COMPOSITIONALITY AND REACHABILITY WITH CONDITIONS ON PATH LENGTHS

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1 compositionlity Interntionl Journl of Foundtions of Computer Science c World Scientific Pulishing Compny COMPOSITIONALITY AND REACHABILITY WITH CONDITIONS ON PATH LENGTHS INGO FELSCHER Lehrstuhl Informtik 7, RWTH Achen University, Ahornstrsse Achen, Germny felscher@utomt.rwth-chen.de felscher/ WOLFGANG THOMAS Lehrstuhl Informtik 7, RWTH Achen University, Ahornstrsse Achen, Germny thoms@utomt.rwth-chen.de thoms/ Received (Dy Month Yer) Accepted (Dy Month Yer) Communicted y (xxxxxxxxxx) In model-checking the systems under investigtion often rise in the form of products. The compositionl method, developed y Fefermn nd Vught in 1959, fits to this sitution nd cn e used to deduce the truth of formul in the product from informtion in the fctors. Building on erlier work of Wöhrle nd Thoms (2004), we study first-order logic with rechility predictes over finitely synchronized products (i.e. synchronized products with finite numer of synchroniztion trnsitions). We extend the rechility predictes y conditions on the length of the corresponding pths, formulted in Presurger rithmetic. For finitely synchronized products with these enhnced rechility predictes we prove composition theorem nd then show tht severe limittions exist for generlistions of this result. Keywords: Fefermn-Vught Theorem; composition theorem; finitely synchronized products; Presurger rithmetic. 1. Introduction The tsk of model-checking is to check whether, given trnsition system K tht is model of (technicl) system nd formul ϕ, modelling property of the system, if ϕ holds in K. A trnsition system is grph with sttes, directed edges (trnsitions) nd lels on the sttes. It is often the cse tht the trnsition system K is composed of smller ones which cn ct independently from ech other ut cn lso e synchronized vi selected trnsitions. An importnt question is whether we cn infer the truth vlue of the formul ϕ, interpreted in the product, from the truth vlues of the formuls in the components. 1

2 compositionlity 2 Ingo Felscher nd Wolfgng Thoms This is especilly of interest for infinite-stte model-checking. Here we study finitely mny infinite components, or infinite products of finite or infinite components. In the field of model theory, Fefermn nd Vught [4] showed tht, given direct (or generlized) product of structures nd first-order formul ϕ, the truth vlue of the formul ϕ in the product cn e deduced from certin formuls α 1,..., α m in the component structures nd formul β which is interpreted in the index structure nd descries in which of the components the formuls α 1,..., α m hold. Mny vrints of this result hve een developed, e.g. reduced or ultrproducts [1] nd ordered sums [17] insted of products. A good overview cn e found in [13]. Further references re [6, 10, 14, 19]. In (infinite-stte) model-checking, this composition theorem (lso clled Fefermn-Vught theorem) cn e used with three min differences: first, here we hve trnsition systems s specil kinds of structures, second, insted of direct or generlized products, the products we tke into ccount hve explicit synchronous nd synchronized ehviour, nd third, stronger logics thn first-order logic hve to e considered which cn express (t lest) certin rechility properties. Although some results hve een otined in the ppliction of the composition theorem in model-checking [20, 15, 16], the method hs some severe limittions, in prticulr regrding the logics which cn e used. An ovious exmple illustrting these limittions is given y the nturl numers together with the successor reltion. This structure hs decidle MSO theory, wheres the MSO theory of the synchronous product of two copies of the nturl numers which is the infinite grid is undecidle [18]. So the composition theorem fils for MSO logic. Unfortuntely, the composition theorem lredy fils for much weker logics. It hs een shown y Rinovich [16] tht it fils for frgments of computtionl tree logic (CTL), nmely for modl logic extended only y the CTL quntifier EG p (mening there exists pth on which glolly p holds) nd n synchronous product of two components. However, it holds for synchronous products nd the extension of modl logic y the CTL quntifier EFp [16]. For finite numer of trnsition systems Wöhrle nd Thoms showed in [20] tht the composition theorem lso holds for FO(R) first order logic extended y rechility predictes nd finitely synchronized products (which re synchronized products with finite numer of synchronized trnsitions). In preliminry work [5] preceding this pper we extended this result to cses with regulr rechility. FO(RegR) is the extension of FO logic y predictes Rech α (x, y) which express tht y is rechle from x vi pth which is lelled y word of the regulr expression α. For FO(Reg1R) we consider only lphets with one element which mounts to djoining rithmeticl constrints expressile in Presurger rithmetic. In this pper we extend this result to cover lso finitely synchronized products with n infinite numer of components. In is known [5] tht the composition theorem fils for ML(RegR) which corresponds to propositionl dynmic logic (PDL) without tests. In this pper we

3 compositionlity Compositionlity nd Rechility with Conditions on Pth Lengths 3 complement this result y showing tht lso the cse of PDL over one-element lphet leds to filure, if we llow tests. The pper is structured s follows: In the next section we formlly introduce the logics nd products we use. In Section 3 the clssicl composition theorem is shown for synchronized products of trnsition systems nd FO logic. In Section 4 we prove the min theorem the composition theorem for the logic FO(Reg1R). The proof is split into two prts: first, we show it for synchronous products nd then for finitelysynchronized products. Afterwrds, we prove tht the composition theorem lredy fils for ML(RegR) if we consider n synchronous product of two components nd llow two letter lphet for the lphet of α in the Rech α (x, y) formuls. Using this known result we show tht it lso fils for PDL with tests for one letter lphets. We conclude with summry nd some remrks on open questions. 2. Technicl Preliminries In this pper we use structured grphs, which we cll trnsition systems. Formlly trnsition system is defined s follows. Definition 1. A trnsition system is lelled grph K = (S, {R Σ}, {P v v V }) where S is set of sttes, R S S is trnsition reltion etween the sttes (for the letter Σ) nd P v S is predicte which holds for the sttes lelled y the letter v V. In the whole pper we use [m] for m N s n revition for the set {1,...,m} Logics In this section we define the logics needed in this pper. We will remind the reder of FO logic, then extend it y reltions tht express rechility vi ny pth, nd finlly extend it y reltions tht express rechility vi pths tht re lelled ccording to regulr expression. Definition 2. First order logic (FO logic) is defined s follows. Let x 1, x 2,... e vriles. For every n-ry reltion symol R, R(x 1,...,x n ) nd x 1 = x 2 re tomic FO formuls. If ϕ, ψ re FO formuls, then ϕ, ϕ ψ, ϕ ψ, x n ϕ(x 1,..., x n 1, x n ) nd x n ϕ(x 1,..., x n 1, x n ) re FO formuls with the usul menings. For trnsition system K = (S, {R Σ}, {P v v V }) the tomic formuls re of the form R (x, y), P v (x) nd x = y. Definition 3. FO logic with rechility (FO(R) logic) is defined s the extension of FO logic y tomic reltions Rech Σ (x, y) for Σ Σ with the mening from x there exists pth to y which is lelled y letters from the set Σ. We define FO logic with regulr rechility (FO(RegR) logic) s the extension of FO logic with reltions Rech β (x, y) for regulr expressions β Reg(Σ) with the

