D. Caucal. Graphes canoniques de graphes algebriques. Rapport de Recherche 872, INRIA, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....programs (they are equipped with a binary sequential operator) This class of processes has been intensively studied by many researchers. Baeten, Bergstra and Klop proved in [BBK87] that bisimilarity is decidable for normed BPA processes. Much simpler proofs of this were later given in [Cau88], HS91] and [Gro91] In [HS91] Httel and Stirling used a tableau decision method and gave also sound and complete equational theory. Hirshfeld, Jerrum and Moller demonstrated in [HJM94a] that the problem is decidable in polynomial time. The decidability result was later extended to the whole ....

....bisimilar to a(Y .X) 3. a = # and the def. equation for X contains two summands of the form b# 1 , b# 2 such that k = Length(# 1 ) Length(# 2 ) 1. It is easy to see that the set CL(Y ) can be constructed in polynomial time for each Y # S(#) The following lemma is due to D. Caucal (see [Cau88]) be normed BPA # processes in GNF and let #, # # Var (#) # ,# #Var(# ) such that # # # and #.# # # .# . Then # # # be normed BPA # processes. Let A 1 , A k #Var (#) X,Y # Var (# ) such that X = Y =1and A 1 . A k # Y .X where l = A 1 . A k 1. Then A k ....

D. Caucal. Graphes canoniques de graphes algebriques. Rapport de Recherche 872, INRIA, 1988.

....between formal language theory and combinatorial group theory ( Ani71, MS83, MS85] see [BB91, p. 95 100] for a survey) but also in algebraic graph theory ( Bau91, Bau92] see [Cou90a] for a survey) in model theory ( Cou89, Cou90b] see [Cou90a] for a survey) and in semantics of processes ( BBK87] [Cau90a], CHS92] HJM94] see [Cau95] for a survey) The aim of this work is to study the structure of Hn Gamma where Gamma is a contextfree graph and H a group acting on Gamma . The study of H itself is done in [Pel95] while, in some sense, the fundamental article [MS83] is treating the case where ....

D. Caucal. Graphes canoniques de graphes alg'ebriques. RAIRO TIA, nr 24-4, pages 339--352, 1990.

....sequential programs (they are equipped with a binary sequential operator) This class of processes has been intensively studied by many researchers. Baeten, Bergstra and Klop proved in [BBK87] that bisimilarity is decidable for normed BPA processes. Much simpler proofs of this were later given in [Cau88,HS91,Gro91]. In [HS91] H#ttel and Stirling Supported by GA #R, grant number 201 97 0456 Preprint submitted to Elsevier Preprint 28 February 1997 used a tableau decision method and gave also sound and complete equational theory. Hirshfeld, Jerrum and Moller demonstrated in [HJM94a] that the problem is ....

....:X) iii) a = and the def. equation for X contains two summands of the form bff 1 , bff 2 such that k = Length(ff 1 ) Length(ff 2 ) Gamma 1. It is easy to see that the set CL(Y ) can be constructed in polynomial time for each Y 2 S ( Delta) The following lemma is due to D. Caucal (see [Cau88]) be nBPA processes in GNF and let ff; fi 2 Var ( Delta) fi 2 Var ( Delta ) such that fi fi and ff:fi ff :fi . Then ff ff be nBPA processes in GNF. Let A 1 ; A k 2 Var ( Delta) X; Y 2 Var ( Delta ) such that jXj = jY j = 1 and A 1 : Delta Delta Delta ....

D. Caucal. Graphes canoniques de graphes algebriques. Rapport de Recherche 872, INRIA, 1988.

....We discuss the question whether these modi cations are strong enough, i.e. whether they always guarantee the existence of a nite witness of bisimilarity. The Caucal base (i.e. a set of pairs that would generate the maximal bisimulation by congruence closure for more details consult [3,4]) is used to semidecide strong bisimulation by enumeration of nite sets for which the Caucal condition is tested. In this way, in the positive case a nite bisimulation (Caucal) base is eventually constructed. The notion of Caucal base can be modi ed into weak Caucal base which serves as ....

