z = x · z + y · z ( x · y ) · z = x · ( y · z ) x + δ = x δ · x = δ ∂ H ( e ) = e if e / ∈ H ∂ H ( e ) = δ if e ∈ H ∂ H ( x + y ) = ∂ H ( x ) + ∂ H ( y ) ∂ H ( x · y ) = ∂ H ( x ) · ∂ H ( y ) x k y = ( x ⌊ ⌊ y + y ⌊ ⌊ x ) + x | y a ⌊ ⌊ x = a · x a · x ⌊ ⌊ y = a · ( x k y ) ( x + y ) ⌊ ⌊ z = x ⌊ ⌊ z + y ⌊ ⌊ z a | b · x = ( a | b ) · x a · x | b · y = ( a | b ) · ( x k y ) ( x + y ) | z = x | z + y | z x | y = y | x ( x | y ) | z = x | ( y | z ) δ | x = δ A ( e ) A ( x ) ∧ A ( y ) ⇒ A ( x | y ) all closed substitution instances of these equa tions a re deriv able. Secondly , the equations a · x | b = ( a | b ) · x and x | ( y + z ) = x | y + x | z ha v e b een r emo v ed. These equations can b e derived using the added equation x | y = y | x . Thirdly , the formulas A ( e ) and A ( x ) ∧ A ( y ) ⇒ A ( x | y ) have be e n added. They expr ess that the pr ocesses deno ted by constants of so r t P are atomic actions and that the pro cesses that result from the commun ication merge o f tw o atomic actions are atomic actions. This does not exclude that ther e are additiona l atomic actions , which is imp ossible in the case of ACP . F or ea ch mo del of the axiom system ACP given in [3], its expa ns ion with the appropria te interpretation of the atomic a ction predicate symbol A is an A CP pro cess a lgebra. Not all pro cesses in an ACP process algebra ha ve to be interpretations of closed terms, even if all a tomic actions are interpretations of close d terms. The pro cesses concerned may b e s olutions of sets of rec ur sion equa tions. It is rec o m- mendable to restr ict the a tt en tion to ACP pr ocess algebr as sa tisfying additiona l axioms by which sets of recursion equa t ions that fulfil a guardedness conditio n hav e unique solutions. F or a comprehensive tr eatmen t of this issue, the reader is referred to [3]. 4 Meado w Enric hed A CP Pro cess Algebras In this section, we intro duce the notion of a meadow enriched A CP pro cess algebra. T his notion is a simple gener alization o f the notion of an ACP pro cess algebra introduced in Sec tio n 3 to pro cesses in whic h data ar e inv olved. The elements of a signed meadow are taken as data. The sig na ture of meadow enriched A CP pro cess algebras is a tw o-sorted signature. It consists of the sorts, co nstan ts and op erators from the signature s 7 of A CP pro cess algebra s and signed meadows a nd in addition the following op erators: – for eac h n ∈ N and e ∈ A , the n -ary data hand ling atomic actio n oper ator e : Q × · · · × Q | {z } n t imes → P ; – the binar y gu ar de d c ommand op erator : → : Q × P → P . W e tak e the v aria bles in U for the v ariables of sort Q and the v ariables in X for the v ariables of sort P . W e assume that the sets U a nd X ar e disjoin t. T erms are built as usual for a ma ny-sorted signature (see e.g. [28,31]). W e use the s ame notational co nven tions as b efore. In addition, we use infix no t ation for the bina ry o perator : → . Let p 1 , . . . , p n and p be close d terms of so r t Q and P be a closed term of so rt P . Intuitiv ely , the additional op erators can b e explained as fo llows: – e ( p 1 , . . . , p n ) is only capable of per fo rming data handling ato mic action e ( p 1 , . . . , p n ) a nd next terminating success fully; – p : → P behaves as the pr ocess P if p equals 0 a nd is not capable of doing anything o therwise. The different gua rded comma nd o perators tha t hav e b een prop osed b efore in the s e t ting of A CP have one thing in common: their first o perand is considered to stand for an element of the domain of a Bo olean algebra (see e .g . [8]). In contrast with those g uarded command o perator s, the fir st o perand of the guarded command o p erator intro duced her e is consider e d to stand for an element of the domain o f a signed meadow. A me adow enriche d ACP pr o c ess algebr a is a n a lgebra with the signatur e of meado w enrich ed ACP pro cess algebr as that satisfies the formulas given in T ables 1 – 4. Like in T able 3, some for m ulas in T able 4 ar e actually sc hemas of formulas: e , e ′ and e ′′ are syntactic v a riables which stand for ar bitrary constants of sor t P different from δ and, in a ddition, n a nd m s t and for arbitr a ry natural nu m ber s. F or meadow enr ic hed ACP pro cess algebras that satisfy the separ ation a xiom and the c a ncellation axiom, the five equa tio ns concerning the gua rded co mma nd op erator on the left-hand side in the upp er half of T able 4 can ea sily b e under - sto od by taking the view th at 0 and 1 represent the Boole a n v alues T and F , resp ectiv ely . In that case, we have that – p/p mo dels the test that yields T if p = 0 and F o therwise; – if b oth p and q are equa l to 0 or 1, then 1 − p mo dels ¬ p , p · q models p ∨ q , and consequently 1 − (1 − p ) · (1 − q ) mo dels p ∧ q . F rom this view, the equations given in the upper ha lf of T a ble 4 differ from the axioms for the most general k ind of guarded command operato r that has bee n prop osed in the setting of ACP (see e.g. [8 ]) on tw o points only . Firstly , the equation u : → x = u/u : → x has b een a dded. This equation formalizes the informal ex planation of the g uarded comma nd given ab o v e. Secondly , the 8 T able 4. Additional axioms for meadow enric hed ACP p rocess algebras 0 : → x = x 1 : → x = δ u : → x = ( u/u ) : → x u : → ( v : → x ) = (1 − (1 − u/u ) · (1 − v /v )) : → x u : → x + v : → x = ( u/u · v /v ) : → x u : → δ = δ u : → ( x + y ) = u : → x + u : → y u : → x · y = ( u : → x ) · y ( u : → x ) ⌊ ⌊ y = u : → ( x ⌊ ⌊ y ) ( u : → x ) | y = u : → ( x | y ) ∂ H ( u : → x ) = u : → ∂ H ( x ) e | e ′ = e ′′ ⇒ e ( u 1 , . . . , u n ) | e ′ ( v 1 , . . . , v n ) = ( u 1 − v 1 ) : → ( · · · : → (( u n − v n ) : → e ′′ ( u 1 , . . . , u n )) · · · ) e | e ′ = δ ⇒ e ( u 1 , . . . , u n ) | e ′ ( v 1 , . . . , v n ) = δ e ( u 1 , . . . , u n ) | e ′ ( v 1 , . . . , v m ) = δ if n 6 = m ∂ H ( e ( u 1 , . . . , u n )) = e ( u 1 , . . . , u n ) if e 6∈ H ∂ H ( e ( u 1 , . . . , u n )) = δ if e ∈ H A ( e ( u 1 , . . . , u n )) equation x | ( u : → y ) = u : → ( x | y ) has been remov ed. This equatio n can b e derived using the equatio n x | y = y | x from T able 3. The equatio ns in T able 4 concerning the co mm