4 compositionlity 4 Ingo Felscher nd Wolfgng Thoms mening from x there exists pth to y which is lelled y word in the lnguge of the regulr expression β. FO(Reg1R) is defined nlogously to FO(RegR) logic, ut the regulr expressions re uilt only over one-element lphet. Let us dd some remrks: First, FO(RegR) is defined like first-order propositionl dynmic logic (FO-PDL) without tests. See [8] for detils on PDL. Second, FO(Reg1R) logic corresponds to FO logic extended y tomic reltions Pth r,k (x, y) which express tht there exists pth from x to y which hs length which is divisile y k with reminder r. This is shown in the following lemm: Lemm 4. Every FO(Reg1R) formul Rech α where α is regulr expression over n one-element lphet cn e trnslted into FO formul with dditionl reltions Pth r,k (x, y). Proof. As the proof is simple consequence of known results, we only give sketch. First we hve to show tht every FO(Reg1R) formul Rech α is equivlent to FO formul with dditionl reltions which express rechility where the pth length is in semiliner set. Let L e the lnguge defined y the regulr expression α. It is well-known tht every regulr lnguge hs semiliner imge [9]. Using the Prikh mpping, the semiliner imge of L cn e written s: ψ(l) = i [m] M i with M i = {k i0 + n i1 k i1 + + n iri k iri n ij 0 for 1 j r i } with k ij N ecuse the lphet hs only one letter. Using the decidility of Presurger rithmetic nd the fct tht the fmily of Presurger sets in N is identicl to the fmily of semiliner sets in N [9], we cn then show tht ech of these M i is union of finite set R nd {z = r + n k z > mx(r), n N}, where k N nd mx(r) is the mximum of the elements in R. So FO logic with dditionl reltions Pth r,k (x, y) suffices. Now, we define MSO logic nd extensions of MSO logic y formuls which count the numer of elements of set vrile modulo numer k, known s counting MSO [2]. Definition 5. The mondic second order logic (MSO logic) is the extension of FO logic y vriles over sets nd quntifiction over these vriles. Every FO formul is MSO formul. Let x 1,...,x n e vriles over elements nd Y 1,...,Y k e vriles over sets then x l Y m (lso written s Y m (x l )), Y k ϕ(x 1,...,x n, Y 1,..., Y k ) nd Y k ϕ(x 1,...,x n, Y 1,..., Y k ) re MSO formuls. For ordered structures let MSO(<) logic denote the extension of MSO logic y formuls x < y. Further, let counting MSO logic (CMSO logic) e MSO logic extended y formuls Crd j,k (Y ) which express tht the numer of elements in the set Y is divisile y k with reminder j. Note tht over linerly ordered structures MSO(<) logic is t lest s expressive s CMSO logic [7, 12], i.e. for every CMSO formul there exists n equivlent

5 compositionlity Compositionlity nd Rechility with Conditions on Pth Lengths 5 MSO(<) formul. Tke e.g. the CMSO formul Crd 0,2 (Y ): it cn e expressed in MSO(<) logic y the formul: x y[y (x) Y (y) Y 0 [(first Y (x) Y 0 (x)) (lst Y (x) Y 0 (x)) (succ Y (x, y) (Y 0 (x) Y 0 (y)))]] where first Y (x) is the formul z(y (z) x z), lst Y is defined nlogously nd succ Y (x, y) is the formul z(y (z) x < z z < y) Products We wnt to define product over trnsition systems K i with i I for n index structure Ind = (I, σ Ind ) with countle index set I. By s we will denote stte of the product nd y s[i] the stte of the i-th component of s. We im t possiility to descrie trnsition reltions in the product y conditions in the components. We use generl description for k-ry reltion y tuple α 1,...,α n of FO formuls nd MSO formul β tht defines (in the structure Ind) in which components the FO formuls hold. We cll the tuple (α 1,..., α k ; β) MSO profile. Definition 6. A MSO profile (for k-ry reltion R) is tuple α 1,..., α m ; β(x 1,...,X m ) where m N nd α l for l [m] re k-ry FO formuls nd β(x 1,...,X m ) is MSO formul. The formuls α l re interpreted in the components nd the MSO formul in the index structure of the product. It descries (vi condition over the sets X 1,..., X m ) in which components the corresponding formuls α 1,..., α m hold. A MSO profile defines k-ry reltion R in the product y ( s 1,..., s k ) R iff (Ind, I 1,...,I m ) β(x 1,..., X m ) where I l := {i I (K i, s 1 [i],..., s k [i]) α l (y 1,..., y k )} for l [m]. Let P i e given (unry) predicte in the signture of the component K i. Vi MSO profile we define predicte P i in the product such tht P i ( s) holds t stte s if P i (s i ) holds in K i, where s i := s[i]. Anlogously, we define predicte P which holds t stte s in the product, if in some component K i the predicte P i (s i ) holds. Afterwrds, we will define synchronous trnsition reltions in the product. Exmple 7. Let K e product over trnsition systems K i with i I nd let P i predicte of K i. We define the predicte P i in the product y the MSO profile P(x); β(x 1 ) with β(x 1 ) = X 1 (i). Then, P i holds t ll sttes of the product where the predicte P holds t the stte in the i-th component. Further, we define the predicte P y P(x); β(x 1 ) with β(x 1 ) = i X 1 (i). It holds t ll sttes of the product where the predicte P holds t the corresponding stte of ny component. For trnsition reltion R in K i the MSO profile R (x, y), x = y; β(x 1, X 2 ) with β(x 1, X 2 ) := i (X 1 (i) j(j i X 2 (j))) defines the synchronous trnsition reltion R in the product. (An -trnsition exists in the product, if it exists in exctly one component.)