D. Caucal. Graphes canoniques de graphes algebriques. In Informatique theorique et Applications, volume 24(4), pages 339-352. RAIRO, 1990.

....Y def = bX; A def = aC b; C def = bAAg. We have that X simA; the reader may want to verify that the relation f(X n ; A n ) j n 0g [ f(Y X n 1 ; CA n ) j n 0g is a bisimulation (where X n here denotes n successive Xs, X 2 V ) 2 The following proposition, originally due to Caucal [8], is essential, providing us with a way of removing suffixes of bisimilar BPA expressions. Proposition 1 If G is a normed BPA process expression and E; F are arbitrary BPA process expressions and EG simFG then E simF . Proof: We show that the relation R = f(E; F ) j EG simFG for some Gg [ ....

....out that this notion of bisimulation up to suffices. As we only consider strings of BPA variables, we shall only need to consider bisimulation up to sequential congruence and do not need to involve the nondeterministic choice operator at all. Such relations, introduced by Didier Caucal in [8] (originally published as [7] are commonly referred to as self bisimulations. Whenever ff simfi, our tableau system will construct a finite self bisimulation, a relation R V ar Theta V ar whose closure under congruence w.r.t. sequential composition is a bisimulation containing (ff; fi) ....

[Article contains additional citation context not shown here]

D. Caucal. Graphes canoniques de graphes alg'ebriques. Informatique th'eorique et Applications (RAIRO), 24(4):339--352, 1990.

....qualities of behavioural equivalences. However, the proof of decidability in [1, 2] is not easy as it relies on isolating a possibly complex periodicity from the transition graphs of these processes. An alternative, more elegant, proof utilizing rewrite techniques is presented by Caucal [7]; a simplified version of this proof is due to Groote [16] The idea is to show that the maximal bisimulation on a transition graph is given as the least congruence of a canonical and strongly normalizing Thue system and that there are only finitely many candidates for such a system. However, the ....

....checking finite and infinite state transition systems [30, 5] The decision procedure yields an upper bound on the depth of a tableau. Moreover, it provides the essential part of the bisimulation relation between two processes which underlies their equivalence, a self bisimulation in the sense of [7]. An important by product of the tableau system is a sound and complete sequent based equational theory for normed BPA processes; the theory emanates from running the tableau method backwards . This result extends Milner s axiomatization of regular processes [26] to the class of context free ....

[Article contains additional citation context not shown here]

D. Caucal. Graphes canoniques de graphes alg'ebriques. Rapport de Recherche 872, INRIA, Juillet 1988.

....that bisimulation equivalence is decidable in the class of normed BPA processes. It was the first result, showing that bisimulation equivalence can remain decidable in a class of processes, in which the language equivalence is undecidable. Much simpler proof of this was later given by Caucal [4] and Groote [5] In [6] Huttel and Stirling used a tableau decision method and gave also sound and complete equational theory for the class of normed BPA processes. This result was later extended to the whole class of BPA processes by Christensen, Huttel and Stirling [7] Another class of ....

....in the normal form, which is bisimilar to the original one. In this section we prove, that if we restrict out attention to the class of normed BPA processes, then the result of [2] can serve as a constructive regularity test for processes of this class. The following lemma is due to D. Caucal [4]: Lemma 5 (Cancelation) Let Delta be a normed BPA process in GNF, ff; fi; fl 2 V ar( Delta) If fffl fifl, then also ff fi. Proof: The set f[ffi 1 ; ffi 2 ] j ffi 1 ; ffi 2 2 V ar( Delta) ffi 1 fl ffi 2 flg is a bisimulation containing the pair [ff; fi] 2 Now it is possible to ....