unication merge of data ha n- dling atomic a ctions formalize the in tuition that tw o da t a handling atomic ac- tions e ( p 1 , . . . , p n ) a nd e ′ ( q 1 , . . . , q m ) ca n b e p erformed s y nc hronously iff e a nd e ′ can b e per formed sy nchronously and n = m and p 1 = q 1 and . . . and p n = q n . The equa tio ns co nce rning the e ncapsulation of data handling atomic actions agree with the wa y in which the enca psulation of da ta handling atomic actions is dea lt with in µ CRL and PSF. The formula concerning the a tomic action pred- icate simply e x presses that data handling atomic actions a re also a to mic actions . Henceforth, w e will write P ⊳ p ⊲ Q for ( p/p ) : → P + (1 − p/p ) : → Q . F o r meadow enric hed A CP pro cess algebras that satisfy the separation axiom and the cancellation axiom, P ⊳ p ⊲ Q can b e read as follows: if p eq uals 0 then P else Q . F or eac h A CP pro cess algebra A ′ and each signed non-trivial cancellatio n meadow A ′′ , there exists an amalgama tion of A ′ and A ′′ , i.e. a mo del of the axioms for b oth ACP pro cess alg e bras and sig ned non-trivial cancella tion mead- ows whose r estriction to the sig nature of ACP pro cess algebr a s is A ′ and whos e restriction to the signature of signed meadows is A ′′ (b y the amalg amation result ab out expa nsions pres e n ted a s Theorem 6.1 .1 in [1 9], adapted to the many-sorted case). F o r each amalgamation o f an A CP pro cess algebr a with a countably in- finite s e t of atomic actions and a sig ned non-trivial cancellation meadow, its expansion with the appropria t e interpretation of the data handling atomic ac- tion o perator s e and the guarded command op erator : → is a meadow enriched A CP pro cess algebra. 9 In s ubs e quen t sections, w e write Σ mp for the sig na ture o f mea do w enriched A CP pro cess algebras . 5 A Calculus for Meado w Enric hed ACP Pro cess Algebras In this se ction, we asso ciate a calculus with mea do w e nr ic hed ACP pro cess al- gebras. F o r that, w e in troduce, for all asso ciativ e op erators fro m the signature of meadow e nric hed ACP pro cess algebras that are not of an auxiliary nature, v ar iable-binding op erators as generaliza tions. T o build terms of the calculus, called binding terms, b oth the constan ts and oper ators fro m the signa t ure of meadow enriched ACP pro cess algebr as and those v ariable-binding o perator s are av ailable. The sets of binding terms o f sorts Q and P , written BT Q and B T P , resp ec- tively , are inductively defined by the following forma tion rules (where S 1 , . . . , S n and S range ov er the so rts from Σ mp ): – if u ∈ U , then u ∈ B T Q ; – if x ∈ X , then x ∈ B T P ; – if c : → S is a co nstan t from Σ mp , then c ∈ B T S ; – if o : S 1 × · · · × S n → S is an o perator from Σ mp and t 1 ∈ B T S 1 , . . . , t n ∈ B T S n , then o ( t 1 , . . . , t n ) ∈ B T S ; – if u ∈ U and t ∈ B T Q , then, for each n ∈ N + , P n u t ∈ B T Q and Q n u t ∈ B T Q ; 1 – if u ∈ U and t ∈ B T P , then, for each n ∈ N + , + n u t ∈ B T P , • n u t ∈ B T P , and k n u t ∈ B T P . P n , Q n , + n , • n , and k n are the v ar iable-binding ope r ators men tioned ab o v e. They bind v a r iables that range ov er all quantities that can b e denoted by numerals k where 0 ≤ k < n (in plain terms, quan tities that co rrespo nd to natural num bers less than n ). Intuitiv ely , P n u t stands for t 1 + · · · + t n , where t i (1 ≤ i ≤ n ) is t with a ll o ccurrences of u repla ced by u − 1 , and analo gously in the ca se o f Q n , + n , • n , and k n . A binding ter m t is a c o mpr ehende d term if it is a binding term of the form ♦ n u t ′ , where ♦ n is a v ariable-binding o p erator. 2 Below, we will give the a xioms of the calculus asso ciated with meadow enriched ACP pro cess algebr a s. W e hav e to do with a calculus b ecause the distinction b et w een free and bo und v ariables is essential in applying the axioms concer ning comprehended terms. A v ariable u ∈ U o ccurs fr e e in a binding term t if there is a n o ccurrence of u in t that is not in a subter m of the form ♦ n u t ′ , where ♦ n is a v ariable-binding op erator. A binding ter m t is close d if it is a binding term in which no v a riable o ccurs free. Substitution of a binding term t ′ of sor t P for a v ar iable x ∈ X in a binding term t , written t [ t ′ /x ], is defined by induction on the structure of t as usual: 1 W e write N + for the set N \ { 0 } . 2 The name comprehended term originates from t h e name comprehended expression introduced in [27]. 10 v [ t ′ /x ] = v , y [ t ′ /x ] =    t ′ y if x ≡ y , 3 otherwise , c [ t ′ /x ] = c , o ( t 1 , . . . , t n )[ t ′ /x ] = o ( t 1 [ t ′ /x ] , . . . , t n [ t ′ /x ]) , ( ♦ n v t ′′ )[ t ′ /x ] =        ♦ n w (( t ′′ [ w/ v ])[ t ′ /x ]) ♦ n v ( t ′′ [ t ′ /x ]) if v o ccurs free in t ′ ( w do es no t o ccur in t ′ , t ′′ ) , otherwise . and substitution of a binding term t ′ of sort Q for a v ariable u ∈ U in a binding term t , written t [ t ′ /u ], is defined by induction on the structure of t as follows: v [ t ′ /u ] =    t ′ v if u ≡ v , otherwise , x [ t ′ /u ] = x , c [ t ′ /u ] = c , o ( t 1 , . . . , t n )[ t ′ /u ] = o ( t 1 [ t ′ /u ] , . . . , t n [ t ′ /u ]) , ( ♦ n v t ′′ )[ t ′ /u ] =              ♦ n v t ′′ ♦ n w (( t ′′ [ w/ v ])[ t ′ /u ]) ♦ n v ( t ′′ [ t ′ /u ]) if u ≡ v , if u 6≡ v , v o ccurs fr ee in t ′ ( w do es no t o ccur in t ′ , t ′′ ) , otherwise . The essentialit y of the distinction b et w een free a nd b ound v ariables in a pply- ing the a x ioms co ncerning comprehended terms originates fr om the s ubstit utions inv olv ed in applying those axioms. The axio ms of the calculus a ssocia ted with meado w enriched A CP pro cess algebras are the fo rm ulas given in T ables 1 – 5. Like some equations in T ables 3 and 4, the equations in T able 5 are actua lly schemas of equations: p a nd P are syntactic v ariables which stand for arbitra ry binding terms of sor t Q and sort P , r espectively , and n s ta nds for an arbitra r y p ositiv e natural num ber. The axioms given in T able 5 a r e ca lled the axioms for c ompr ehende d terms . They consist of three ax ioms, including an α -con v ersion axio m, for each of the v ar iable-binding o perator s of the calculus. F or each co mprehended term, we can derive fro m these axioms that the comprehended term is equal to a term ov er the s ig nature of meadow enriched ACP pro cess algebra s. 3 W e write ≡ for syntactic identit y . 