6 compositionlity 6 Ingo Felscher nd Wolfgng Thoms We will now introduce the synchronized product of the components K i. In the product we distinguish explicitly etween synchronous trnsitions where trnsition is tken independently of the other components nd synchronized trnsitions which re used to model synchroniztion etween the trnsitions of some components. For the synchronous trnsitions we cll the trnsitions which re used in the components locl. We use two disjoint lphets in ech component. The first one is used for the locl trnsitions nd the second one for the definition of the synchronized trnsitions. Definition 8. Let Ind = (I, σ Ind ) e n index structure nd let K i = (S i, {R i Σ l } {R c c Σ s }, {Pv i v V }) (i I) e trnsition systems, where the lphet Σ l is used for locl trnsitions with synchronous ehviour in the product nd the lphet Σ s is used for the definition of synchronized trnsitions. We ssume tht the lphets Σ l nd Σ s re disjoint. Further, let C e n lphet for the synchronized trnsitions in the product. We ssign to ech c C tuple τ c := (c 1,...,c m ) with m N nd c j Σ s for j [m] nd MSO formul β τc (C 1,..., C m ), which specifies condition in which component which of the c 1,..., c m -trnsitions hve to e tken. The synchronized product of the components K i (i I) is defined y the following trnsition system K = ( S, { R Σ l } { R c c C}, { P v i v V }): The stte set S is the product of the component stte sets: S := i I S i. (We write s[i] for the stte of the i-th component of s S.) The synchronous trnsition reltion R is defined y the MSO profile R (x, y), x = y; β(x 1, X 2 ) with β(x 1, X 2 ) := i (X 1 (i) j i X 2 (j)). The synchronized trnsition reltion R c with τ c := (c 1,..., c m ) is defined y the MSO profile R c1 (x, y),..., R cm (x, y), x = y; β(x 1,..., X m, X m+1 ) where β(x 1,...,X m+1 ) = disjoint(x 1,..., X m+1 ) i(( j [m] X j(i)) X m+1 (i)) β τc (X 1,...,X m )] nd disjoint(x 1,...,X m+1 ) is the MSO formul j,k [m+1],j k ( i (X j(i) X k (i)). The predicte P i v is the set { s s[i] P i v }. A synchronized product is clled finitely synchronized if the trnsition reltion R c is finite for every symol c C. An synchronous product is synchronized product without synchronized trnsitions. 3. Composition method In this section we introduce the composition method. Before we stte the composition theorem, we need lemm, which llows us to construct specil MSO profiles out of existing ones, such tht the disjunction over certin FO formuls is stisfied, ut the conjunction over ny two of the FO formuls does not hold. This lemm Note tht β(x 1,..., X m+1 ) is formlly not n MSO formul, ut it cn e trnslted into one, ecuse the sets [m],[m + 1] which re used in the conjunctions re finite.

7 compositionlity Compositionlity nd Rechility with Conditions on Pth Lengths 7 will e needed in the proof of the composition theorem for the negtion step. The construction is nlogous to [16]. Lemm 9. For every MSO profile α 1,...,α m ; β(x 1,..., X m ) there exists MSO profile α 1,..., α p; β (X 1,..., X p) tht descries the sme reltion, where p i=1 α i is stisfied nd α i α j is not stisfied for ll i, j [p]. Proof. We define for every suset H of [m] formul which sttes tht exctly the formuls α i with indices in H hold: α H := i H α i i H α i. Note tht there re 2 m of such formuls α H. We oserve tht H [m] α H holds, ecuse there exists one set H [m] such tht for ll i [m]: i H α i stisfied. For two different sets H 1, H 2 [m] there exists t lest one index i H 1 nd i H 2 (or vice vers), so α H 1 α H 2 contins the conjunctive elements α i nd α i which implies tht α H 1 α H 2 is unstisfile. It holds tht {H i H} α H = α i, ecuse {H i H} ( j H α j j H α j) = j [m],j i ( {H j H} α j {H j H} α j) {H i H} α i = α i. Let h : [2 m ] Pot([m]) e ijection. Now the MSO profile α 1,..., α 2 m; β (X 1,..., X 2 m) with the MSO formul β (X 1,..., X 2 m) = X 1,..., X n n i:=1 ( t : X i (t) {k i h(k)} X k (t)) β(x 1,...,X m ) descries the sme reltion s the given MSO profile. Now we show the composition theorem for synchronized products nd FO logic. We shll distinguish only etween synchronous nd synchronized trnsitions, ecuse we need this distinction in Section 4. We wnt to mention tht Theorem 10 cn e strengthened to generlized products, where we llow ny reltion R(x, y) in the product which cn e defined y MSO profile. The following proof is nlogous to the proof of the composition theorem for generlized products nd modl logic in [16]. Theorem 10. Composition theorem for synchronized products nd FO logic Let Ind = (I, σ I ) e n index set with some signture σ I. For every FO formul ϕ( x 1,..., x r ) we cn effectively compute MSO profile α 1,..., α m ; β(z 1,..., Z m ) with FO formuls α j (x 1,...,x r ) (j [m]) nd MSO formul β(z 1,..., Z n ) interpreted in the index structure Ind, such tht for every synchronized product K := i I K i = ( S, { R Σ l } { R c c C}, { P i v v V }) of trnsition systems K i = (S i, {R i Σ l } {R i c c Σ i s}, {P i v v V }) (i I), where ϕ( x 1,..., x r ) cn e interpreted in the product nd the formuls α j (j [m]) cn e interpreted in the components nd for every stte s 1,..., s r : ( i I K i, s 1,..., s r ) = ϕ( x 1,..., x r ) Ind = β(i 1,..., I n ) with I k = {i I (K i, s 1 [i],..., s r [i]) = α k (x 1,..., x r )} for k [n]. Proof. We use structurl induction to construct MSO profiles for the formuls interpreted in the product. In the induction strt, we consider the tomic formuls.