[Article contains additional citation context not shown here]

D. Caucal. Graphes canoniques de graphes algebriques. Rapport de Recherche 872, INRIA, Juillet 1988.

....result indicating that decidability issues for bisimilarity are rather different from the ones for language equivalence is due to Baeten, Bergstra and Klop. They proved in [BBK87, BBK93] that bisimilarity is decidable for normed BPA systems. Much simpler proofs of this were later given in [Cau88], HS91] and [Gro92] It is well known result by Christensen, H#ttel and Stirling that the bisimulation equivalence is decidable in the class of all BPA systems # [CHS92] The proof consists of two semidecidable procedures running in parallel. Burkart, Caucal and Steffen demonstrated in [BCS95] ....

D. Caucal. Graphes canoniques de graphes algebriques. Rapport de Recherche 872, INRIA, 1988.

....first result indicating that decidability issues for bisimilarity are rather different from the ones for language equivalence is due to Baeten, Bergstra and Klop. They proved in [BBK87, BBK93] that bisimilarity is decidable for normed BPA systems. Much simpler proofs of this were later given in [Cau88], HS91] and [Gro92] It is well known result by Christensen, Httel and Stirling that the bisimulation equivalence is decidable in the class of all BPA systems [CHS92] The proof consists of two semidecidable procedures running in parallel. Burkart, Caucal and Steffen demonstrated in [BCS95] ....

D. Caucal. Graphes canoniques de graphes algebriques. Rapport de Recherche 872, INRIA, 1988.

....processes. Moreover, designed algorithms are practically usable. This is no more true if one moves to process classes which contain also processes with infinitely many states (up to bisimilarity) Some problems can remain decidable for example, bisimilarity is known to be decidable for BPA (see [BBK87, Cau88, Gro91, HS91, CHS92]) and BPP (see [CHM93] 1 processes. The same problem becomes undecidable for labelled Petri nets (see [Jan94] But even if a given property is decidable, the algorithm is usually not interesting from the practical point of view due to its complexity. Before running a complex algorithm, it is a ....

D. Caucal. Graphes canoniques de graphes algebriques. Rapport de Recherche 872, INRIA, 1988.

....different from the ones for language equivalence is due to Baeten, Bergstra, and Klop. They proved in [BBK87, BBK93] that bisimilarity is decidable for context free grammars in GNF (this class of processes is also known under the name normed BPA ) Much simpler proofs of this were later given in [Cau88], HS91] and [Gro91] In [HS91] Httel and Stirling used a tableau decision method and gave also sound and complete equational theory. If we replace the binary sequential operator with the parallel operator, we obtain BPP processes. They can thus be seen as simple parallel programs. Christensen, ....

D. Caucal. Graphes canoniques de graphes algebriques. Rapport de Recherche 872, INRIA, 1988.

....The first result indicating that decidability issues for bisimilarity are rather different from the ones for language equivalence is due to Baeten, Bergstra and Klop. They proved in [BBK87, BBK93] that bisimilarity is decidable for normed BPA . Much simpler proofs of this were later given in [Cau88], HS91] and [Gro92] It is well known result by Christensen, Httel and Stirling that the bisimulation equivalence is decidable in the class of all BPA systems [CHS92] The proof consists of two semidecidable procedures running in parallel. Burkart, Caucal and Steffen demonstrated in [BCS95] ....

D. Caucal. Graphes canoniques de graphes algebriques. Rapport de Recherche 872, INRIA, 1988.

.... technique onto weak bisimilarities (where a special action, called silent action, is distinguished from the others and partially ignored in the bisimilarity clause) has been studied by Milner and Sangiorgi [SM92] Two special cases of the up to context technique had been previously put forward: In [Cau90], Caucal 4 defines a notion of self bisimulation in the setting of BPA processes (they can be viewed as the processes generated by a context free grammar) which allows him to eliminate common prefixes and suffixes in the derivatives of two processes. Self bisimulations have been used in [Cau90] ....

....[Cau90] Caucal 4 defines a notion of self bisimulation in the setting of BPA processes (they can be viewed as the processes generated by a context free grammar) which allows him to eliminate common prefixes and suffixes in the derivatives of two processes. Self bisimulations have been used in [Cau90], as well as in a number of other papers (e.g. CHS92, HJM95] to establish decidability results for the classes of BPA and BPA processes (roughly, the latter differ from the former in that the composition operator is commutative) Another form of up to context technique is Milner, Parrow and ....