11 T able 5. A xioms for comprehend ed t erms P n u p = P n v ( p [ v /u ]) if v do es n o t o ccur free in p P 1 u p = p [0 /u ] P n +1 u p = p [0 /u ] + P n u ( p [ u + 1 /u ]) Q n u p = Q n v ( p [ v /u ]) if v do es n o t o ccur free in p Q 1 u p = p [0 /u ] Q n +1 u p = p [0 /u ] · Q n u ( p [ u + 1 /u ]) + n u P = + n v ( P [ v/u ]) if v does n ot occur free in P + 1 u P = P [0 /u ] + n +1 u P = P [0 /u ] + + n u ( P [ u + 1 /u ]) • n u P = • n v ( P [ v/u ]) if v do es not occu r free in P • 1 u P = P [0 /u ] • n +1 u P = P [0 /u ] · • n u ( P [ u + 1 /u ]) k n u P = k n v ( P [ v/u ]) if v do es not o ccur free in P k 1 u P = P [0 /u ] k n +1 u P = P [0 /u ] k k n u ( P [ u + 1 /u ]) Theorem 1 (Elimi nat ion). F or al l c ompr ehende d t erms t , ther e exists a term t ′ over the signatur e of me adow enriche d ACP pr o c ess algebr as such that t = t ′ is derivable fr om the axioms for c ompr ehende d terms. Pr o of. If t is o f the for m P n u t ′′ , Q n u t ′′ , + n u t ′′ , • n u t ′′ or k n u t ′′ , where t ′′ is a term ov er the signature of meadow enr iched ACP pro cess a lgebras of the right so rt, then it is easy to pr o v e by induction on n that there exis ts a term t ′ ov er the signature of meadow enr ic hed ACP pro cess a lgebras such that t = t ′ is deriv able from the axioms for co m prehended terms. Using this fact, the general case is easily prov ed by induction on the depth of t . ⊓ ⊔ The comprehended ter ms of the calculus associa ted with meado w enriched A CP pro cess algebra s ar e finitary co mprehended terms b ecause the v ariable- binding op erators of the calculus bind v ar iables with a finite range only . This is a pre r equisite for eliminatio n of v ariable-binding op erators. 6 The I n terpretation of T er m s of the Calculus In this se ction, we define the in terpretation of terms of the calculus asso ciated with mea do w enr ic hed ACP pro cess algebr as. W e ass ume that a fixed but arbi- trary meadow enr iched ACP pro cess algebra A has be e n g iv en. 12 W e write σ A , where σ in Σ mp , for the interpretation of σ in A . Mor eo v er, we write f + 1, wher e f : Q A → Q A or f : Q A → P A , for the function f ′ : Q A → Q A or f ′ : Q A → P A , r espectively , defined by f ′ ( q ) = f ( q + A 1 A ). The terms of the calculus in tro duced ab ov e ca n b e directly in terpreted in A . T o achiev e that, we asso ciate with each v a riable-binding o perator ♦ n of the calculus a function ♦ n A : ( Q A → Q A ) → Q A or ♦ n A : ( Q A → P A ) → P A as follows: P 1 A ( f ) = f (0 A ) , P n +1 A ( f ) = f (0 A ) + A P n A ( f + 1) , Q 1 A ( f ) = f (0 A ) , Q n +1 A ( f ) = f (0 A ) · A Q n A ( f + 1 ) , + 1 A ( f ) = f (0 A ) , + n +1 A ( f ) = f (0 A ) + A + n A ( f + 1 ) , • 1 A ( f ) = f (0 A ) , • n +1 A ( f ) = f (0 A ) · A • n A ( f + 1) , k 1 A ( f ) = f (0 A ) , k n +1 A ( f ) = f (0 A ) k A k n A ( f + 1) . The interpretation of a ter m of the calculus in A dep ends on the elements of Q A and P A that are asso ciated with the v a r iables that o ccur fr ee in it. W e mo del such asso ciations by functions ρ : ( U ∪ X ) → ( Q A ∪ P A ) such that u ∈ U ⇒ ρ ( u ) ∈ Q A and x ∈ X ⇒ ρ ( x ) ∈ P A . Thes e functions are called assignments in A . W e w r ite A ss A for the set of all assignments in A . F or e a c h assignment ρ ∈ A ss A , u ∈ U a nd q ∈ Q A , w e write ρ ( u → q ) for the unique assig nmen t ρ ′ ∈ A ss A such that ρ ′ ( v ) = ρ ( v ) if v 6≡ u a nd ρ ′ ( u ) = q . The in terpretation of terms of the calculus in a meadow enriched A CP pro cess algebra A is g iv en by the function [ [ ] ] A : ( B T Q ∪ B T P ) → ( A ss A → ( Q A ∪ P A )) defined a s follows: [ [ u ] ] A ( ρ ) = ρ ( u ) , [ [ x ] ] A ( ρ ) = ρ ( x ) , [ [ c ] ] A ( ρ ) = c A , [ [ o ( t 1 , . . . , t n )] ] A ( ρ ) = o A ([ [ t 1 ] ] A ( ρ ) , . . . , [ [ t n ] ] A ( ρ )) , [ [ ♦ n u t ] ] A ( ρ ) = ♦ n A ( f ) , where f is defined by f ( q ) = [ [ t ] ] A ( ρ ( u → q )) . The axio ms of the calculus a ssocia ted with meado w enriched A CP pro cess algebras a re s o und with resp ect to the in terpretation o f the terms of the calculus given a bov e. Theorem 2 (Soundness). F or all e quations t = t ′ that b elong to the axioms of the c alculus asso ciate d with me adow enriche d A CP pr o c ess a lgebr as, we have that [ [ t ] ] A ( ρ ) = [ [ t ′ ] ] A ( ρ ) for al l assignments ρ ∈ A ss A . Pr o of. F o r all equations t = t ′ that belong to the a xioms for meadow enriched A CP pro cess algebra s, the so undness follows immediately from the fact that A is 13 a mea do w enriched A CP pro cess algebra . F or a ll equations t = t ′ that belo ng to the axioms for comprehended terms, the so undness is easily proved by inductio n on the structure of t . ⊓ ⊔ Because the terms of the ca lculus asso ciated with meadow enriched A CP pro cess algebr as ca n be directly interpreted in meadow enriched A CP pro cess algebras , w e consider the v ariable-binding operato rs of the ca lculus to consti- tute a pro cess algebr aic feature. Fitting them in an algebraic fra mew ork doe s not inv olve any s erious theor etical complicatio n. It is muc h more difficult to fit the v ariable-binding op erators fro m µ CRL and P SF that g eneralize a ssociative op erators of ACP, but do no t giv e rise to finitary comprehended ter ms, in an algebraic framework (see e.g . [22]). 