Decidability of the interval temporal logic ABBar over the natural numbers

Symposium on Theoretical Aspects of Computer Science 2010 (Nancy , Fr ance), pp. 597-608 www .st acs-conf .org DECIDAB ILITY OF THE INTER V AL TEMPORAL LOGIC AB B O VER THE NA TURAL NUMBERS ANGELO MONT AN ARI 1 AND GABRIELE PUPPIS 2 AND PIETR O SALA 1 AND GUIDO SCIA VICCO 3 1 Department o f Mathematics and Computer Science, Ud ine Unive rsity , Ital y E-mail addr ess : {angelo.mo ntanari|pie tro.sala}@dimi.uniud.it 2 Computing Lab oratory , Oxford Un ivers ity , E ngland E-mail addr ess : Gabriele.P uppis@comla b.ox.ac.uk 3 Department o f Information, Engineering and Communications, Murcia Universi ty , Spain E-mail addr ess : guido@um.e s Abstra ct. In this paper, we focus our attention on the interv a l temporal logic of the Allen’s relations “meets”, “b egins”, and “begun by” ( AB B for short), interpreted ov er natural n umbers. W e first introduce t he logic and w e show that it is expressive enough to mod el distinctive in terv al properties, such as accomplishment conditions, to capt u re basic mod alities of p oint-based temporal logic, su ch as the un til operator, and to en code relev an t metric constrain ts. Then, w e prov e that the satisfiabilit y p roblem for AB B ov er n atu ral num bers is decidable by p roviding a small mo del theorem based on an original contractio n metho d . Finally , we p ro ve the EX PS P ACE-completeness of th e problem. 1. In tro duction In terv al temp oral logics are mo dal logics th at allo w one to represen t and to reason ab out time interv als. It is w ell kno wn that, on a linear ord er in g, one among thirteen differen t bi- nary relations ma y hold b et ween an y p air of interv als, namely , “ends”, “dur ing”, “b egins”, “o ve rlaps”, “meets”, “b efore”, together with their inv erses, and the relation “equals” (the so-calle d Allen’s relations [1]) 1 . Allen’s relations giv e r ise to resp ectiv e unary mo dal op era- tors, thus defin ing the mo dal logic of time in terv als HS in tro d uced by Halp ern and Shoham 1998 ACM Subje ct Classific ation: F.3: logics and meaning of programs; F.4: mathematical logic and formal languages. Key wor ds and phr ases: interv al temp oral logics, compass structures, decidabilit y , complexity . The work has b een partially supp orted by the GNCS pro ject: “Logics, automata, and games for th e formal veri fication of complex systems”. Guido Sciavicco h as also b een supp orted by the Spain/South Africa Integrated Action N. HS2008-0006 on: “Metric interv al temp oral logics: Theory and Applications”. 1 W e d o not consider here the case of ternary relations. Amongst the multitude of ternary relations among interv als there is on e of particular imp ortance, which corresp onds t o the binary op eration of concatenation of meeting in terv als. The logic of suc h a ternary interv al relation has b een inv estigated by V enema in [20]. A sy stematic analysis of its fragments has b een recently given by Ho dkinson et al. [13]. c  A. Monta nari, G. Puppis, P . Sala, and G. Sciavicco CC  Creative Commons Attribution- NoDerivs License 598 A. MONT ANARI, G. PUPPIS, P . SALA, AND G. SCIA VICCO in [12]. S ome of these mo dal op erators are actually defin able in terms of others; in p artic- ular, if sin gleton interv als are included in the structure, it suffices to c ho ose as basic the mo dalities corresp ond ing to the r elations “b egins” B and “ends” E , and their transp oses B , E . HS turns out to b e highly u ndecidable under v ery we ak assumptions on the class of in terv al structures ov er wh ic h its formulas are interpreted [12 ]. In particular, un decidabilit y holds for an y class of interv al stru ctures o ve r linear ord erings that con tains at least one linear ordering with an infinite ascending or descend in g c hain, thus includin g the natural time fl o ws N , Z , Q , and R . Undecidabilit y of HS o ve r fin ite structures directly follo ws from results in [15]. In [14], Lo da ya sharp ens the undecidabilit y of HS sh o wing that the t wo mo dalities B , E s u ffice for und ecidabilit y o ver dense lin ear orderings (in fact, the r esult applies to the class of all linear orderings [11]). Eve n though HS is v ery natural and the meaning of its op erators is qu ite in tuitive, for a long time such s w eeping