8 compositionlity 8 Ingo Felscher nd Wolfgng Thoms For the predicte P i v ( x) we tke the MSO profile P v(x); β(x 1 ) with β(x 1 ) = X 1 (i). For R ( x, ȳ) nd R c ( x, ȳ) the MSO profiles re given y the definition of the product. For x = ȳ we tke the MSO profile x = y; β(x 1 ) with β(x 1 ) = ix 1 (i). For the induction step we hve to consider negtion, disjunction nd existentil quntifiction. For ϕ( x 1,..., x r ) = ψ( x 1,..., x r ) let α 1,..., α m ; β(x 1,..., X m ) e the MSO profile ssigned to ψ, then we ssign the MSO profile α 1,...,α m ; β(x 1,...,X m ) to ϕ. For ϕ( x 1,..., x r ) = ψ( x 1,..., x r ) ψ ( x 1,..., x r ) let α 1,..., α m ; β(x 1,...,X m ) nd α 1,...,α m ; β (X 1,..., X m ) e the MSO profiles ssigned to ψ respectively ψ. W.l.o.g. we cn ssume tht the common vriles of ψ nd ψ re the vriles x 1,..., x t with 0 t m. Then we ssign the MSO profile α 1,...,α t, α t+1,..., α m, α t+1,..., α m ; β (Y 1,..., Y m+m t) to ϕ( x 1,..., x r ) where β (Y 1,..., Y m+m t) is the formul β(y 1,...,Y m ) β (Y 1,..., Y t, Y m+1,..., Y m+m t). For ϕ( x 1,..., x r ) = x r+1 ψ( x 1,..., x r, x r+1 ) let the MSO profile for ψ e α 1,..., α m ; β(x 1,..., X m ). By Lemm 9 we cn ssume tht m i=1 α i is stisfied nd α i α j is not stisfied for ll i, j [m]. We cn ssign to ϕ the following MSO profile: x r+1 α 1 (x 1,..., x r+1 ),..., x r+1 α m (x 1 (,..., x r+1 ); β (Y 1,...,Y m ) where β (Y 1,..., Y m ) is defined y Z 1... Z m k [m] Z k Y k k,k [m] Z k Z k = β(z 1,..., Z m ) ). k k For the existentil quntifiction we prove correctness nd completeness. For the completeness let (K, s 1,..., s r ) x r+1 ψ( x 1,..., x r+1 ), then there exists stte s such tht (K, s 1,..., s r, s) ψ( x 1,..., x r+1 ). By induction hypothesis MSO profile α 1 (x 1,...,x r+1 ),...,α m (x 1,...,x r+1 ); β(x 1..., X m ) is ssigned to ψ nd Ind β(x 1,...,X m ) holds. Rememer, tht we wnt to prove tht β (Y 1,..., Y m ) holds. Actully, we prove tht the X k re in fct the sets Z k in β for k [m]. Becuse of X k = {i I K i, s 1 [i],..., s r [i], s[i] α k (x 1,..., x r+1 )} nd Y k = {i I K i, s 1 [i],..., s r+1 [i] x r+1 α k (x 1,..., x r+1 )}, X k Y k holds nd y Lemm 9 we ssumed tht α k α k is not stisfied, so X k X k = for ll k, k [m]. Thus, Ind β (Y 1,..., Y m ). For the correctness we hve given the MSO profile x r+1 α 1 (x 1,..., x r+1 ),..., x r+1 α 1 (x 1,...,x r+1 ); β (Y 1,..., Y m ) nd Ind β (Y 1,..., Y m ) with Y k = {i I (K i, s 1 [i],..., s r [i]) x r+1 ψ(x 1,..., x r+1 )}. From Z k Y k follows tht for ll i Z k (k [m]) (K i, s 1 [i],..., s r [i]) x r+1 ψ(x 1,..., x r+1 ) holds. Thus, for ll i Z k (k [m]) there exists stte s i S i, such tht (K i, s 1 [i],..., s r [i], s i ) ψ(x 1,..., x r+1 ). Becuse of Z k Z k = nd m k:=1 Z k = I there exists exctly one stte s i S i for ll i I, which define ll together stte s of the product with The formul S m k:=1 Z k = I holds ecuse of Ind = β(z 1,..., Z k ) nd Lemm 9

9 compositionlity Compositionlity nd Rechility with Conditions on Pth Lengths 9 s[i] := s i. By pplying the induction hypothesis (K, s 1,..., s r, s) ψ( x 1,..., x r+1 ) nd thus (K, s 1,..., s r ) x r+1 ψ( x 1,..., x r+1 ) holds. Corollry 11. If the composition theorem holds for specific logic nd product nd if the model-checking prolem in the components is decidle, then the modelchecking prolem is lso decidle in the product. Anlogous corollries follow from the min theorems in the next section, however we will not stte them explicitly. 4. Min Results In this section, we show our min results. First, we show tht, if we consider synchronous products, the composition theorem lso holds for the logic FO(Reg1R), which llows us to express modulo counting over pth lengths. Then, we generlize this result to finitely synchronized products. Theorem 12. Composition theorem for synchronous products nd FO(Reg1R) Let ϕ( x 1,..., x r ) e FO(Reg1R) formul. Then there exist n effectively computle CMSO-profile α 1,...,α m ; β(z 1,...,Z m ) where α i (x 1,...,x r ) (i [m]) re FO(Reg1R) formuls which re interpreted in the components, nd CMSO formul β(z 1,...,Z n ), interpreted in the index structure Ind, such tht for every synchronous product K := i I K i = (S, { R Σ l }, { P v i v V }) of trnsition systems K i = (S i, {R i Σ l}, {Pv i v V }) (i I), where ϕ( x 1,..., x r ) cn e interpreted in the product nd the formuls α j (j [m]) cn e interpreted in the components nd for every stte s 1,..., s r : ( i I K i, s 1,..., s r ) = ϕ( x 1,..., x r ) Ind = β(i 1,..., I n ) with I k = {i I (K i, s 1 [i],..., s r [i]) = α k (x 1,..., x r )} for k [n]. Proof. The proof is n extension of the proof of Theorem 10. The cses for the induction step (disjunction, negtion nd existentil quntifiction) re nlogous ecuse synchronous products re specil cses of synchronized products. We hve to consider the dditionl tomic cse Rech α ( x, ȳ) for regulr expression α Reg({}) with {} Σ. Becuse of Lemm 4, insted of Rech α ( x, ȳ) we only hve to consider formuls Pth l,k ( x, ȳ). Note tht for the existence of pth in the product from x to ȳ with pth length l (mod k) there must exist finite numer of components, such tht the sum of the pth segments in these components is l (mod k). Let Y k denote the numer of elements of Y modulo k. For the tomic formul Pth p,k ( x, ȳ) we use the CMSO profile Pth 0,k (x, y),..., Pth k 1,k (x, y);