D. Caucal. Graphes canoniques de graphes alg'ebriques. Informatique Th'eorique et Applications (RAIRO), 24(4):339--352, 1990. 32

....b; Y def = bX; A def = aC b; C def = bAAg. We have that X A; the reader may want to verify that the relation f(X n ; A n ) j n 0g [ f(Y X n 1 ; CA n ) j n 0g is a bisimulation (where X n here denotes n successive Xs, X 2 V ) 2 The following proposition, originally due to Caucal [8], is essential, providing us with a way of removing suffixes of bisimilar BPA expressions. Proposition 1 If G is a normed BPA process expression and E; F are arbitrary BPA process expressions and EG FG then E F . Proof: We show that the relation R = f(E; F ) j EG FG for some Gg [ f(ffl; ....

....out that this notion of bisimulation up to suffices. As we only consider strings of BPA variables, we shall only need to consider bisimulation up to sequential congruence and do not need to involve the nondeterministic choice operator at all. Such relations, introduced by Didier Caucal in [8] (originally published as [7] are commonly referred to as self bisimulations. Whenever ff fi, our tableau system will construct a finite self bisimulation, a relation R V ar Theta V ar whose closure under congruence w.r.t. sequential composition is a bisimulation containing (ff; fi) ....

[Article contains additional citation context not shown here]

D. Caucal. Graphes canoniques de graphes alg'ebriques. Informatique th'eorique et Applications (RAIRO), 24(4):339--352, 1990.

....qualities of behavioural equivalences. However, the proof of decidability in [1, 2] is not easy as it relies on isolating a possibly complex periodicity from the transition graphs of these processes. An alternative, more elegant, proof utilizing rewrite techniques is presented by Caucal [7]; a simplified version of this proof is due to Groote [16] The idea is to show that the maximal bisimulation on a transition graph is given as the least congruence of a canonical and strongly normalizing Thue system and that there are only finitely many candidates for such a system. However, the ....

....checking finite and infinite state transition systems [30, 5] The decision procedure yields an upper bound on the depth of a tableau. Moreover, it provides the essential part of the bisimulation relation between two processes which underlies their equivalence, a self bisimulation in the sense of [7]. An important by product of the tableau system is a sound and complete sequent based equational theory for normed BPA processes; the theory emanates from running the tableau method backwards . This result extends Milner s axiomatization of regular processes [26] to the class of context free ....

[Article contains additional citation context not shown here]

D. Caucal. Graphes canoniques de graphes alg'ebriques. Rapport de Recherche 872, INRIA, Juillet 1988.

....Moreover they demonstrate that the family of pushdown processes is the smallest extension of context free processes with this closure property. Baeten, Bergstra and Klop proved that bisimulation equivalence is decidable for normed context free processes [1, 2] Simpler proofs were developed in [5, 11, 17, 13], and [14] showed that there is even a polynomial time decision procedure. The decidability result was generalized in [10] to encompass unnormed processes, and then refined in [4] to give upper bounds. Groote and Huttel proved that other standard equivalences on processes (traces, failures, ....

Caucal, D. (1990). Graphes canoniques de graphes alg'ebriques. Informatique Th'eorique et Applications (RAIRO), 24,339-352.

....Moreover they demonstrate that the family of pushdown processes is the smallest extension of context free processes with this closure property. Baeten, Bergstra and Klop proved that bisimulation equivalence is decidable for normed context free processes [2, 3] Simpler proofs were developed in [7, 15, 22, 17], and [19] showed that there is even a polynomial time decision procedure. The decidability result was generalized in [14] to encompass unnormed processes, and then refined in [5] to give upper bounds. Groote and H uttel proved that other standard equivalences on processes (traces, failures, ....