undecidabilit y results ha ve discouraged the searc h for practical app lications and furth er inv estigatio ns in the field. A renewed interest in in terv al temp oral logics has b een r ecen tly stim u lated by the identificat ion of some decidable fragmen ts of HS , whose d ecidabilit y do es n ot dep end on simplifying semantic assump tions such as lo calit y and h omogeneit y [11]. Th is is the case with the fragments B B , EE (logics of the “b egins/b egun by” and “ends/ended by” relations) [11], A , A A (logics of temp oral neighborh o o d, whose mo d alities capture the “meets/met b y” relations [10]), and D , D D (logic s of the subinterv al/sup erin terv al relations) [3, 16]. In this pap er, we fo cus our atten tion on th e pro du ct logic AB B , obtained from the join of B B and A (the case of AEE is fully sym metric), in terpreted o ve r the linear order N of the natural num b ers (or a finite prefi x of it). The decidabilit y of B B can b e prov ed by translating it in to the p oint -based prop ositional temp oral logic of linear time with temp oral mo dalities F (sometime in the future) and P (sometime in the past), whic h has the finite (pseudo-)mo del prop ert y and is decidable, e.g., [9]. In general, suc h a r eduction to p oin t- based temp oral logics d o es not wo rk: form ulas of inte rv al temp oral logics are ev aluated o ver pairs of p oin ts and tran s late into binary relations. F or instance, this is the case with A . Unlik e the case of BB , when d ealing with A one cannot abstract w ay from the left endp oint of inte rv als, as con trad ictory formulas ma y h old o ver interv als with the same right endp oint and a different left end p oint . Th e d ecidabilit y of AA , and thus th at of its f ragmen t A , ov er v arious classes of linear order in gs has b een p r o ved b y Bresolin et al. b y reducing its satisfiabilit y problem to that of the t wo -v ariable fragmen t of first-order logic o ver the same classes of structures [4], whose decidabilit y has b een pro v ed by O tto in [18]. Optimal tableau metho ds for A with resp ect to v arious classes of interv al stru ctures can b e found in [6, 7]. A decidable metric extension of A o ve r the natur al n umbers has b een pr op osed in [8]. A num b er of u ndecidable extensions of A , and A A , hav e b een giv en in [2, 5]. AB B retains the simp licit y of its constituents BB and A , but it impro v es a lot on their expressiv e p o wer (as w e shall sh o w, such an increase in exp ressiv en ess is ac hiev ed at the cost of an increase in complexit y). First, it allo ws one to express assertions that ma y b e true at certain in terv als, but at n o subinte rv al of them, suc h as the conditions of accomplishmen t. Moreo v er, it mak es it p ossib le to easily enco de the until op erator of p oint- based temp oral logic (this is p ossible neither with B B nor w ith A ). Finally , meaningful metric constrain ts ab out th e length of inte rv als can b e expressed in ABB , th at is, one can constr ain an interv al to b e at least (resp., at most, exactly) k p oin ts long. W e pro ve the decidabilit y of A B B in terpreted ov er N b y providing a small mo del theorem based on an original cont raction metho d. T o pr o ve it, w e tak e adv an tage of a natural (equiv alen t) in terpretation of A B B form ulas o ver grid-like stru ctures based on a bijection b et ween the set of int erv als o v er N DECIDABI LITY OF THE INTER V AL TE MPORAL LOGIC ABB 599 and (a suitable su b set of ) the set of p oin ts of the N × N grid. In add ition, w e pr o ve that the satisfiabilit y problem for AB B is EXPSP A C E-complete (that for A is NEXPTIME- complete). In the p ro of of hardness, we use a redu ction from the exp onen tial-corridor tiling problem. The pap er is organized as follo ws. In Section 2 we introdu ce AB B . In Section 3, we pro v e the d ecidabilit y of its satisfiabilit y prob lem. W e first describ e the app lication of the con traction metho d to finite mo dels and then we generalize it to in finite ones. In S ection 4 we deal with computational complexity issu es. Conclusions provide an assessment of the w ork and outline futur e researc h directions. Missing p ro ofs can b e found in [17]. 