10 compositionlity 10 Ingo Felscher nd Wolfgng Thoms β(x 0,...,X k 1 ) where the formul β(x 0,..., X k 1 ) is defined s follows: k 1 β(x 0,...,X k 1 ) = Y 0... Y k 1 ( Y j X j disjoint(y 0,..., Y k 1 ) k 1 j:=1 k 1 finite c (Y j ) ( i( j:=1 j:=0 nd the formul γ(y 1,..., Y k 1 ) expresses tht Y j (i)) Y 0 (i)) γ(y 1,...,Y k 1 )) Y 1 k + 2 Y 2 k + + (k 1) Y k 1 k = p (mod k). (3) The formul β(x 0,...,X k 1 ) sttes tht there exists finite numer of components where the pth segments re not divisile y k without reminder nd for ll other components the pth segments re divisile y k without reminder (which includes the cse tht the pth is empty (x = y)), descried y the FO formul Pth 0,k (x, y). The formul γ(y 1,..., Y k 1 ) ensures tht for the components where the pth segments re divisile y k with reminder different from 0, the pth segments sum up to p (mod k). Note tht every component, in which the pth segment hs length j (mod k) dds j to the length of the pth in the product (j [k 1]), so we need j Y j k s summnd in (3). For ll j [k 1] the vlues Y j k re etween 0 nd k 1, so there re only finitely mny solutions of the eqution (3). We denote the set of solutions y L. Formlly L is defined y: L := {(l 1,..., l k 1 ) l (k 1) l k 1 = p (mod k) j : l j [k 1]} Then the formul γ(y 1,...,Y k ) cn e written s (l 1,...,l k 1 ) L (Crd l 1,k(Y 1 )... Crd lk 1,k(Y k 1 )). Thus, β(x 0,...,X k 1 ) is defined completely. Let us now generlize the result from Theorem 12 to finitely synchronized products. Aprt from the restriction tht there hs to e finite numer of synchronized trnsitions, the proof now depends on the ctul product or to e more specific on the synchronized trnsitions. However the method is uniform for ll finitely synchronized products. Theorem 13. Composition theorem for finitely synchronized products nd FO(Reg1R) Given finitely synchronized product i I K i = ( S, { R Σ l } { R c c C}, { P i v v V }) of components K i = (S i, {R i Σ l} {R c c Σ s }, {P i v v V }) nd FO(Reg1R) formul ϕ( x 1,..., x r ), which is interpreted in the product, there exists n effectively computle CMSO-profile α 1,..., α m ; β(z 1,..., Z m ) with FO c The formul finite(y j ) cn e expressed y W k 1 r:=0 Crd r,k(y j ).

11 compositionlity Compositionlity nd Rechility with Conditions on Pth Lengths 11 formuls α i (x 1,..., x r ) (i [m]) interpreted in the components, nd CMSO formul β(z 1,..., Z n ) interpreted in the index structure Ind such tht for ll sttes s 1,..., s r : ( i I K i, s 1,..., s r ) = ϕ( x 1,..., x r ) Ind = β(i 1,..., I n ) with I k = {i I (K i, s 1 [i],..., s r [i]) = α k (x 1,..., x r )} for k [n]. Proof. As in Theorem 12 we only hve to consider the dditionl tomic cse Pth p,k ( x, ȳ). The proof given here is n dption of the proof in [5]. It depends on the numer of synchronized trnsitions in the product. We denote the synchronized trnsitions y r 1,...,r q. First, we descrie the proof ide: We will define formuls which descrie pths using ny synchronous trnsition ut (inductively) only the synchronized trnsitions up to r m (m q). This definition works nlogously to the construction of regulr expressions from finite utomton in the proof of Kleene s Theorem [11]. Then, we show tht we cn limit the usge of ech of these synchronized trnsitions to finite numer of times, ecuse we re only interested in the pth length modulo k. Finlly, we use formuls like the ones descried ove together with this restriction t the sme time in ll components to find the cses when the pth segments in the components sum up to pth length modulo k in the product. We introduce new trnsitions in the product lelled with new symol, such tht s s, if for ny lel Σ l s s. For every synchronized trnsition r m (m [q]) we define s m,t m to e the sttes of r m = (s m, t m ). We define H := {x, y} m [q] {s m, t m }. (i) We construct regulr expressions Rh,l m with h, l H which descrie ll pths from stte h to stte l which use only the synchronized trnsitions r 1,..., r m (m q) nd ny synchronous trnsitions. By induction we finlly get the set R q h,l which descries ll pths. For ny synchronized trnsition (s, t) we define the lel of the trnsition y L s,t := c if s c t. Induction strt: The regulr expression Rh,l 0 descries tht stte l is rechle from stte h vi synchronous trnsitions. We define Rh,l 0 := for h l nd Rh,h 0 := ǫ. For the induction step (m 1 m) we tke the regulr expression: R m h,l := R m 1 h,l + R m 1 h,s m L sm,t m (Rt m 1 m,s m L sm,t m ) R m 1 t m,l We explin this definition for m = 1: Either l is rechle from h y synchronous trnsitions (R 0 h,l ) or the pth uses the synchronized trnsition r 1 nd is structured s follows: h s 1 c t1 ( s 1 c t1 ) l where the rrow descries the usge of synchronous trnsitions. For m > 1 the rrow descries the usge of synchronous trnsitions nd synchronized trnsitions up to r m 1. (ii) We prove tht it is sufficient to replce every str in R m h,l y k (with α k := α 0 α k ) to get ll possile pth lengths modulo k. Tke the synchronized trnsition s m t m. If there exists pth from t m ck to s m with pth length