.... Type 1 1 2 ff a Gamma fi where jffj = 2 and jfij 0 Type 2 X a Gamma fi Type 3 X a Gamma Y or X a Gamma ffl This Caucal hierarchy is implicit in Caucal s work on understanding context free graphs, and understanding when the monadic second order theory of graphs is decidable [8, 7, 9, 6, 10]. With respect to language equivalence, there is no distinction between Type 2 and Type 2. With respect to bisimulation equivalence this is not the case. However Caucal showed in [8] that Type 0 processes coincide (up to isomorphism of their transition graphs) with pushdown processes (and hence ....

Caucal, D. (1990). Graphes canoniques de graphes alg ebriques. Informatique Theorique et Applications (RAIRO), 24,339-352.

....Y def = bX; A def = aC b; C def = bAAg. We have that X A; the reader may want to verify that the relation f(X n ; A n ) j n 0g[f(Y X n 1 ; CA n )jn 0g is a bisimulation (where X n here denotes n successive Xs, X 2 V ) 2 The following proposition, originally due to Caucal [7] is essential, providing us with a way of removing suffixes of bisimilar BPA expressions. Proposition 1 If G is a normed BPA process expression and E; F are arbitrary BPA process expressions and EG FG then E F . Proof: We show that the relation R = f(E; F ) j EG FG for some Gg [ f(ffl; ....

....However, whenever ff fi, our tableau system will construct a self bisimulation, a finite relation R V ar Theta V ar whose closure under congruence w.r.t. sequential composition is a bisimulation containing (ff; fi) The notion of self bisimulation was introduced by Didier Caucal in [7] (originally published as [6] Here the notion of a least congruence is essential. Definition 3 For any binary relation R on V ar , R is the least precongruence w.r.t. sequential composition that contains R, R the symmetric closure of R and R the reflexive and transitive ....

[Article contains additional citation context not shown here]

D. Caucal. Graphes canoniques de graphes alg'ebriques. Informatique th'eorique et Applications (RAIRO), 24(4):339--352, 1990.

....qualities of behavioural equivalences. However, the proof of decidability in [1, 2] is not easy as it relies on isolating a possibly complex periodicity from the transition graphs of these processes. An alternative, more elegant, proof utilizing rewrite techniques is presented by Caucal [6]; a simplified version of this proof is due to Groote [14] The idea is to show that the maximal bisimulation on a transition graph is given as the least congruence of a canonical and strongly normalizing Thue system and that there are only finitely many candidates for such a system. However, the ....

....checking finite and infinite state transition systems [26, 5] The decision procedure yields an upper bound on the depth of a tableau. Moreover, it provides the essential part of the bisimulation relation between two processes which underlies their equivalence, a self bisimulation in the sense of [6]. An important by product of the tableau system is a sound and complete sequent based equational theory for normed BPA processes; the theory emanates from running the tableau method backwards . We see this as an initial step in extending Milner s axiomatization of regular processes [23] to this ....

[Article contains additional citation context not shown here]

D. Caucal. Graphes canoniques de graphes alg'ebriques. Rapport de Recherche 872, INRIA, Juillet 1988.

....of Mathematics and Computer Science, Aalborg University Centre, Fredrik Bajersvej 7E, 9220 Aalborg , Denmark. Their proof is rather lengthy and hard to grasp; it ultimately relies on showing a periodicity for any transition graph generated from normed context free processes. Caucal presented in [8] a more elegant (and shorter) proof of the same result utilising rewrite techniques. Finally, in [16] Huttel and Stirling presented yet another proof of the decidability result by appealing to the tableau method. The tableau based approach also supports a sound and complete sequent based ....

....problem (whether or not ff n fi) is decidable. This means that bisimulation inequivalence is semidecidable via the simple procedure which seeks the least i such that ff 6 i fi. Therefore we just need to establish the semi decidability of bisimulation equivalence. The proof of this (inspired by [6, 7, 8]) relies on showing that there is a finite self bisimulation relation which generates the bisimulation equivalence. 3.1 Self bisimulations The notion of self bisimulation was introduced by Didier Caucal in [8] originally published as [7] Here the notion of a least congruence is essential. ....