2. The in terv al temp oral logic A B B In this section, w e b r iefly introd uce syntax and s eman tics of the logic AB B , w h ic h fea- tures three mo d al op erators h A i , h B i , and h B i corresp onding to the thr ee Allen’s relations A (“meets”), B (“b egins”), and B (“b egun by”), r esp ectiv ely . W e show that ABB is expressive enough to capture the n otion of accomplishmen t, to d efine the standard until op erator of p oint -based temp oral logics, and to enco de metric conditions. Th en, we int ro du ce the basic notions of atom, typ e, and dep endency . W e conclude the section by p ro vidin g an alternativ e in terpretation of ABB o ve r lab eled grid -lik e structures. 2.1. Syn tax and seman t ics Giv en a set P r op of prop ositional v ariables, form u las of A BB are built up from P r op using the b o olean connectiv es ¬ and ∨ a nd the un ary mo dal op erators h A i , h B i , h B i . As usual, we shall tak e adv an tage of shorthands lik e ϕ 1 ∧ ϕ 2 = ¬ ( ¬ ϕ 1 ∨ ¬ ϕ 2 ) , [ A ] ϕ = ¬ h A i ¬ ϕ , [ B ] ϕ = ¬ h B i ¬ ϕ , ⊤ = p ∨ ¬ p , and ⊥ = p ∧ ¬ p , with p ∈ P r op . Hereafter, w e denote b y | ϕ | the size of ϕ . W e interpret formulas of AB B in in terv al temp oral structur es o ver natural num b ers endo wed with th e r elations “meets”, “b egins”, and “b egun by” . Precisely , we id entify any giv en ordinal N 6 ω with the prefix of length N of the linear order of the natural num b ers and we accordingly define I N as the set of all non-singleton closed in terv als [ x , y ] , with x , y ∈ N and x < y . F or any pair of inte rv als [ x , y ] , [ x ′ , y ′ ] ∈ I N , the Allen’s relatio ns “meets” A , “b egins” B , and “b egun by” B are defined as follo ws (note th at B is the in verse relation of B ): • “meets” relation: [ x , y ] A [ x ′ , y ′ ] iff y = x ′ ; • “b egins” rela tion: [ x , y ] B [ x ′ , y ′ ] iff x = x ′ and y ′ < y ; • “b egun by” relation: [ x , y ] B [ x ′ , y ′ ] iff x = x ′ and y < y ′ . Giv en an interval structur e S = ( I N , A , B , B , σ ) , where σ : I N → P ( P r op ) is a lab eling function that maps in terv als in I N to sets of prop ositional v ariables, and an initial interv al I , we defin e the seman tics of an ABB formula as follo ws: • S , I  a iff a ∈ σ ( I ) , f or any a ∈ P r op ; • S , I  ¬ ϕ iff S , I 6  ϕ ; • S , I  ϕ 1 ∨ ϕ 2 iff S , I  ϕ 1 or S , I  ϕ 2 ; • for ev ery relation R ∈ { A , B , B } , S , I  h R i ϕ iff there is an inte rv al J ∈ I N suc h that I R J and S , J  ϕ . 600 A. MONT ANARI, G. PUPPIS, P . SALA, AND G. SCIA VICCO Giv en an interv al structure S and a f orm u la ϕ , we sa y that S satisfies ϕ if there is an in terv al I in S suc h that S , I  ϕ . W e sa y that ϕ is satisfiable if there exists an interv al structure that satisfies it. W e defi ne the satisfiability pr oblem for AB B as the problem of establishing whether a giv en AB B -form ula ϕ is satisfiable. W e conclude the section with some examples that accoun t f or AB B expressiv e p o w er. The firs t one shows ho w to enco de in AB B conditions of accomplishment (think of form ula ϕ as the assertion: “Mr. Jones flew f rom V enice to Nancy”): h A i  ϕ ∧ [ B ]( ¬ ϕ ∧ [ A ] ¬ ϕ ) ∧ [ B ] ¬ ϕ  . F orm ulas of p oin t-based temp oral logics of the form ψ U ϕ , using the standard unt il op erator, can b e enco ded in ABB (wher e atomic interv als are tw o-p oin t in terv als) as follo ws: h A i  [ B ] ⊥ ∧ ϕ  ∨ h A i  h A i ([ B ] ⊥ ∧ ϕ ) ∧ [ B ]( h A i ([ B ] ⊥ ∧ ψ ))  . Finally , metric conditions like : “ ϕ holds ov er a right neigh b or inte rv al of length greater than k (resp., less than k , equ al to k )” can b e captured by the follo win g ABB f orm u la: h A i  ϕ ∧ h B i k ⊤  (resp., h A i  ϕ ∧ [ B ] k − 1 ⊥  , h A i  ϕ ∧ [ B ] k ⊥ ∧ h B i k − 1 ⊤  ) 2 . 2.2. A toms, types, and dep endencies Let S = ( I N , A , B , B , σ ) b e an interv al structur e and ϕ b e a formula of ABB . In the sequel, we shall compare inte rv als in S with resp ect to the set of subformulas of ϕ they satisfy . T o do that, w e introduce the k ey notions of ϕ -atom, ϕ -t yp e, ϕ -cluster, and ϕ - shading. First of all, we define the closur e C l ( ϕ ) of ϕ as th e set of all subformulas of ϕ and of their negatio ns (we identify ¬¬ α with α , ¬ h A i α w ith [ A ] ¬ α , etc.). F or tec hnical reasons, w e also in tro duce the extende d closur e C l + ( ϕ ) , which is defin ed as the set of all formulas in C l ( ϕ ) p lus all formulas of the forms h R i α