12 compositionlity 12 Ingo Felscher nd Wolfgng Thoms l (mod k) with l {0,..., k 1} nd we repet the loop s m t m s m i-times, the pth length is i (l + 1) (mod k). We oserve tht {i (l + 1) (mod k) i N} is the sme set s {i (l + 1) (mod k) 0 i k}, so the restriction to k is justified. (iii) We now use formuls like the ones defined in (i) with the restriction descried in (ii) in every component. To rgue out the length of the segments of the pth in the components, we replce the regulr expressions R 0 h,l = for the synchronous trnsitions y vriles X h,l. To chieve pth in the product which hs the length p (mod k) we hve to ensure the following: for c-lelled trnsition the corresponding trnsitions in the components lelled with c 1,...,c n re tken t the sme time the sum of the numer of the synchronous trnsitions in the components plus the numer of used synchronized trnsitions is p (mod k) Let J = (j 1,..., j m ) with 0 j g k (g [m]) e tuple of indices for the repetitions of the loops of the synchronized trnsitions r 1,...,r m (m q). We define for ech component K i (i I) regulr expression R h,l ( L(i), X(i), J). Let the tuple X(i) := (X t1,s 1 (i),..., X x,y (i)). These vriles will e used for the synchronous pths etween the sttes t e /x nd s f /y (with e, f [q]) in H. Further let the tuple L(i) := (L s1,t 1 (i),..., L sq,t q (i)) where L sm,t m (i) is the lel which is induced c y the synchronized trnsition r m = s m tm in the component K i. Formlly let R c e defined y R c1 (x, y),..., R cn (x, y), x = y; β(x 1,...,X n, X n+1 ), then L sm,t m (i) = c j, if i X j for j [n], respectively L sm,t m (i) = ǫ, if i X n+1. The sets R h,l ( L(i), X(i), J) re generlistion of the sets Rh,l m. For the induction strt let R h,l ( L(i), X(i), ) = X h,l (i) if h l nd R h,h ( L(i), X(i), ) = ǫ. Induction step (m 1 m): Let J = (j 1,..., j m ) nd J = (j 1,..., j m 1 ). Then R h,l ( L(i), X(i), J) = ( ) R h,l ( L(i), X(i), J ), if j m = 0 ( ) = R h,sm ( L(i), X(i), J ) L sm,t m (i) [R tm,s m ( L(i), X(i), J ) L sm,t m (i)] jm 1 R tm,l( L(i), X(i), J ), if 1 j m k Finlly we get R x,y ( L(i), X(i), J) with J := (j 1,...,j q ) which descries ll pths in the component K i from x to y with the repetitions d j 1,...,j q of the synchronized trnsitions r 1,...,r q. We now look t fixed tuple (j 1,..., j q ) nd fixed component K i : for every numer θ (0 θ k) there re only finitely mny possile ssignments of the vriles X t1,s 1 (i),..., X x,y (i) y regulr expressions f ( k ) with 0 f k 1 such tht the regulr expression R x,y ( L(i), X(i), J) with these ssignments descries pths of the length θ (mod k) in the component K i. d In fct r q is repeted j q times, r q 1 is repeted j q j q 1 times n so on.

13 compositionlity Compositionlity nd Rechility with Conditions on Pth Lengths 13 Let α θ ( L(i)) (0 θ k) e the disjunction over ll R x,y ( L(i), X(i), J) with ssignments for X(i) tht descrie pth length θ (mod k). e With these α θ we cn uild CMSO profile..., α θ,...;β (..., X θ,... ), where we comine the formul β(x 0,...,X k 1 ) from the CMSO profile Pth 0,k (x, y),..., Pth k 1,k (x, y); β(x 0,...,X k 1 ) in the proof of Theorem 12 nd the MSO profiles for the definition of the synchronized trnsitions. Then we hve constructed CMSO profile which expresses ll possile pths in the components, such tht the pth length is divisile y k with reminder q for one tuple (j 1,..., j q ) of repetitions of the synchronized trnsition r 1,...,r q. We cn now cpture ll (finitely mny) tuples (j 1,...,j q ) (with 0 j g k for g [q]) y disjunction over the CMSO profiles for these tuples. This gives us CMSO profile for the formul Pth p,k ( x, ȳ) in the product. 5. Limits In this section we will show limittions of the composition theorem. First we will consider the logic FO(RegR) insted of FO(Reg1R) for finitely synchronized products nd fterwrds we will consider propositionl dynmic logic over one-element lphet, where we llow tests. We will prove tht in oth cses the composition theorem lredy fils for the specil cse of n synchronous product of two trnsition systems. Afterwrds we will show tht for the logic FO(Reg1R) the composition theorem fils if we consider synchronized products where we llow n infinite numer of synchroniztion trnsitions. To prove the filure of the composition theorem, we use proof schem developed in [16] which sttes tht the composition theorem fils for formul ψ if we cn find two infinite sequences of trnsition systems such tht the product of two trnsition systems with the sme index stisfies ψ wheres the product of two trnsition systems with different indices does not stisfy ψ: Theorem 14. [16] Given formul ψ nd two infinite sequences of trnsition systems {C k k N} nd {D l l N}, which ll hve stte in common: s S(C k ) nd s D(D l ), where S(C k ) nd S(D l ) re the stte sets of C k nd D l. Further let the product C k nd D l e defined in common wy for k, l N, then the composition theorem fils, if k N : C k D k ψ nd k, l N with k l : C k D l ψ. Using Theorem 14 we show tht the composition theorem lredy fils for n synchronous product of two components nd model logic extended y regulr rechility over two letter lphet. Following [5] we use the formul e Note tht α θ ( L(i)) is depended on the trnsitions (lelled with c j or with ǫ) of the component K i which re used for the synchronized trnsitions in the product. However, ecuse the definition of synchronized trnsition uses only finite numer of different lels for the trnsitions in the components nd there re only finitely mny synchronized trnsitions in the product, there cn e only finitely mny different formuls α θ ( L(i)) for ll components K i together.