7 The B in ary V ariable- Binding Op e rators F ull elimination of all v ariable- binding op erators o ccurring in a co mprehended term can lead to a combinatorial explosio n. In this sectio n, we show that no com- binatorial ex plosion takes place if v a riable-binding op erators that bind v ariables with a tw o-v alued range are still p ermitted in the re sulting ter m . W e b egin by lo oking at an example. F rom the axioms for comprehended terms, we ea sily derive the equation P 7 u p = p [0 /u ] + · · · + p [6 / u ] . This suggests that, on full elimination of v ariable-binding op erators, the size of the r esulting term gr o ws r apidly as the s ize of the orig inal ter m incr eases (there are seven subs t itution insta nces of p and they hav e increasing s izes). Using the axioms for co mpr ehended terms as well as other a xioms of the ca lc ulus , we derive the following: p [0 /u ] + · · · + p [6 /u ] = p [0 /u ] + · · · + p [6 /u ] + 0 = (0 ⊳ 1 − s ( u − 6) ⊲ p )[0 /u ] + · · · + (0 ⊳ 1 − s ( u − 6) ⊲ p )[7 /u ] = P 2 u  P 2 v  P 2 w  (0 ⊳ 1 − s ( u − 6 ) ⊲ p )[2 2 · w + 2 1 · v + 2 0 · u /u ]  = P 2 u  P 2 v  P 2 w  ((0 ⊳ 1 − s ( u − 6 ) ⊲ p )[2 · v + u /u ])[2 · w + v /v ]  . This suggests that, if v a riable-binding op erators tha t bind v a riables with a t wo- v alued r ange are still p ermitted in the resulting ter m, its size gr o ws far less rapidly as the size o f the o riginal term incr eases (there is only o ne s ubstit ution instance of p ). How ev er, a count erpart of the first step in the deriv ation ab ove do es not exist for co mprehended terms of the for ms • n u p and k n u p be c ause identit y elements for s equen tial and par allel comp osition ar e missing . Henceforth, w e will use the term binary variable-bi nding op er ators for the v ar iable-binding op erators that bind v ariables with a tw o-v a lued range and the 14 term non-binary vari able-binding op er ators for the o th er v a riable-binding op er- ators. The s ize of binding terms is giv en by the function size : ( B T Q ∪ B T P ) → N defined a s follows: size ( u ) = 1 , size ( x ) = 1 , size ( c ) = 1 , size ( o ( t 1 , . . . , t n )) = size ( t 1 ) + · · · + size ( t n ) + 1 , size  ♦ n u ( t )  = size ( t ) + log 2 ( n ) + 1 . 4 The summand log 2 ( n ) o ccurs in the equation for the size of a term of the form ♦ n u ( t ) be cause having (the cardina lit y o f ) the range of u enco ded in the v ar iable- binding oper ator is an artifice that m ust b e tak en into acco un t using the most efficient wa y in which n could b e repres en ted by a binding term. It follows from Prop osition 1 for m ulated b elow that the size of this term is o f order log 2 ( n ). The imp ortan t insights relev an t to elimination o f non-binar y v ariable-binding op erators are br ough t to gether in the following prop osition. Prop osition 1. F ro m the axioms of the c alculus asso ciate d with me adow en- riche d A CP pr o c ess algebr as, we c an derive the e quations fr o m T able 6 for e ach binding t erm p of sort Q , binding term P of sort P , and n , m ∈ N + . Pr o of. It follows immediately fro m the axio ms for comprehended terms that the first t w o eq uations for P n are deriv able. It is eas y to prov e by inductio n o n n that P 2 · n u p = P n u ( p [2 · u/ u ]) + P n u ( p [2 · u + 1 /u ]) is deriv a ble. F rom this it follows easily that the third equa tion for P n is deriv- able. It is eas y to prove by ca se distinctio n b et w een n = 1 a nd n > 1 that P n u (0 ⊳ 1 − s ( u − 0 ) ⊲ p ) = p [0 /u ] is deriv able. Using this fact, it is eas y to prov e b y induction on n that for all m ≥ n + 1 : P n +1 u p = P m u (0 ⊳ 1 − s ( u − n ) ⊲ p ) is der iv able. F rom this it follows ea s ily that the fourth equation for P n is der iv - able. The pro ofs for the equatio ns for Q n , + n , • n and k n go a nalogously , with the exception o f the fourth equation for • n and k n . It is easy to prove by induc- tion on n that for a ll m < n : • n u P = • m u P · • n − m u ( P [ m + u/u ]) 4 W e use t he conv entio n that, whenev er we write log 2 ( n ) in a context requiring a natural num ber, ⌈ log 2 ( n ) ⌉ is implicitly meant. 15 T able 6. D eri ved eq uations for comprehended terms P 1 u p = p [0 /u ] P 2 u p = p [0 /u ] + p [1 /u ] P 2 n +1 u p = P 2 u  P 2 n v ( p [2 · v + u/u ])  P n +1 u p = P 2 m u (0 ⊳ 1 − s ( u − n ) ⊲ p ) if n + 1 ≤ 2 m Q 1 u p = p [0 /u ] Q 2 u p = p [0 /u ] · p [1 /u ] Q 2 n +1 u p = Q 2 u  Q 2 n v ( p [2 · v + u/u ])  Q n +1 u p = Q 2 m u (1 ⊳ 1 − s ( u − n ) ⊲ p ) if n + 1 ≤ 2 m + 1 u P = P [0 /u ] + 2 u P = P [0 /u ] + P [1 /u ] + 2 n +1 u P = + 2 u  + 2 n v ( P [2 · v + u/u ])  + n +1 u P = + 2 m u ( δ ⊳ 1 − s ( u − n ) ⊲ P ) if n + 1 ≤ 2 m • 1 u P = P [0 /u ] • 2 u P = P [0 /u ] · P [1 /u ] • 2 n +1 u P = • 2 u  • 2 n v ( P [2 · v + u/u ])  • n +1 u P = • 2 m u P · • ( n +1) − 2 m u ( P [2 m + u/u ]) if 2 m < n + 1 < 2 m +1 k 1 u P = P [0 /u ] k 2 u P = P [0 /u ] k P [1 /u ] k 2 n +1 u P = k 2 u  k 2 n v ( P [2 · v + u/u ])  k n +1 u P = k 2 m u P k k ( n +1) − 2 m u ( P [2 m + u/u ]) if 2 m < n + 1 < 2 m +1 is deriv a ble. F rom this it follows easily that the fourth equa tion for • n is deriv a ble. The pro of for the fourth equation for k n go es analog o usly . ⊓ ⊔ The axio ms for comprehended terms give r is e to a corolla r y ab out full e lim- ination o f all v ariable-binding op erators. Corollary 1. L et t b e a c ompr ehende d term without c ompr ehende d terms as pr op er su bterms, and let k = size ( t ) . Then ther e ex i sts a term t ′ over the sig- natur e of me ado w enriche d A CP pr o c ess algebr as such that t = t ′ is derivabl e fr om the axio ms of the c alculus asso ciate d with me ado w enriche d A CP pr o c ess algebr as and – size ( t ′ ) = O ( k 2 · 2 k ) ; 16 – size ( t ′ ) = Ω ( k · 2 k − 2 ) if t is a t erm of the form P n u t ′′ or Q n u t ′′ and the numb er of times t h at u o c cur s fr e e in t ′′ is gr e ater than zer o; – size ( t ′ ) = Ω ( k · 2 k − 3 ) if t is a term of the form + n u t ′′ , • n u t ′′ or k n u t ′′ and the nu mb er of times that u o c curs fr e e in t ′′ is gr e ater than zer o. Pr o of. T e rm t is a binding ter m of the form ♦ n u t ′′ , w he r e ♦ n is a v ar iable- binding op erator. Let k ′ = s i ze ( t ′′ ), le t k ′′ be the num ber of times that u o ccurs free in t ′′ , and let l i (0 ≤ i < n ) be the size of the smallest term p o ver the signature o f meadow e nric hed A CP pr o cess alg ebras such that p = i . Then size ( t ′ ) = n · k ′ + P n − 1 i =0 ( k ′′ · l i ) + n − 1 . Beca use k = k ′ + log 2 ( n ) + 1, we k no w that k ′ < k , log 2 ( n ) < k and n < 2 k . Moreov er, we know that k ′′ < k ′ and l i = Θ (log 2 ( i + 1 )) . Hence size ( t ′ ) = O ( k 2 · 2 k ). W e a lso know that k ′ ≥ 1 and, b ecause k = k ′ + lo g 2 ( n ) + 1, lo g 2 ( n ) ≥ k − 2 and n ≥ 2 k − 2 if t is of the form P n u t ′′ or Q n u t ′′ ; and that k ′ ≥ 2 and, because k = k ′ + log 2 ( n ) + 1, log 2 ( n ) ≥ k − 3 and n ≥ 2 k − 3 if t is o f the form + n u t ′′ , • n u t ′′ or k n u t ′′ . Hence, in the case wher e k ′′ ≥ 1 , size ( t ′ ) = Ω ( k · 2 k − 2 ) if t is of the form P n u t ′′ or Q n u t ′′ and size ( t ′ ) = Ω ( k · 2 k − 3 ) if t is of the form + n u t ′′ , • n u t ′′ or k n u t ′′ . ⊓ ⊔ Prop osition 1 g iv es r ise to a co rollary ab out full elimination of all non-binary v ar iable-binding op erators. Corollary 2. L et t b e a c ompr ehende d term without c ompr ehende d terms as pr op er subterms, and let k = size ( t ) . Th en ther e exists a binding term t ′ with- out non-binary varia ble-binding op er ators such that t = t ′ is derivab le fr om the axioms of t h e c alc ulus asso ci ate d with me adow en ri che d A CP pr o c ess algebr as and – size ( t ′ ) = O ( k 3 ) if t is a term o f the form P n u t ′′ , Q n u t ′′ or + n u t ′′ ; – size ( t ′ ) = Ω ( k 2 ) if t is a term o f the form P n u t ′′ , Q n u t ′′ or + n u t ′′ ; – size ( t ′ ) = O ( k 4 ) if t is a term o f the form • n u t ′′ or k n u t ′′ ; – size ( t ′ ) = Ω ( k 3 ) if t is a t erm of the form • n u t ′′ or k n u t ′′ and the num b er of times t h at u o c curs fr e e in t ′′ is gr e ater than zer o. Pr o of. Firstly , we co nsider the case wher e t is a term of the fo r m P n u t ′′ , Q n u t ′′ or + n u t ′′ . Let k ′ = s i ze ( t ′′ ), let k ′′ be the num ber of times that u o ccurs free in t ′′ , and let l ′ n be the size of the smallest term p over the sig nature o f mea do w enriched ACP pro cess algebr as such that p = 1 − s ( u − n ). Then size ( t ′ ) = k ′ + P log 2 ( n ) i =0 ( k ′′ · (6 · i ))+log 2 ( n ) · (log 2 ( n )+1)+4 · l ′ n +6. B ecause k = k ′ +log 2 ( n )+1, we know that k ′ < k and log 2 ( n ) < k . Moreov er, w e k now that k ′′ < k ′ and l ′ n = Θ (log 2 ( n + 1)). Hence size ( t ′ ) = O ( k 3 ). W e also kno w that k ′ ≥ 1 and, bec ause k = k ′ + log 2 ( n ) + 1, log 2 ( n ) ≥ k − 2 if t is o f the form P n u t ′′ or Q n u t ′′ ; and that k ′ ≥ 2 and, b ecause k = k ′ + log 2 ( n ) + 1, log 2 ( n ) ≥ k − 3 if t is of the form + n u t ′′ . Hence, size ( t ′ ) = Ω ( k 2 ). Secondly , we consider the case where t is a term of the form • n u t ′′ or k n u t ′′ . Let k ′ = size ( t ′′ ), and let k ′′ be the n um ber of times that u o ccurs free in t ′′ . Then size ( t ′ ) ≤ P log 2 ( n ) i =0 ( k ′ + P log 2 ( i ) j =0 ( k ′′ · (6 · j )) + log 2 ( i ) · (log 2 ( i ) + 1 )). Because k = k ′ + log 2 ( n ) + 1, we know that k ′ < k and lo g 2 ( n ) < k . Moreov er, 17 we know that k ′′ < k ′ . Hence size ( t ′ ) = O ( k 4 ). W e also have that size ( t ′ ) ≥ k ′ + P log 2 ( n ) i =0 ( k ′′ · (6 · i )) + log 2 ( n ) · (log 2 ( n ) + 1). Becaus e k = k ′ + log 2 ( n ) + 1 and k ′ ≥ 2 , we als o know that log 2 ( n ) ≥ k − 3. Hence, in the ca se where k ′′ ≥ 1, size ( t ′ ) = Ω ( k 3 ). ⊓ ⊔ Corollar ies 1 and 2 show that muc h o f the compactness that can b e achiev ed with the v a r iable-binding op erators of the calculus asso ciated with meadow en- riched ACP pro cess algebra s can a lready b e achieved with the binary v ariable- binding op erators. In Cor o llary 2, size ( t ′ ) is O ( k 4 ) instea d of O ( k 3 ) if t is of the form • n u t ′′ or k n u t ′′ . The o rigin of this is that A CP proc e ss alg ebras do not hav e iden tit y elements for sequent ial and parallel comp osition. In the se tting of A CP, the ident it y element for seq uen tial compo sition, as well as parallel co mposition, is known a s the empty pro cess. 8 Adding an Identit y Elemen t for Sequen tial Comp osition In this section, we inv estigate the effect of adding a n iden tit y element for sequen- tial comp osition to ACP pro cess alg ebras on the result conce r ning elimination of non-binary v ariable-binding op erators pr esen ted a bov e. The signature of these alge br as is the signatur e of ACP pro cess algebras extended with the following: – the empty pr o c ess constant ǫ : → P ; – the unar y t ermi nation o p erator √ : P → P . Let P b e a closed term o f sort P . Intuitiv ely , the additional constant and op erator can b e ex pla ined as follows: – ǫ is only capable of terminating successfully; – √ ( P ) is only capable o f termina tin g succ e ssfully if P is capable of terminat- ing successfully and is not capable of doing anything otherwise. In the setting of ACP, the addition o f the empt y pr ocess cons t ant has been treated in several w a ys. The tre a tmen t in [2 1 ] yields a non-asso ciative par allel comp osition op erator. The firs t treatment that yields an asso ciative parallel comp osition op erator [30] is fro m 1986, but w as not published un til 1997. The treatment in this pap er is base d on [1]. An ACP pr o c ess algebr a with an identity eleme nt fo r se quential c omp o sition is an alg ebra with the sig nature of A CP pro cess a lgebras with a n identit y element for seq uen tial co mp osition that satisfies the formulas given in T able 3 with the exception o f x k y = ( x ⌊ ⌊ y + y ⌊ ⌊ x ) + x | y and the formulas given in T able 7. W e could disp ense with the equatio ns a ⌊ ⌊ x = a · x and a | b · x = ( a | b ) · x from T able 3 b ecause they hav e b ecome deriv able from the other equations. In spite of the replacement of the equa tio n x k y = ( x ⌊ ⌊ y + y ⌊ ⌊ x ) + x | y by the eq uation x k y = (( x ⌊ ⌊ y + y ⌊ ⌊ x ) + x | y ) + √ ( x ) · √ ( y ), the equations characterizing ACP pro cess a lgebras with an iden tit y ele men t for seq uen tial comp osition constitute 18 T able 7. R epla cing and additional ax ioms for empty pro cess constant x · ǫ = x ǫ · x = x x k y = (( x ⌊ ⌊ y + y ⌊ ⌊ x ) + x | y ) + √ ( x ) · √ ( y ) x ⌊ ⌊ ǫ = x ǫ ⌊ ⌊ x = δ ǫ | x = δ ∂ H ( ǫ ) = ǫ √ ( ǫ ) = ǫ √ ( a ) = δ √ ( x + y ) = √ ( x ) + √ ( y ) √ ( x · y ) = √ ( x ) · √ ( y ) √ ( x ) · √ ( y ) = √ ( y ) · √ ( x ) x + √ ( x ) = x a conserv ative ex tension o f the equations c haracterizing ACP pro cess alg ebras. The e q uation √ ( x ) · √ ( y ) = √ ( y ) · √ ( x ) is of impo rtance b ecause it makes the equation ( x k y ) k z = x k ( y k z ) der iv able. The equa tio n x + √ ( x ) = x is of impo rtance be cause it ma k es the eq ua tion x k ǫ = x deriv able. Meadow enriched ACP pr ocess a lg ebras with an identit y element for sequen- tial comp osition a re defined like meadow