and ¬ h R i α , with R ∈ { A , B , B } and α ∈ C l ( ϕ ) . A ϕ -atom is any non-empt y set F ⊆ C l + ( ϕ ) su c h that (i) for ev ery α ∈ C l + ( ϕ ) , w e hav e α ∈ F iff ¬ α 6∈ F and (ii) for ev ery γ = α ∨ β ∈ C l + ( ϕ ) , we hav e γ ∈ F iff α ∈ F or β ∈ F (in tu itively , a ϕ -atom is a maximal lo cally consistent set of formulas chosen fr om C l + ( ϕ ) ). Note that the cardinalities of b oth sets C l ( ϕ ) and C l + ( ϕ ) are linear in the n umber | ϕ | of subformulas of ϕ , while the num b er of ϕ -atoms is at most exp onen tial in | ϕ | (precisely , we ha ve |C l ( ϕ ) | = 2 | ϕ | , |C l + ( ϕ ) | = 14 | ϕ | , and there are at most 2 7 | ϕ | distinct atoms). W e also asso ciate with eac h in terv al I ∈ S the set of all formulas α ∈ C l + ( ϕ ) such that S , I  α . Su c h a set is called ϕ -typ e of I and it is den oted b y T yp e S ( I ) . W e hav e that eve ry ϕ -t yp e is a ϕ -atom, b ut not vice v ersa. H ereafter, w e shall omit the argumen t ϕ , th us calling a ϕ -atom (resp., a ϕ -t yp e) simp ly an atom (resp., a type). Giv en an atom F , w e denote b y O bs ( F ) the set of all observables of F , namely , the form ulas α ∈ C l ( ϕ ) su c h that α ∈ F . Similarly , giv en an atom F and a relation R ∈ { A , B , B } , w e denote by R e q R ( F ) the set of all R -r e quests of F , namely , the form u las α ∈ C l ( ϕ ) suc h that h R i α ∈ F . T aking adv ant age of the ab o v e sets, we can define th e follo win g t w o r elations b et w een atoms F and G : F A − → G iff R e q A ( F ) = O bs ( G ) ∪ R e q B ( G ) ∪ R e q B ( G ) ; F B − → G iff  O bs ( F ) ∪ R e q B ( F ) ⊆ R e q B ( G ) ⊆ O bs ( F ) ∪ R e q B ( F ) ∪ R e q B ( F ) , O bs ( G ) ∪ R e q B ( G ) ⊆ R e q B ( F ) ⊆ O bs ( G ) ∪ R e q B ( G ) ∪ R e q B ( G ) . 2 It is n ot difficult to show that AB B subsumes the metric ex tension of A given in [8]. A simple game- theoretic argument shows that the former is in fact strictly more expressive than the latter. DECIDABI LITY OF THE INTER V AL TE MPORAL LOGIC ABB 601 p 0 p 1 p 2 p 3 I 0 I 1 I 2 I 3 Figure 1: Corresp on d ence b et w een in terv als and p oin ts of a d iscr ete grid. Note that the relation B − → is transitive , wh ile A − → is not. Moreo ver, b oth A − → and B − → satisfy a vi ew-to-typ e dep endency , namely , for every pair of int erv als I , J in S , we h av e that I A J imp lies T yp e S ( I ) A − → T yp e S ( J ) I B J implies T yp e S ( I ) B − → T yp e S ( J ) . Relations A − → and B − → will come into play in the d efinition of consistency conditions (see Definition 2.1). 2.3. Compass structures The logic A B B can b e equiv alen tly in terpr eted o ve r grid-lik e str u ctures (the so-called compass structures [20]) by exploiting the existence of a natural bijection b et wee n the in terv als I = [ x , y ] and the p oin ts p = ( x , y ) of an N × N grid such that x < y . As an example, Figure 1 depicts four in terv als I 0 , ..., I 3 suc h that I 0 A I 1 , I 0 B I 2 , and I 0 B I 3 , together with the corresp ondin g p oin ts p 0 , ..., p 3 of a discrete grid (n ote that the three Allen’s relations A , B , B b etw een int erv als are mapp ed to corresp onding spatial relations b et w een p oint s; f or th e sak e of readabilit y , w e name the latter ones as the former ones). Definition 2.1. Given an AB B form ula ϕ , a (consisten t and fulfi lling) c omp ass ( ϕ -) structur e of length N 6 ω is a pair G = ( P N , L ) , where P N is the set of p oin ts p = ( x , y ) , with 0 6 x < y < N , and L is function that maps an y p oin t p ∈ P N to a ( ϕ -)atom L ( p ) in such a wa y that • for eve ry pair of p oints p , q ∈ P N and every relation R ∈ { A , B } , if p R q holds, then L ( p ) R − → L ( q ) f ollo ws ( consistency ); • for every p oint p ∈ P N , ev ery relation R ∈ { A , B , B } , and ev ery formula α ∈ R e q R  L ( p )  , there is a p oin t q ∈ P N suc h th at p R q and α ∈ O bs  L ( q )  ( ful- fillmen t ). W e sa y that a compass ( ϕ -)structure G = ( P N , L ) fe atur es a form ula α if th ere is a p oin t p ∈ P N suc h that α ∈ L ( p ) . T he follo wing prop osition implies that the satisfiabilit y pr oblem 602 A. MONT ANARI, G. PUPPIS, P . SALA, AND G. SCIA VICCO for A B B is reducible to the problem of deciding, for any giv en formula ϕ , whether there exists a ϕ -compass structur e that features ϕ . Prop osition 2.2. An