14 compositionlity 14 Ingo Felscher nd Wolfgng Thoms () (p 1 p 2 ), which expresses tht there exists pth to stte where (p 1 p 2 ) holds nd tht this pth is lelled with nd in lterntion nd ends with -trnsition. Theorem 15. The composition theorem fils for ML(RegR) (nd for hence FO(RegR)) nd synchronous products. Proof. Let the trnsition systems C k nd D l for k, l 2 e defined s C k := ([k], R, P 1 ) nd D l := ([l], R, P 2 ) where R nd R re the successor reltions up to k, respectively l, nd P 1 := {k} nd P 2 := {l}. As synchronous product C k D l we get finite grid from the stte (1, 1) up to the (k, l). The formul ϕ := p 1 p 2 holds only t the stte (k, l). In Figure 1 the synchronous products C 4 D 4 nd C 4 D 6 re shown nd the stte where ϕ holds is mrked s ox. Figure 1. Asynchronous products C 4 D 4 nd C 4 D 6 We wnt to determine the truth vlue of the formul ψ := () ϕ t the stte (1, 1). We oserve tht for every k 2 in C k D k there exists the pth (1, 1) (1, 2) (2, 2)... (k 1, k) (k, k), so the formul ψ holds in C k D k. This is shown on the left side of Figure 1 for k = 4. If we consider the product C k D l for l > k we oserve tht the sme pth exists up to the stte (k, k), ut we cnnot extend it with nd in lterntion (nd ending with ), ecuse we hve lredy reched the lst stte k of the component C k, so there exist not further -trnsitions from (k, k) onwrds. This is shown in Figure 1 on the right side for k = 4 nd l = 6. By n nlogous rgument, we get the sme result for l < k. So, we hve proven tht k N : C k D k ψ nd k, l N with k l : C k D l ψ. By pplying Theorem 14, we hve shown tht the composition theorem fils for ML(RegR) nd synchronous products. Becuse synchronous products re finitely synchronized products without ny synchroniztion trnsition, we get s n immedite consequence tht the composition theorem fils for FO(RegR) nd finitely synchronized products.

15 compositionlity Compositionlity nd Rechility with Conditions on Pth Lengths 15 As mentioned ove ML(RegR) corresponds to PDL without tests. As we hve just seen tht the step from ML(Reg1R) to ML(RegR) leds to the filure of the composition theorem, one my sk wht hppens if we extend ML(Reg1R) to PDL with tests over one-element lphet. We will now see tht this extension lso results in filure of the composition theorem for synchronous products. Theorem 16. The composition theorem fils for PDL over one-element lphet with tests nd synchronous products. The following proof involves the sme trnsition system s defined in [16], used there for showing the filure of the composition theorem for modl logic extended y the CTL quntifier EG nd lso synchronous products. Proof. For k N let C k = ([3 k], R, Q 1 0, Q1 1, Q1 2, P 1 ), where R is the successor reltion, P 1 := {k} nd the Q 1 r re predictes which hold t the sttes divisile y 3 with reminder r for 0 r 2 (Q 1 r := {s [3 k] s = r mod 3}). For the trnsition system D k = ([3 k], R, Q 2 0, Q2 1, Q2 2, P 2 ) the predictes nd reltions re defined nlogously. As synchronous product C k D l we get gin the finite grid from the stte (1, 1) up to the stte (k, l). We use the formul ϕ := p 1 p 2 which holds t the stte (k, l) nd the formul ϑ := [p 1 0 (p 2 0 p1 2 )] [p 1 1 (p 2 1 p 2 2)] [p 1 2 (p 2 2 p 2 0)]. The formul ϑ holds t the sttes which re mrked in lck in Figure 2. Figure 2. Asynchronous products C 2 D 2 nd C 2 D 3 Now consider the PDL formul ψ := (ϑ?) ϕ. This formul descries tht there exists pth to the lst stte, such tht fter ech trnsition of the pth the formul ϑ holds, which mens tht ll sttes of the pth hve to e mrked sttes. Note tht the formul ψ ctully simultes the CTL formul E(ϑUϕ). We oserve tht from stte (1, 1) there exists only one pth such tht ll sttes re mrked nd for k = l this pth reches the stte (3 k, 3 k), ut for l > k this pth does not rech the stte (3 k, 3 l) ecuse from the stte (3 k, 3 k + 1) there is no