enriched ACP pro cess algebra s . W e can asso ciate a calc ulus with meado w enriched ACP pr ocess algebra s with an iden- tit y element for sequent ial comp osition like we did befor e for mea do w enr ic hed A CP pro cess algebras . By the addition o f an identit y element for seq uen tial comp osition, the pr op- erties o f • n and k n with respect to elimination of no n-binary v aria ble-binding op erators b ecome co mparable to the prop erties of P n , Q n and + n with resp ect to elimination o f no n-binary v ariable-binding op erators. Prop osition 2. F ro m the axioms of the ab ove -mentione d c alculus, we c a n d e- rive the fol lowing e quat ions for e ach binding term P of sort P and n, m ∈ N + : • n +1 u P = • 2 m u ( ǫ ⊳ 1 − s ( u − n ) ⊲ P ) if n + 1 ≤ 2 m , k n +1 u P = k 2 m u ( ǫ ⊳ 1 − s ( u − n ) ⊲ P ) if n + 1 ≤ 2 m . Pr o of. The proo fs for these equations go a nalogously to the proofs for the last equations for P n , Q n and + n in the pr o of of P r oposition 1. ⊓ ⊔ Prop osition 2 gives rise to a cor o llary a bout full elimination of the non- binary v ar iable-binding op erators for sequential and para llel c o mposition in the presence of an identit y element for s equen tial comp osition. Corollary 3. L et t b e a c ompr ehende d term of the form • n u t ′′ or k n u t ′′ without c ompr ehende d t erms as pr op er subterms, and let k = size ( t ) . Then ther e exists a binding t erm t ′ without non-binary variable-bindi ng op er a tors such that t = t ′ is derivable fr om the axioms of the ab ove-mentione d c alculus and size ( t ′ ) = O ( k 3 ) and size ( t ′ ) = Ω ( k 2 ) . Pr o of. The pro of go es analogo usly to the cas e where t is of the form P n u t ′′ , Q n u t ′′ or + n u t ′′ in the pro of o f Coro lla ry 2. ⊓ ⊔ 19 Corollar ies 2 and 3 imply that, on full elimina t ion o f the non-binary v ariable- binding oper ators for se quen tial and parallel co mposition, the addition of an ident it y element for seq ue ntial compos ition to ACP pro cess alg ebras gives rise to po lynomially s ma ller terms. 9 Adding Pr ocess Sequences In this section, we in tro duce pr ocess sequences to demonstra t e that there is an alter nativ e to in troducing v ariable-binding o perators for s ev eral asso ciativ e op erators on pr ocesses . The signature of ACP pro cess algebra s with a n identit y e lemen t for se q uen tial comp osition and pro cess seq uences is the s ignature of A CP pro cess algebr a s with an identit y element for sequential comp osition extended with the sort PS of pr o c ess se quenc es and the following constants and o perator s: – the empty pr o c ess se quenc e constant h i : → PS ; – the unar y singleton pr o c ess se quenc e op erator h i : P → PS ; – the binar y pr o c ess se quenc e c onc atenation op erator y : PS × PS → PS ; – the unar y gener alize d alternative c omp osition opera tor + : PS → P ; – the unar y gener alize d se quent ial c omp o sition opera tor • : PS → P ; – the unar y gener alize d p ar allel c omp osition oper a tor k : PS → P . W e assume that there is a countably infinite set V of v a riables of so r t PS , which contains α , β and γ , with and without subscr ipts. W e use the sa me notational conv en tions as b efore. In additio n, we use infix nota tion fo r the binary op erator y and mixfix nota tio n for the unary op erator h i . The co nstan t and the first tw o op erators introduced ab o ve ar e the usual ones for sequences, which g iv es a n appropr ia te intuition ab out them. The remaining three op erators introduced abov e g eneralize a lternativ e, sequential and par allel comp osition to an a rbitrary finite num ber of pro cesses. An ACP pr o c ess algebr a with an identity eleme nt fo r se quential c omp o sition and pr o c ess se quenc es is an a lg ebra with the signature of ACP pro cess algebr as with an ide ntit y element for sequential comp osition and pr o cess sequences that satisfies the fo rm ulas given in T able 3 with the exception o f x k y = ( x ⌊ ⌊ y + y ⌊ ⌊ x ) + x | y and the for m ulas g iv en in T ables 7 and 8. If we would introduce pro cess sequences in the absence o f an identit y element for se quen tial comp o sition, we s hould consider no n-empt y pro cess sequences o nly . Meadow enriched ACP pr ocess a lg ebras with an identit y element for sequen- tial comp osition and pro cess sequences are defined lik e mea do w enr ic hed ACP pro cess algebr a s. W e ca n asso ciate a calculus with meado w enriched ACP pro- cess algebras with an identit y element for sequential comp osition and pro cess sequences like we did b efore for mea do w enr ic hed ACP pro cess algebr as. More - ov er, we can e x tend the resulting calculus with v ariable- binding op erators that generalize the pro cess s e quence concatenation op erator. F or the terms o f the extended calculus, we need the following additional formatio n rule: – if u ∈ U and t ∈ B T PS , then, for each n ∈ N + , y n u t ∈ B T PS . 20 T able 8. A dditional axioms for p rocess sequences α y h i = α h i y α = α ( α y β ) y γ = α y ( β y γ ) + ( h i ) = δ + ( h x i ) = x + ( h x i y α ) = x + + ( α ) • ( h i ) = ǫ • ( h x i ) = x • ( h x i y α ) = x · • ( α ) k ( h i ) = ǫ k ( h x i ) = x k ( h x i y α ) = x k k ( α ) T able 9. A dditional axioms for compreh ended terms of sort PS y n u S = y n v ( S [ v /u ] ) y 1 u S = S [0 /u ] y n +1 u S = S [0 /u ] y y n u ( S [ u + 1 /u ]) The a xioms of the extended ca lc ulus are the form ulas g iven in T ables 1 – 5 and 7 – 9. Like some equa tions in T ables 3 – 5, the equations in T able 9 are a ctu ally schemas o f equations: S is a s y n tactic v ariable which stands for an ar bitrary binding term of s ort PS , and n stands for an ar bitrary p ositive natural num ber . The prop erties of y n with resp ect to elimination of non-binar y v ariable- binding ope r ators a re compara ble to the pr operties of + n , • n and k n with resp ect to elimination o f non-binary v ariable-binding op erators. Prop osition 3. F ro m the axioms of t h e extende d c alculus, we c an derive t h e fol lowi ng e qu ations for e ach binding term S of sort PS and n, m ∈ N + : y 1 u S = S [0 /u ] , y 2 u S = S [0 /u ] y S [1 /u ] , y 2 n +1 u S = y 2 u  y 2 n v ( S [2 · v + u/u ])  , y n +1 u S = y 