AB B -formula ϕ is satisfie d by some interval structur e if and only if it i s fe atur e d by some ( ϕ -)c omp ass structur e. 3. Deciding the satisfiabilit y problem for AB B In th is s ection, we prov e that the satisfiabilit y problem for AB B is decidable by pro- viding a “small-mo del theorem” for the satisfiable formulas of the logic. F or the sak e of simplicit y , we firs t sh o w that the satisfiability p roblem for AB B in terp r eted o ve r finite in- terv al structur es is d ecidable and then w e generalize suc h a result to all (fi nite or infinite) in terv al structures. As a preliminary step, w e in tro d uce the k ey notion of shad in g. Let G = ( P N , L ) b e a compass s tructure of length N 6 ω and let 0 6 y < N . T he shading of the r ow y of G is th e set S hading G ( y ) =  L ( x , y ) : 0 6 x < y  , namely , the set of th e atoms of all p oint s in P N whose v ertical co ordin ate has v alue y (basically , we in terp r et d ifferen t atoms as differen t colors). Clearly , for ev ery pair of atoms F and F ′ in S hading G ( y ) , w e hav e R e q A ( F ) = R e q A ( F ′ ) . 3.1. A small-mo del theorem for finite structures Let ϕ b e an AB B formula. Let us assu m e that ϕ is featured b y a finite compass structure G = ( P N , L ) , with N < ω . In fact, without loss of generalit y , w e can assume that ϕ b elongs to the atom asso ciated with a p oint p = ( 0, y ) of G , with 0 < y < N . W e pro ve that we can restrict our atten tion to compass stru ctur es G = ( P N , L ) , wh ere N is b oun ded b y a doub le exp onential in | ϕ | . W e start with the follo wing lemma that pro ves a simple, but crucial, p r op ert y of the relations A − → and B − → (the pro of can b e found in [17 ]). Lemma 3.1. If F A − → H and G B − → H hold for some atoms F , G , H , then F A − → G holds as wel l. The next lemma sho ws th at, under su itable conditions, a giv en compass stru cture G ma y b e red u ced in length, p reserving th e existence of atoms featuring ϕ . Lemma 3.2. L e t G b e a c omp ass structur e fe aturing ϕ . If ther e exist two r ows 0 < y 0 < y 1 < N in G such that S hading G ( y 0 ) ⊆ S hading G ( y 1 ) , then ther e exists a c omp ass structur e G ′ of length N ′ < N that fe atur es ϕ . Pr o of. Supp ose th at 0 < y 0 < y 1 < N are t wo r o ws of G suc h that S had ing G ( y 0 ) ⊆ S hading G ( y 1 ) . Then, there is a function f : { 0, ..., y 0 − 1 } → { 0, ..., y 1 − 1 } suc h that, for ev ery 0 6 x < y 0 , L ( x , y 0 ) = L ( f ( x ) , y 1 ) . Let k = y 1 − y 0 , N ′ = N − k ( < N ), and P N ′ b e the p ortion of the grid that consists of all p oin ts p = ( x , y ) , with 0 6 x < y < N ′ . W e extend f to a function that maps p oin ts in P N ′ to p oints in P N as follo ws: • if p = ( x , y ) , with 0 6 x < y < y 0 , then we simply let f ( p ) = p ; • if p = ( x , y ) , with 0 6 x < y 0 6 y , then w e let f ( p ) = ( f ( x ) , y + k ) ; • if p = ( x , y ) , with y 0 6 x < y , then w e let f ( p ) = ( x + k , y + k ) . DECIDABI LITY OF THE INTER V AL TE MPORAL LOGIC ABB 603 F 1 F 2 F 3 F 2 F 3 F 1 F 4 F 3 F 2 y 0 y 1 f f f G F 1 F 2 F 3 G ′ Figure 2: Cont raction G ′ of a compass structur e G . W e d enote by L ′ the lab eling of P N ′ suc h that, for every p oin t p ∈ P N ′ , L ′ ( p ) = L ( f ( p )) and we denote b y G ′ the resulting structure ( P N ′ , L ′ ) (see Figure 2 ). W e ha ve to pro v e that G ′ is a consisten t and fulfilling compass structur e that f eatures ϕ (see Definition 2.1). First, w e show that G ′ satisfies th e consistency conditions for the relations B and A ; then w e show that G ′ satisfies the fulfi llmen t conditions for the B -, B -, and A -requests; finally , w e sho w that G ′ features ϕ . Consistency with re la tion B . Consider t wo p oint s p = ( x , y ) and p ′ = ( x ′ , y ′ ) in G ′ suc h that p B p ′ , i.e., 0 6 x = x ′ < y ′ < y < N ′ . W e p ro ve that L ′ ( p ) B − → L ′ ( p ′ ) by distinguishing among the follo w ing three cases (note that exactly one of suc h cases holds ): (1) y < y 0 and y ′ < y 0 , (2) y > y 0 and y ′ > y 0 , (3) y > y 0 and y ′ < y 0 . If y < y 0 and y ′ < y 0 , then, b y constru ction, we h av e f ( p ) = p and f ( p ′ ) = p ′ . Since G is a (consisten t) compass stru ctur e, w e immediately obtain L ′ ( p ) = L ( p ) B − → L ( p ′ ) = L ′ ( p ′ ) . If y > y 0 and y > y 0 , then, by