Stirling, C. and D. Walker. "Local Model Checking in the Modal Mu-Calculus." In Proceedings of TAPSOFT '89, Lecture Notes in Computer Science 351. Springer-Verlag, Berlin, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....generation in the direction of the shorter path. Current implementation of KEGVis supports static exploration only; the dynamic exploration mode is still under development. 13 5 Related Work The proof system developed in this paper is similar to the tableaux used for local modelchecking [SW91]. In fact, the automated proof generation technique can be seen as a simulation of a run of a local model checker, where the information collected from the run of a global model checker is used to guide the construction of the proof. Several other researches have explored the idea of generating ....

C. Stirling and D. Walker. Local model-checking in the modal mu-calculus. Theoretical Computer Science, 89, 1991.

....Of course, to make the check effective, we require a finite representation of . This issue is addressed shortly in the guise of a (finite )state transition system that generates all traces in M, including . With this in hand, we can apply, for example, the tableau technique of Stirling and Walker [46] to systematically decompose OE and decide j= OE. If we plan to check universal or existential properties of a state, s, then we require a manageable (i.e. finite) representation of M # s and indeed, of M itself. The standard approach is to encode M as a (finite) transition relation, ....

....technique calculates the state sets that satisfy the fixed points. This iterative approach resembles the one taken by iterative data flow analysis [9,23,26] and an overt connection will be made later in the paper [39,41,43,44] When OE contains alternating fixed points, a tableau method [34,46] can be used to decide specific goals of form, a j= OE. 5.3 Soundness and completeness of universal state checking Because the universal trace abstraction, hff ; fli, is a Galois connection, the standardized soundness result of Cousot and Cousot [12] applies to the semantics of Figure ....

C. Stirling and D. Walker. Local model checking in the modal mu-calculus. In J. Diaz and F. Orejas, editors, Proc. TAPSOFT '89, volume 351 of Lecture Notes in Computer Science, pages 369--383. Springer-Verlag, 1989.

....is independent of the language of programs (process terms) One would like proof systems for such logics to maintain this desirable separation. The second approach to including xed points in the logics is to adopt a tableaubased approach to derivations, in uenced by local model checking [25]. Under this approach, one simply includes unfolding rules for xed points, e.g. p : A[ X:A=X] p : X:A; The power of the method is achieved by identifying global combinatorial discharge conditions on derivation trees, involving repetitions of sequents, that suce for the ....

C.P. Stirling and D. Walker. Local model checking in the modal mu-calculus. Theoretical Computer Science, 89:161-177, 1991.

....is to simplify the proof search; the second consists of the transformation rules and allows to introduce new operators in terms of the existing ones; finally, the third class of rules deals with formulae belonging to different worlds and can introduce modifications in the set R of relations. In [49] a tableau system for the modal mu calculus, an extension of the dynamic logic, is presented. Like in our proposal, the tableau calculus in [49] makes use of prefixed formulae and represents the underlying model construction during a proof as a graph (the transition system) However, as in dynamic ....

....ones; finally, the third class of rules deals with formulae belonging to different worlds and can introduce modifications in the set R of relations. In [49] a tableau system for the modal mu calculus, an extension of the dynamic logic, is presented. Like in our proposal, the tableau calculus in [49] makes use of prefixed formulae and represents the underlying model construction during a proof as a graph (the transition system) However, as in dynamic logic, no axioms of the form we have considered are allowed. More recently, in [11] Castilho et al. present a modal tableau calculus, whose ....

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C. Stirling and D. Walker. Local model checking in the modal mucalculus. Theoretical Computer Science, 89:161--177, 1991.

....di#erent proof skeletons for model checking problem presented in di#erent formalism, and indicate that solving model checking problem is equivalent to establishing these proof skeleton. These skeletons include decision function in calculus [46] tableau proof in tableau based mode checkers [45], or a conjunctive (disjunctive) boolean equation system as a proof skeleton for a boolean equation system [33] Nevertheless, few work has been done on practical aspects of these skeleton as evidences for justifying model checking results. For example, the research emphasizes on the existences of ....

C. Stirling and D. Walker. Local model checking in the modal mu-calculus. Theoretical Computer Science, 89(1), 1991.

....with the converse operator, thus obtaining the first tableaux calculus for CPDL. The tableaux based technique we propose here combines in a natural way a number of intuitions and techniques that have been developed for validity satisfiability checking [9, 17, 20, 22, 27, 28] and model checking [3, 17, 25, 24] with prefixed tableaux [11, 13, 18] for modal logics. Indeed, the work in this paper confirms that the combination of model checking and theorem proving techniques, as witnessed also by the recent CAV conference [1] may be very fruitful also for purely deductive techniques. 1.1 Plan of the ....