2 m u ( h i ⊳ 1 − s ( u − n ) ⊲ S ) if n + 1 ≤ 2 m . Pr o of. The pro of go es analogo usly to the case of the equations for P n in the pro of of Prop osition 1. ⊓ ⊔ Prop osition 3 gives rise to a cor o llary a bout full elimination of the non- binary v ar iable-binding op erators fo r pr ocess sequence concatenation. Corollary 4. L et t b e a c o mpr ehende d term of the form y n u t ′′ without c om- pr ehende d terms as pr op er su bterms, and let k = size ( t ) . Then ther e exists a binding term t ′ without non-binary variable -binding op er ators such that t = t ′ is derivable fr om the axioms of the extende d c alculus and size ( t ′ ) = O ( k 3 ) and size ( t ′ ) = Ω ( k 2 ) . Pr o of. The pro of go es analogo usly to the cas e where t is of the form P n u t ′′ , Q n u t ′′ or + n u t ′′ in the pro of o f Coro lla ry 2. ⊓ ⊔ 21 In the pre s ence of the op erators + , • and k and the v ariable-binding op erator y n , the v ariable-binding op erators + n , • n , a nd k n are sup e rfluous. Prop osition 4. F ro m the axioms of t h e extende d c alculus, we c an derive t h e fol lowi ng e qu ations for e ach binding term P of sort P and n ∈ N + : + n u P = +  y n u h P i  , • n u P = •  y n u h P i  , k n u P = k  y n u h P i  . Pr o of. This is easy to prov e by inductio n on n . ⊓ ⊔ If we would intro duce quantit y sequences as well, we could g et a similar result for the v ariable-binding op erators P n and Q n . Prop osition 4 shows that ther e is an alterna tiv e to introducing v ariable- binding op erators for a lter nativ e, sequential and para llel comp osition. How ev er, this pr oposition also gives rise to a c orollary ab out full elimination of the non- binary v a r iable-binding op erators for alternative, sequential and parallel comp o- sition. Corollary 5. L et t b e a c ompr ehende d t e rm of the form + n u t ′′ , • n u t ′′ or k n u t ′′ without c ompr ehende d terms as pr op er su bterms, and let k = size ( t ) . Then ther e exists a binding term t ′ without n o n-binary variable-binding op er ators such that t = t ′ is derivable fr om t h e axioms of the extende d c alculus and size ( t ′ ) = O ( k 3 ) and size ( t ′ ) = Ω ( k 2 ) . Pr o of. This is a direct consequenc e of Corolla r y 4 and P r oposition 4. ⊓ ⊔ Corollar y 5 implies that in the presenc e of an identit y element for sequential comp osition, on full elimina tio n of the non- binary v a riable-binding o perators for alternative, sequential and parallel compo sition, the addition of process se- quences to A CP pro cess algebras do es not giv e rise to significantly smaller o r larger terms. 10 Concluding Remarks W e hav e introduce d the notion of an A CP pr o cess algebra . The set of equations that ha v e b een tak en to c haracterize A CP pro cess algebras is a revision of the axiom system ACP. W e cons ider this revision worth mentioning of itself, if only bec ause it removes the need to have a cons ta n t for each atomic a ction. W e have also in tro duced the notion o f a meadow enriched A CP pr ocess algebra. This notion is a simple gener alization of the notion of an A CP pro cess algebra to pro cesses in which data are inv olv ed, the mathematical structure o f da t a b eing a meadow. The primar y mathematical structure for calcula tions is unquestiona bly a field, and a meadow differs fro m a field only in that the multiplicativ e inv erse op eration is ma de total by imp osing that the m ultiplicative inv erse of z e r o is zero. Therefore, w e co nsider the co m bination o f ACP pro cess alg ebras a nd meadows made in this pap er, a combination with p otentially many applications. 22 F or all asso ciative o perators from the signature of mea do w enr ic hed A CP pro cess algebr a s that are not o f an auxilia ry nature, we have intro duced v ariable- binding op erators as ge ne r alizations. Thus, we hav e o btained a pro cess c a lculus whose terms can b e in terpreted in all meadow enr ic hed ACP pro cess algebras. W e hav e shown that the use of v ariable-binding operato r s that bind v ariables with a tw o-v a lued range can already hav e a ma jor impa ct on the size o f terms, and that the impact can b e further increased if we add an identit y element fo r sequential comp osition to meadow enriched ACP pr ocess algebr a s. In addition, we have demonstrated that ther e is a n alternative to in tro ducing v ar iable-binding op erators for several asso ciative op erators on pro cesses if we a dd a sort of pro cess sequences and suitable op e rators o n pro cess sequences to meadow enr ic hed A CP pro cess a lgebras. All v ar ia ble-binding o perator s of the calculus asso ciated with meadow en- riched ACP pro cess alg ebras can be eliminated fro m all terms of the calculus by mea ns of its a xioms, and a ll terms of the calculus can b e dir ectly in terpreted in meadow enriched ACP pro cess algebras. Therefor e, although they yield a calculus, we conside r these v ariable-binding op erators to constitute a pro cess algebraic fea ture. Fitting them in an alg e braic framework do es no t in volv e any serious theor etical complication. Different from the v ariable-binding op erators in tro duced in this pap er, the v ar iable-binding op erators fro m µ CRL and PSF that genera lize asso ciative op- erators of ACP do not give rise to finitar y comprehended terms. It is muc h mor e difficult to fit the v ariable- binding op erators from those formalis ms in an alge- braic fra mew ork, see e.g. [22]. This also ho lds for the integration op erator, which is found in extensio ns of the axiom s ystem ACP co ncerning timed pr ocesses to al- low for the alter nativ e co mposition of a co n tin uum of differently timed pro cesses to be expr e s sed (see e.g. [2]). It is worth men tioning that in effective µ CRL, a restriction of µ CRL for which a simulator is fea sible (see e .g . [1 7]), the v ar iable bo und by the v ariable binding op erator that generalizes alter nativ e co mposition m ust have a finite rang e. W e hav e also attempted to fit v ariable-binding o perator s that bind v ariables with an infinite r ange in an algebr aic framework. W e hav e lo oked a t binding algebras [29], which ar e second-o rder algebras of a sp ecific kind that co vers v ar iable-binding op erators. The