construction, w e ha v e either f ( p ) = ( f ( x ) , y + k ) or f ( p ) = ( x + k , y + k ) , dep ending on whether x < y 0 or x > y 0 . S imilarly , w e ha ve either f ( p ′ ) = ( f ( x ′ ) , y ′ + k ) = ( f ( x ) , y ′ + k ) or f ( p ′ ) = ( x ′ + k , y ′ + k ) = ( x + k , y ′ + k ) . This implies f ( p ) B f ( p ′ ) and thus, since G is a (consisten t) compass structure, w e ha ve L ′ ( p ) = L ( f ( p )) B − → L ( f ( p ′ )) = L ′ ( p ′ ) . If y > y 0 and y ′ < y 0 , then, since x < y ′ < y 0 , w e h a ve by construction f ( p ) = ( f ( x ) , y + k ) and f ( p ′ ) = p ′ . Moreo v er, if we consider the p oint p ′′ = ( x , y 0 ) in G ′ , we easily see that (i) f ( p ′′ ) = ( f ( x ) , y 1 ) , (ii) f ( p ) B f ( p ′′ ) (whence L ( f ( p )) B − → L ( f ( p ′′ )) ), (iii) L ( f ( p ′′ )) = L ( p ′′ ) , and (iv) p ′′ B p ′ (whence L ( p ′′ ) B − → L ( p ′ ) ). It th u s follo ws that L ′ ( p ) = L ( f ( p )) B − → L ( f ( p ′′ )) = L ( p ′′ ) B − → L ( p ′ ) = L ( f ( p ′ )) = L ′ ( p ′ ) . Finally , b y ex- ploiting th e trans itivity of the relation B − → , w e obtain L ′ ( p ) B − → L ′ ( p ′ ) . 604 A. MONT ANARI, G. PUPPIS, P . SALA, AND G. SCIA VICCO Consistency with rela tion A . Consider t w o p oin ts p = ( x , y ) and p ′ = ( x ′ , y ′ ) suc h that p A p ′ , i.e., 0 6 x < y = x ′ < y ′ < N ′ . W e defin e p ′′ = ( y , y + 1 ) in su c h a w ay that p A p ′′ and p ′ B p ′′ and w e distingu ish b et ween the follo wing tw o cases: (1) y > y 0 , (2) y < y 0 . If y > y 0 , th en , b y construction, we h a ve f ( p ) A f ( p ′′ ) . Since G is a (consisten t) compass structure, it follo ws th at L ′ ( p ) = L ( f ( p )) A − → L ( f ( p ′′ )) = L ′ ( p ′′ ) . If y < y 0 , then, by construction, w e ha v e L ( p ′′ ) = L ( f ( p ′′ )) . Again, since G is a (consistent ) compass structure, it follo ws that L ′ ( p ) = L ( f ( p )) = L ( p ) A − → L ( p ′′ ) = L ( f ( p ′′ )) = L ′ ( p ′′ ) . In b oth cases we ha ve L ′ ( p ) A − → L ′ ( p ′′ ) . No w, we recall that p ′ B p ′′ and th at, by p revious argumen ts, G ′ is consisten t with the r elation B . W e th u s hav e L ′ ( p ′ ) B − → L ′ ( p ′′ ) . Fin ally , b y app lying Lemma 3.1, we obtain L ′ ( p ) A − → L ′ ( p ′ ) . Fulfillme nt of B -reques ts. Consider a p oin t p = ( x , y ) in G ′ and some B -request α ∈ R e q B  L ′ ( p )  asso ciated with it. Since, by construction, α ∈ R e q B  L ( f ( p ))  and G is a (fulfilling) compass stru cture, w e know that G con tains a p oint q ′ = ( x ′ , y ′ ) such that f ( p ) B q ′ and α ∈ O bs  L ( q ′ )  . W e pr ov e that G ′ con tains a p oin t p ′ suc h th at p B p ′ and α ∈ O bs  L ′ ( p ′ )  b y distinguish ing among the follo wing three cases (note that exactl y one of suc h cases holds): (1) y < y 0 (2) y ′ > y 1 , (3) y > y 0 and y ′ < y 1 . If y < y 0 , then, by constru ction, we ha ve p = f ( p ) and q ′ = f ( q ′ ) . Therefore, we simply defi n e p ′ = q ′ in such a wa y that p = f ( p ) B q ′ = p ′ and α ∈ O bs  L ′ ( p ′ )  ( = O bs  L ( f ( p ′ ))  = O bs  L ( q ′ )  ). If y ′ > y 1 , then, b y constru ction, we ha v e either f ( p ) = ( f ( x ) , y + k ) or f ( p ) = ( x + k , y + k ) , dep endin g on whether x < y 0 or x > y 0 . W e d efine p ′ = ( x , y ′ − k ) in suc h a wa y that p B p ′ . Moreo v er, w e observ e that either f ( p ′ ) = ( f ( x ) , y ′ ) or f ( p ′ ) = ( x + k , y ′ ) , dep en ding on wh ether x < y 0 or x > y 0 , and in b oth cases f ( p ′ ) = q ′ follo ws. T h is shows that α ∈ O bs  L ′ ( p ′ )  ( = O bs  L ( f ( p ′ )  = O bs  L ( q ′ )  ). If y > y 0 and y ′ < y 1 , then we define p = ( x , y 0 ) and q = ( x ′ , y 1 ) and we observ e that f ( p ) B q , q B q ′ , and f ( p ) = q . F rom f ( p ) B q and q B q ′ , it f ollo ws that α ∈ R e q B  L ( q )  and hence α ∈ R e q B  L ( p )  . Sin ce G is a (fulfilling) compass structur e, w e kno w that there is a p oin t p ′ suc h that p B p ′ and α ∈ O bs  L ( p ′ )  . Moreo v er, since p B p ′ , w e h a ve f ( p ′ ) = p ′ , from w hic h we obtain p B p ′ and α ∈ O bs  L ( p ′ )  . Fulfillme nt of B -reques ts. The pro of that G ′ fulfills all B -requests of its atoms is symmetric with r esp ect to the previous one. Fulfillme nt of A -reque sts. Consider a p oint p = ( x , y ) in G ′ and some A -request α ∈ R e q A  L ′ ( p )  asso ciated with p in G ′ . Since, b y previous argumen ts, G ′ fulfills