16 compositionlity 16 Ingo Felscher nd Wolfgng Thoms outgoing trnsition to mrked stte. This is shown in Figure 2 for k = 2 nd l = 3 on the right side. Anlogously for l < k the stte (3 k, 3 l) is not rechle vi mrked pth. We conclude tht for ll k (C k D k ) ψ nd for ll k l (C k D l ) ψ. So we meet the requirements for Theorem 14 nd thus the composition theorem fils for synchronous products nd PDL over one-element lphet, if we llow tests. We hve shown tht for finitely synchronized products oth the extensions of FO(Reg1R) to either two letter lphets for the regulr expressions or to tests included in the regulr expressions led to filure of the composition theorem. If we remin in the logic FO(Reg1R), ut consider synchronized products where we llow n infinite numer of synchronized trnsitions, the composition theorem lso fils. In [16] it ws shown tht the composition theorem lredy fils for the direct product of two components nd ML(R). As direct products re specil kinds of synchronized products without synchronous trnsitions nd ML(R) formuls cn e written s FO(Reg1R) formuls, the filure of the composition theorem follows immeditely. 6. Conclusion In this pper we considered new cses for the composition theorem in the re of (infinite-stte) model checking. In contrst to the clssicl composition theorem we looked t logics which cn express t lest certin rechility properties. In prticulr, we looked t n extension of FO logic y reltions tht express rechility vi lelled pths corresponding to regulr expressions. The specil cse of oneelement lphet for the regulr expressions is equivlent to modulo counting over pth lengths. As products we considered synchronous nd finitely synchronized products of ( possily infinite numer of) trnsition systems f. We proved tht for oth types of products the composition theorem holds for the specil cse of modulo counting over pth lengths. (For finitely synchronized products, we needed the ctul trnsition system defined y the product or, to e more specific, the finitely mny synchronized trnsitions.) We lso showed tht these results cnnot e improved in severl spects. First, the composition theorem fils if we omit the strong restriction to finite numer of synchronized trnsitions, ecuse it lredy fils for synchronized product of two components nd FO(R)/ML(R) [16]. Second, it lredy fils if we look t n synchronous product of two components nd llow rechility vi pths descried y regulr expressions with either n lphet of t lest two letters or n lphet of only one letter, ut use PDL with tests. We wnt to mention some relted work nd open questions. First, the complexity of the composition theorem is quite prohiitive. For FO logic, it ws proven in [3] f In contrst to [5] where we considered only products of finite numer of components.

17 compositionlity Compositionlity nd Rechility with Conditions on Pth Lengths 17 tht the numer of formuls which hve to e interpreted in the components grows t lest non-elementry in the size of the formul which is interpreted in the product. It is open whether this non-elementry lower complexity ound lso holds for modl logic. Second, it would e interesting to show composition theorem for forms of rechility. Third, we might im t stronger results (e.g. towrds computtion time logic) where the signture of the index structure is more expressive. Another interesting cse refers to products where ll the components re the sme trnsition system, leding us to n infinite power. We hve lredy seen tht we needed to e le to express modulo counting in the index structure to otin the results in this pper. A nturl sequel would e to consider the order of the nturl numers s index structure or other structures, such s ring or the infinite grid. Acknowledgements We wish to thnk Alexnder Rinovich for improvements nd corrections on preliminry version of this pper. Biliogrphy [1] C. C. Chng nd H. J. Keisler. Model Theory. North Hollnd, Amsterdm, third edition, [2] B. Courcelle. The mondic second-order logic of grphs : Definle sets of finite grphs. In Grph-Theoretic Concepts in Computer Science, volume 344 of Lecture Notes in Computer Science, pges 30 53, Berlin/Heidelerg, Springer. [3] A. Dwr, M. Grohe, S. Kreutzer, nd N. Schweikrdt. Model theory mkes formuls lrge. In ICALP 07: 34th Interntionl Colloquium on Automt, Lnguges nd Progrmming, volume 4596 of Lecture Notes in Computer Science, pges Springer Verlg, [4] S. Fefermn nd R. Vught. The first-order properties of products of lgeric systems. Fundment Mthemtice, 47:57 103, [5] I. Felscher. The compositionl method nd regulr rechility. In V. Hlv nd I. Potpov, editors, Proceedings of the Workshop on Rechility Prolems RP 08, Electronic Notes in Theoreticl Computer Science, pges , Liverpool, UK, Septemer [6] D. Gy nd V. Shehtmn. Products of modl logics, prt 1. Logic Journl of IGPL, 6(1):73 146, [7] T. Gnzow nd S. Ruin. Order-invrint mso is stronger thn counting mso in the finite. In S. Alers nd P. Weil, editors, 25th Interntionl Symposium on Theoreticl Aspects of Computer Science (STACS 2008), pges , Schloss Dgstuhl, Germny, Leiniz-Zentrum für Informtik. [8] D. Hrel, D. Kozen, nd J. Tiuryn. Dynmic Logic. MIT Press, [9] M. A. Hrrison. Introduction to Forml Lnguge Theory. Series in Computer Science. Addison Wesley, Boston, MA, USA, [10] W. Hodges. Model theory, volume 42 of Encyclopedi of Mthemtics nd its Applictions. Cmridge University Press, Cmridge, [11] J. E. Hopcroft, R. Motwni, nd J. D. Ullmn. Introduction to Automt Theory, Lnguges, nd Computtion. Addison Wesley, 2nd edition, [12] L. Likin. Elements of Finite Model Theory, volume XIV of Texts in Theoreticl Computer Science. An EATCS Series. Springer, 2004.

18 compositionlity 18 Ingo Felscher nd Wolfgng Thoms [13] J. A. Mkowsky. Algorithmic uses of the fefermn-vught theorem. Annls of Pure nd Applied Logic, 126(1-3): , [14] A. Mostowski. On direct products of theories. The Journl of Symolic Logic, 17(1):1 31, [15] A. Rinovich. Selection nd uniformiztion in generlized product. Logic Journl of IGPL, 12(2): , [16] A. Rinovich. On compositionlity nd its limittions. ACM Trnsctions on Computtionl Logic, 8(1), Jn [17] S. Shelh. The mondic theory of order. The Annls of Mthemtics, 102(3): , [18] W. Thoms. Automt on infinite ojects. In J. vn Leeuwen, editor, Hndook of Theoreticl Computer Science B: Forml Models nd Semntics, pges MIT Press, Cmridge, MA, USA, [19] W. Thoms. Ehrenfeucht gmes, the composition method, nd the mondic theory of ordinl words. In J. Mycielski, G. Rozenerg, nd A. Slom, editors, Structures in Logic nd Computer Science, A Selection of Essys in Honor of A. Ehrenfeucht, volume 1261 of Lecture Notes in Computer Science, pges , Berlin-Heidelerg- New York, Springer Verlg. [20] S. Wöhrle nd W. Thoms. Model checking synchronized products of infinite trnsition systems. In Proceedings of the 19th Annul IEEE Symposium on Logic in Computer Science, Lecture Notes in Computer Science, pges 2 11, Wshington, DC, USA, IEEE Computer Society.