problem is that the theory of binding algebras is insufficiently e laborate for our purp ose. F or example, it is no t known whether the impo r tan t c haracterization results from the theory of first-order algebras, i.e. B irkhoff ’s v ariety result and Malcev ’s quasi-v a riet y result (see e .g . [14,26 ]), hav e gener alizations fo r binding a lgebras. It is k no wn that many impo rtan t results from the theory of first-or der alg e- bras, including the ab o ve-men tioned ones, hav e genera lizations for higher -order algebras as considered in the theory of g eneral higher -order algebra s develop ed in [24,20,25]. Therefore, we hav e also considered the replacement of v a r iable- binding o perator s by higher-o rder op erators that give r is e to suc h hig her-order algebras . How ev er, owing to the abs e nc e of bound v ariables, additional higher - order o p erators ar e needed which serve the sa me purp ose as the combinators of 23 combinatory logic [18]. Thus, this leads to the line taken ear lier with combinatory pro cess a lgebra [4 ]. References 1. Baeten, J.C.M., v an Glabb eek, R .J .: Merge and termination in pro cess algebra. In : Nori, K.V. (ed.) Proceedings 7th Conference on F oundations of Softw a re T echnol- ogy and Theoretical Computer Science. L ectu re Notes in Computer Science, vol. 287, p p. 153–172. Springer-V erlag (1987) 2. Baeten, J.C.M. , Middelbu rg, C.A.: Pro cess Algebra with Timing. Monographs in Theoretical Computer S ci ence, A n EA TCS S er ies, Sprin ger-V erla g, Berlin (2002) 3. Baeten, J.C.M., W eijland, W.P .: Pro cess A lg ebra, Cam bridge T racts in Theoretical Computer Science, vol . 18. Cambridge Un i versi ty Press, Cambridge (1990) 4. Bergstra, J.A., Bethke, I., Po nse, A.: Process algebra with com binators. In: B¨ orger, E., Gurevich, Y., Meinke, K. (eds.) CSL ’93. Lecture Notes in Computer Science, vol . 832, pp . 36–65. Sp ringe r-V erlag ( 1 994) 5. Bergstra, J.A., Bethke, I., Ponse, A.: Cancellation meado ws: A ge neric b a sis the- orem and some applications. Computer Journal 56(1), 3–14 (2013) 6. Bergstra, J.A., Hirshfeld, Y., T uck er, J.V.: Meadows and the equ ati onal sp ecifica- tion of division. Theoretical Computer Science 410(12–13), 1261–1271 (2009) 7. Bergstra, J.A., Klop, J.W.: Process algebra for synchronous communication. In- formation an d Control 60(1–3), 109–137 ( 1984) 8. Bergstra, J.A., Middelburg, C.A.: S pli tting bisimulations and retrospective condi- tions. In f ormation and Comput a tion 204(7), 1083–1138 (2006) 9. Bergstra, J.A., Middelburg, C.A.: Meadow en ri c hed ACP p ro cess algebras. arXiv: 0901.3012v 2 [math.RA] (2009) 10. Bergstra, J.A., Middelbu rg , C.A.: A pro cess calculus with finitary comprehended terms. arXiv:0903.2914v1 [cs.LO] (2009) 11. Bergstra, J.A., Middelburg, C.A.: Inversiv e meadows and divisive meadow s. Jour- nal of App li ed Logic 9(3), 203–220 (2011) 12. Bergstra, J.A., T ucker, J.V.: Elementary alge braic specifications of the rational complex num b ers . In : F utatsugi, K., et al. (eds.) Goguen F estschrift. Lecture Notes in Computer Science, vol. 4060, pp . 459–475. S p ringer-V erlag (2006) 13. Bergstra, J.A., T uck er, J.V.: The rational numbers as an abstract data typ e. Jour- nal of the ACM 54(2), Article 7 (2007) 14. Burris, S., Sank appana v a r, H.P .: A Course in Universal Algebra, Graduate T exts in Mathematics, vol . 78. Springer-V erl ag, Berlin (1981) 15. F okkink, W.J.: In trod u ctio n to Pro cess Algebra. T ex ts in Theoretical Computer Science, A n EA TCS Series, Sprin ger-V erla g, Berlin (2000) 16. Groote, J.F., Ponse, A.: Pro o f th e ory for µ CRL: A language for p rocesses with data. In: Andrews, D.J., Gro ote, J.F., Middelburg, C.A. (eds.) Semantics of Sp ecification Languages. p p. 232–251. W orkshops in Comput in g S eries, S pringer-V erlag (1994) 17. Groote, J.F., Ponse, A . : The syntax and semantics of µ CRL. I n : Ponse, A ., V erhoef, C., va n V l ijmen, S .F.M. ( eds. ) Algebra of Communicating Pro cesses 1994. pp. 26– 62. W orkshops in Computing Series, Springer-V erl ag (1995) 18. Hindley , J.R., Seldin, J.P .: Introdu ct ion to Com binators and λ -calculus. Cam bridge Universit y Press, Cambridge ( 1986) 19. Ho dges, W.A.: Model Theory , Encyclop edia of Mathematics and I ts Applications, vol . 42. Cambridge Un i versi ty Press, Cambridge (1993) 24 20. Kosiuczenko, P ., Meinke, K.: On t he p o w er of higher-order algebraic sp ecification metho d s. Information and Computation 124(1), 85–101 (1996) 21. Koymans, C.P .J., V ranck en, J.L.M.: Extending process algebra with the emp t y process ǫ . Logic Group Preprint Series 1, Department of Philosophy , Utrech t Un i- versi ty , Utrech t (1985) 22. Luttik, S .P .: Choice Q uan tification in Pro cess A lg ebra. Ph.D. thesis, Programming Researc h Group, Universit y of A ms terdam, Amsterdam (2002) 23. Mauw , S., V eltink, G.J.: A p rocess sp ecificatio n formalism. F u ndamen ta In f ormat- icae 13(2), 85–139 (1990) 24. Meinke, K.: Un iv ersa l algebra in higher types. Theoretical Computer Science 100(2), 385–417 (1992) 25. Meinke, K.: Pro of th eory of higher-order equations: Conserv ativity , normal forms and term rewriting. Journal of Computer and Sy stem Sciences 67(1), 127–173 (2003) 26. Meinke, K., T uck er, J.V.: Universal algebra. In : Abramsky , S., Gabbay , D.M., Maibaum, T.S.E. (eds.) Handb ook of Logic in Computer Science, vol. I , pp. 189– 411. O xfo rd Un iv ersi ty Press, O x fo rd (1992) 27. RAIS E Language Group : The RA ISE Sp ecification Language. Prentice-Hall, En- glew oo d Cliffs (1992) 28. Sannella, D., T arlec ki, A.: Algebraic p rel iminaries. In: Astesiano, E., Kreo wski, H.J., K rie g-Br¨ uc kner, B. (eds.) A lg ebraic F oun d atio ns of Systems Specification, pp. 13–30. S pringer-V erlag , Berlin (1999) 29. Sun Y ong: An algebraic generalizatio n of Frege structures – Binding algebras. Theoretical Computer S ci ence 211(1–2), 189–232 ( 1999) 30. V ranck en, J. L.M.: The algebra of comm unicating pro ces ses with emp t y p rocess. Theoretical Computer S ci ence 177(2), 287–328 (1997) 31. Wirsing, M.: Algebraic sp ecificatio n. I n: v an Leeuw en, J. (ed.) H a ndb ook of The- oretical Comput e r Science, vol. B, pp. 675–788. Elsevier, Amsterdam (1990) 25