all B -requests of its atoms, it is suffi cien t to pro v e that either α ∈ O bs  L ′ ( p ′ )  or α ∈ R e q B  L ′ ( p ′ )  , where p ′ = ( y , y + 1 ) . This can b e easily pro v ed by distinguishin g among th e three cases y < y 0 − 1, y = y 0 − 1, and y > y 0 . Fea tured formulas. Recall that, by previous assumptions, G con tains a p oint p = ( 0, y ) , with 0 < y < N , such that ϕ ∈ L ( p ) . If y 6 y 0 , then, by constru ction, we ha ve DECIDABI LITY OF THE INTER V AL TE MPORAL LOGIC ABB 605 ϕ ∈ L ′ ( p ) ( = L ( f ( p )) = L ( p ) ). Otherwise, if y > y 0 , w e d efi ne q = ( 0, y 0 ) and we observe that q B p . Sin ce G is a (consisten t) compass structure and h B i ϕ ∈ C l + ( ϕ ) , w e h a ve that ϕ ∈ R e q B  L ( q )  . M oreo ver, b y construction, we hav e L ′ ( q ) = L ( f ( q )) and hen ce ϕ ∈ R e q B  L ′ ( q )  . Finally , since G ′ is a (fu lfi lling) compass structure, we kn o w that there is a p oin t p ′ in G ′ suc h th at f ( q ) B p ′ and ϕ ∈ O bs  L ′ ( p ′ )  . ✷ On the ground s of the ab o ve result, we can p ro vide a suitable upp er b ound for th e length of a min imal finite interv al structure that satisfies ϕ , if there exists any . This yields a straigh tforward, b ut inefficien t, 2EXPSP A CE algorithm that decides whether a giv en AB B -form ula ϕ is satisfiable o v er finite in terv al structures. Theorem 3.3. A n AB B -formula ϕ is satisfie d by some finite interval structur e iff it is fe atur e d by some c omp ass structur e of leng th N 6 2 2 7 | ϕ | (i.e., double exp onential in | ϕ | ). Pr o of. One direction is trivial. W e pro ve the other one (“only if ” part). S upp ose that ϕ is satisfied by a finite interv al structur e S . By Prop osition 2.2, there is a compass structure G that features ϕ and has finite length N < ω . Without loss of generalit y , w e can assu m e that N is minimal among all finite compass structures that feature ϕ . W e recall from Section 2.2 that G con tains at most 2 7 | ϕ | distinct atoms. This imp lies that there exist at most 2 2 7 | ϕ | differen t s hadings of the form S hading G ( y ) , with 0 6 y < N . Finally , by applying Lemma 3.2, we obtain N 6 2 2 7 | ϕ | (otherwise, there would exist tw o ro ws 0 < y 0 < y 1 < N such that S hading G ( y 0 ) = S hading G ( y 1 ) , w hic h is against the h yp othesis of minimalit y of N ). ✷ 3.2. A small-mo del theorem for infinite structures In general, compass structur es that feature ϕ m a y b e in finite. Here, w e prov e that, without loss of generalit y , w e can restrict our atten tion to sufficient ly “regular” infinite compass s tr uctures, whic h can b e repr esen ted in doub le exp onen tial space with resp ect to | ϕ | . T o do that, we introdu ce the notion of p erio dic compass structur e. Definition 3.4. An infin ite compass structur e G = ( P ω , L ) is p e rio dic , with thr eshold e y 0 , p e rio d e y , and binding e g : { 0, ..., e y 0 + e y − 1 } → { 0, ..., e y 0 − 1 } , if the follo w ing conditions are satisfied: • for every e y 0 + e y 6 x < y , we hav e L ( x , y ) = L ( x − e y , y − e y ) , • for every 0 6 x < e y 0 + e y 6 y , we ha v e L ( x , y ) = L ( e g ( x ) , y − e y ) . Figure 3 giv es an example of a p erio dic compass structure (the arr o ws represent some relationships b et ween p oin ts induced by the bind ing fun ction e g ). Note that an y p erio dic compass structure G = ( P ω , L ) can b e fi nitely represent ed b y sp ecifying (i) its th reshold e y 0 , (ii) its p erio d e y , (iii) its bind ing e g , and (iv) the lab eling L r estricted to th e p ortion P e y 0 + e y − 1 of the domain. The follo wing th eorem leads immediately to a 2EXPSP A CE algorithm that d ecides whether a giv en AB B -form ula ϕ is satisfiable ov er infinite in terv al structures (the pro of is pro vided in [17]). Theorem 3.5. A n ABB -formula ϕ is satisfie d by an infinite interval structur e iff it is fe atur e d by a p erio dic c omp ass structur e with thr eshold e y 0 < 2 2 7 | ϕ | and p erio d e y < 2 | ϕ | · 2 2 7 | ϕ | · 2 2 7 | ϕ | . 606 A. MONT ANARI, G. PUPPIS, P . SALA, AND G. SCIA VICCO ... e y 0 e y 0 + e y e y 0 + 2 e y e g e g e g e g e g e g Figure 3: A p erio d ic compass structure with thr eshold e y 0 , p erio d e y , and bind ing e g . 