[Article contains additional citation context not shown here]

D. Caucal. Graphes canoniques de graphes alg'ebriques. Informatique th'eorique et Applications (RAIRO), 24(4):339--352, 1990.

....problem (whether or not ff n fi) is decidable. This means that bisimulation inequivalence is semidecidable via the simple procedure which seeks the least i such that ff 6 i fi. Therefore we just need to establish the semi decidability of bisimulation equivalence. The proof of this (inspired by [6, 7, 8]) relies on showing that there is a finite self bisimulation relation which generates the bisimulation equivalence. 3.1 Self bisimulations The notion of self bisimulation was introduced by Didier Caucal in [8] originally published as [7] Here the notion of a least congruence is essential. ....

....equivalence. The proof of this (inspired by [6, 7, 8] relies on showing that there is a finite self bisimulation relation which generates the bisimulation equivalence. 3. 1 Self bisimulations The notion of self bisimulation was introduced by Didier Caucal in [8] originally published as [7]) Here the notion of a least congruence is essential. Definition 3.1 For any binary relation R on V ar , R is the least precongruence w.r.t. sequential composition that contains R, R the symmetric closure of R and R the reflexive and transitive closure of R and thus ....

D. Caucal. Graphes canoniques de graphes alg'ebriques. Rapport de Recherche 872, INRIA, Juillet 1988.

....tableau for ff = fi indicates that ff b fi. This follows from the fact that the tableau system tries to construct an essential part of a branching bisimulation , namely a branching bisimulation up to sequential congruence that contains the root, a notion originating in work by Caucal [Cau88]. If the tableau is successful this relation exists and is finite, consisting of all pairs compared in the tableau. Here the notion of the least congruence under sequential composition becomes important: Definition 3.4 The least precongruence under sequential composition of a relation R, R , ....

.... that q m q m1 Delta Delta Delta q m n(m) a q 0 with p m R q mj for 1 j n(m) The reason why a bisimulation up to sequential congruence can be said to be an essential part of a bisimulation lies in the following result which is our version of a result used by Caucal [Cau88]: Lemma 3.1 If R is an sbb then R is a bb. Corollary 3.1 ff b fi iff there is an sbb R such that ffRfi. Proof: From the above and from the fact that any bb is an sbb. 2 We now have Theorem 3.3 If ff = fi has a successful tableau T then RT = f(ff 0 ; fi 0 ) j ff 0 = fi 0 or fi ....

D. Caucal. Graphes canoniques de graphes alg'ebriques. Rapport de Recherche 872, INRIA, Juillet 1988.

....between formal language theory and combinatorial group theory ( Ani71, MS83, MS85] see [BB91, p. 95 100] for a survey) but also in algebraic graph theory ( Bau91, Bau92] see [Cou90a] for a survey) in model theory ( Cou89, Cou90b] see [Cou90a] for a survey) and in semantics of processes ( BBK87] [Cau90a], CHS92] HJM94] see [Cau95] for a survey) The aim of this work is to study the structure of Hn Gamma where Gamma is a contextfree graph and H a group acting on Gamma . The study of H itself is done in [Pel95] while, in some sense, the fundamental article [MS83] is treating the case where ....

D. Caucal. Graphes canoniques de graphes alg'ebriques. RAIRO TIA, nr 24-4, pages 339--352, 1990.

....Klop [4, 5] presented the first such decidability result, that bisimilarity between normed BPA is decidable. Their lengthy proof exploits the periodicity which exists in normed BPA transition systems, and several simpler proofs exploiting structural properties were soon recorded, notably by Caucal [31], Huttel and Stirling [80] and Groote [60] Huynh and Tian [81] demonstrated that this problem has a complexity of Sigma P 2 by providing a nondeterministic algorithm which relies on an NP oracle; Hirshfeld, Jerrum and Moller [69, 70] refined this result by providing a polynomial algorithm, ....

D. Caucal. Graphes canoniques de graphes alg'ebriques. RAIRO, 24(4):339--352, 1990.

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D. Caucal. Graphes canoniques de graphes algebriques. Rapport de Recherche 872, INRIA, 1988.

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