....: with oe:A:n already present in the branch oe:A :n : hAi with oe already present in the branch Fig. 2. Transitional rules for CPDL modal logics [18] we use both forward (F) and backward (B) rules for necessities. The rules for iteration combines prefixed tableaux with the ideas of [25, 24] for model checking fixpoints in the modal calculus (Fig. 3) In practice when an iterated eventuality hae i is found, we introduce a new propositional variable X (possibly with indices) set a side condition X i , and use the X rule for further reductions. The set X of propositional ....

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C. Stirling and D. Walker. Local model checking in modal mu-calculus. Theor. Comp. Sci., 89:161--177, 1991.

....and the verification, it helps the verification task to be integrated with the the design process. Several techniques have been proposed for verifying processes against calculus properties. Most of these algorithms, tableau systems and proof systems are based on global state space exploration ([17, 39, 5, 11] and many more) Compositional systems have been developed for example in [27] or [2] However, most of these techniques are only applicable to finite state system. When modelling a concurrent system in a process algebra like CCS one easily encounters infinite state processes like unbounded ....

David Walker and Colin Stirling. Local model checking in the modal mucalculus. Theoretical Computer Science, 89(1):161--177, October 1991.

....0 where r is the root of a modal graph M, the model checking problem is to determine if p j= f , i.t. if p 2 [ r 0 ] In this section we assume G and M are over a finite data domain Val . The model checking algorithm is presented in Figure 2. It belongs to the family of local algorithms ([6, 1, 12, 3]) as it only visits the part of search space needed for the computation. States are generated in a demand driven fashion. In particular input variables are instantiated on the fly , when input modalities are checked. A state s in the underlying search space S is a pair consisting of a process and ....

C. Stirling and D. Walker. Local model checking in the modal mu-calculus. Theoretical Computer Science, 89:161-- 177, 1991.

....assuming r j= X where X is a solution of the equation X = OE 2. r j= X:OE if and only if r j= X can be established from the assumption that X is a solution of the equation X = OE 25 We present a proof system based on the tableau system for the socalled modal calculus due to Stirling and Walker [52]. Closely related approaches are due to Streett and Emerson [53] and Vardi [57] The key idea is to introduce constants U (V , W ) to name occurrences of fixed point formulas. In particular, a constant is a constant if it names a least fixed point formula. Then, to show r j= OE it suffices to ....

....is in a certain sense characterised by a unique constant, and then we proceed to show soundness and completeness. Lemma 13.5 Let be any infinite path . through a tableau . Then there is a unique constant U s.t. OE i = U for infinitely many i Proof: We roughly follow [52]. Define the degree of OE, d(OE) in the following way: d(X) d( X) d(U) 0 d( OE) d(a:OE) d(a : OE) d(oeX:OE) 1 d(OE) d(OE ) d(OE ) 1 max(d(OE) d( and then d(r Delta;V OE) d(OE) if OE is not a constant d( Delta(OE) otherwise Let be an infinite path as ....

C. Stirling and D. Walker. Local model checking in the modal mu-calculus. Theoretical Computer Science, 89:161--177, 1991.

....with arbitrary systems whose state space is likely to be infinite. One very appealing way of reducing such a limitation is to check whether a state satisfies a property without computing all the states of the model that satisfy the property. This approach is referred to as local model checking [60, 58, 89, 22, 92, 83, 84, 59, 85] and has the clear advantage that only the necessary part of the model is explored. Winskel [92] proposes a very elegant technique for avoiding the global computation of fixpoints by exploiting some interesting properties concerning the unfolding of fixpoints. We propose a compositional labelled ....

....of this algorithm, where the goal is of avoiding repeating all the proof, when the system is possibly modified. The main limitation of this algorithmic approach is due to the state explosion problem, since it cannot cope with arbitrary systems whose state space is infinite or large. In contrast [60, 58, 89, 22, 92, 83, 84, 59, 85] advocate the use of either a proof system or a set of reduction rules for performing local model checking, where some interesting properties of fixpoints are exploited. One advantage of this proof theoretic approach is that it may be possible to prove properties also for infinite state systems. ....

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C. Stirling and D. Walker. Local Model Checking in the Modal mu-Calculus. Theoretical Computer Science, 89:161--177, 1991.

....both the branching time temporal logic CTL and the linear time temporal logics L(F ) and L(X;U;S) over K bounded Petri nets are PSPACE complete. 1 Introduction Formal verification techniques of distributed systems have received much attention, see for example [Lam80, SC85, CES86, Lar88, Mil89, SW89, Val90, WG93] A predominant technique is known as model checking. The approach is as follows. The systems one considers either explicitly or implicitly specify a state space which can be regarded as a (labelled) graph. Viewing these graphs as models (Kripke structures) for temporal logics, one ....