4. Tigh t complexit y b ounds to t he satisfiabilit y problem for AB B In this section, we sho w that the satisfiabilit y p roblem for AB B interpreted o ver (either finite or in fi nite) inte rv al temp oral structures is E XPS P A CE-complete. The EXPSP ACE-hardness of the satisfiabilit y problem for AB B follo ws from a re- duction from the exp onential-c orridor tiling pr oblem , which is kno wn to b e EXPSP A CE- complete [19]. F ormally , an instance of the exp onent ial-corridor tiling problem is a tuple T = ( T , t ⊥ , t ⊤ , H , V , n ) consisting of a finite set T of tiles, a b ottom tile t ⊥ ∈ T , a top tile t ⊤ ∈ T , tw o binary relations H , V ov er T (sp ecifying the horizon tal and vertica l constrain ts), and a p ositiv e natural n umber n (r ep resen ted in un ary notation). The p roblem consists in deciding whether there exists a tiling f : N × { 0, ..., 2 n − 1 } → T of the in fi nite d iscrete corridor of heigh t 2 n , that asso ciates the tile t ⊥ (resp., t ⊤ ) with the b ottom (resp ., top) ro w of the corridor and that resp ects th e h orizon tal and ve rtical constrain ts H and V , namely , i) for every x ∈ N , we hav e f ( x , 0 ) = t ⊥ , ii) f or ev ery x ∈ N , we ha v e f ( x , 2 n − 1 ) = t ⊤ , iii) for every x ∈ N and every 0 6 y < 2 n , w e h av e f ( x , y ) H f ( x + 1, y ) , iv) for every x ∈ N and every 0 6 y < 2 n − 1, we ha v e f ( x , y ) V f ( x , y + 1 ) . The pro of of the follo win g lemma, which reduces the exp onen tial-corridor tiling problem to the satisfiabilit y pr oblem for ABB , can b e found in [17]. Intuitiv ely , such a r eduction exploits (i) the corresp ondence b et wee n the p oint s p = ( x , y ) inside th e infinite corridor N × { 0, ..., 2 n − 1 } and the int erv als of the form I p = [ y + 2 n x , y + 2 n x + 1 ] , (ii) | T | pr op ositional v ariables whic h represent the tiling fu n ction f , (iii) n add itional prop ositional v ariables whic h r epresen t (the binary expansion of ) the y -co ordinate of eac h ro w of the corr id or, and DECIDABI LITY OF THE INTER V AL TE MPORAL LOGIC ABB 607 (iv) the mo d al op erators h A i and h B i b y means of w hic h one can enf orce the lo cal constrains o ver the tiling fu nction f (as a matter of fact, this sh ows that the satisfiabilit y problem for the AB fragmen t is already hard for EXPSP A CE ). Lemma 4.1. Ther e is a p olynomial-time r e duction fr om the exp onential-c orridor tiling pr oblem to the satisfiability pr oblem for AB B . As for th e EXPSP ACE-co mpleteness, we claim that the existence of a compass stru cture G that features a give n form ula ϕ can b e d ecided b y verifying suitable local (and stronger) consistency conditions ov er all pairs of con tiguous r o ws. In fact, in ord er to c h ec k that these lo cal conditions hold b etw een tw o con tiguous ro ws y and y + 1, it is sufficient to store in to memory a b ound ed amount of information, namely , (i) a counter y that r anges ov er  1, ..., 2 2 7 | ϕ | + | ϕ | · 2 2 7 | ϕ |  , (ii) the tw o guessed shadings S and S ′ asso ciated with the rows y and y + 1, and (iii) a function g : S → S ′ that captures the h orizon tal alignmen t relation b et w een p oint s with an asso ciated atom from S and p oin ts w ith an associated atom from S ′ . This sh o ws that the satisfiabilit y problem for AB B can b e decided in exp onen tial space, as claimed by the follo wing lemma. F urther details ab out the decision pr o cedure, includ ing soundness and completeness pr o ofs, can b e found in [17]. Lemma 4.2. Ther e is an EXP SP ACE non-deterministic pr o c e dur e that de cides whether a given formula of A BB is satisfiable or not. Summing u p, we obtain the follo wing tigh t complexit y result. Theorem 4.3. The satisfiability pr oblem for AB B i nterpr ete d over (pr efixes of ) natur al numb ers is EXP SP ACE-c omplete. 5. Conclusions In this pap er, we pr o ve d that the satisfiabilit y problem for AB B interpreted o ve r (p r e- fixes of ) the natural num b ers is EXPSP ACE-complete . W e restricted our atten tion to these domains b ecause it is a common commitment in computer science. Moreo v er, this gav e us the p ossibilit y of expressing meaningful metric constraint s in a fairly natural wa y . Nev er- theless, w e b eliev e it p ossible to extend our r esults to the class of all linear orderings as w ell as to relev ant sub classes of it. 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