....than the net. We will call such systems compact systems. K bounded Petri nets [JLL77] and synchronised automata [WG93] are examples of models which are widely use to specify and implement concurrent systems. Verification techniques for these and related systems have been presented in [Lar88, SW89, Val90, WG93, ES92, Esp93, BCM 92] Whereas the work in [Lar88, SW89] focuses on algorithms (tableau systems) for solving the model checking problem, the work in [Val90, WG93, ES92, Esp93, BCM 92] is mainly motivated by the state space explosion problem and how to overcome this problem ....

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Colin P. Stirling and David Walker. Local model checking in the modal mu-calculus. Technical Report ECS--LFCS--89--78, Laboratory for Foundations of Computer Science, Department of Computer Science -- University of Edinburgh, May 1989.

....#, then it is considered good practice for the model checker to provide a counter example, that is, the trace of a possible execution in M which does not satisfy #. There are di#erent approaches to and thus various tools for doing model checking. For instance, the method may be based on tableaux [70, 125], state 8 CHAPTER 1. INTRODUCTION labeling and graph algorithms [25, 26, 86, 123] automata theory [65, 130] partial order semantics [39, 129] BDDs [18, 20] or partial model checking [6] Mona Our contribution, the design and development of the first BDD based version of Mona, belongs to the ....

Colin P. Stirling and David Walker. Local model checking in the modal mu-calculus. Technical Report ECS-LFCS-89-78, LFCS, Department of Computer Science, University of Edinburgh, May 1989.

....of calculus terms can be described within the modal mu calculus. A natural next step is to investigate how one can use the mu calculus to verify interesting properties of objects. The notion of model checking, that is, algorithmically checking whether a term satisfies a given modal formula [SW89], is already well understood in the context of process calculi. It remains to be seen how far we can proceed within the calculus. We have also shown a correspondence between the type system Ob 1 : for the calculus and the modal mu calculus, which captures both type assignment and subtyping. ....

C. Stirling and D. Walker. Local model checking in the modal mu-calculus. In LNCS 351, pages 369--383. Springer-Verlag, 1989.

....that it will still be a guarded formula. In our completeness proof we will need a result from [17] which gives a characterisation of the validity of the calculus formulas by means of infinite tableaux. We will briefly recall the result here. First we introduce the concept of a definition list [21] which will name the fixpoint subformulas of a given formula in order of their nesting. We extend vocabulary of the calculus by a countable set Dcons of fresh symbols that will be referred to as definition constants and usually denoted U; V; These new symbols are now allowed to appear ....

Colin P. Stirling and David J. Walker. Local model checking in the modal mu-calculus. In International Joint Conference in Theory and Practice of Software Development, volume 351 of LNCS, pages 369--382. SpringerVerlag, 1989.

.... have the same complexity as the model checking procedure described in [CES] It would also be interesting to see how the algorithm in this paper could be extended to handle the full modal mu calculus, which includes alternating fixed points [Ko] Algorithms of this generality can be found in [C, EL, SW, Wi]. However, only Emerson and Lei [EL] give a complexity analysis. Their algorithm is exponential in ad 1, where ad, the alternation depth of the formula, is a measure of the degree of mutual recursion among greatest and least fixed points. Our algorithm outperforms this algorithm in the special ....

Stirling, C., and Walker, D. "Local Model Checking in the Modal Mu-Calculus." In Proceedings CAAP'89, Lecture Notes in Computer Science 351, pp. 369 - 383, 1989.

....###########] Testing and Debugging debugging aids 1. INTRODUCTION Tableau based proof systems are used for deductivereasoning in a variety of computing applications, including automated theorem proving [14] and in speci cation and veri cation of temporal properties of concurrent systems [4, 29, 33]. Such systems are typically presented as a set of proof rules.Given a set of proof rules and a goal (which is a proof Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for ....

C.P. Stirling and D.J. Walker. Local model checking in the modal mu-calculus. In Proceedings of TAPSOFT, LNCS 351, pages 369-382, 1989.

....X.#(X) #X.#(X) #X.#(X) X.#(X) to produce an equivalent positive formula. It is easy to see that it will be still a guarded formula. Next we introduce some tools which allow us to deal with occurrences of subformulas of a given formula. These tools are very similar to those used in [4] or [7]. We would like to have a di#erent name (which will be a variable) for every fixpoint subformula of a given formula. We will also introduce a notion of a binding function which will associate subformulas to names. Definition 3 (Binding) We call a formula well named i# every variable is 7 bound ....

Colin S. Stirling and David J. Walker. Local model checking in the modal mu-calculus. Theoretical Computer Science, 89:161--171, 1991. 39

....that it will still be a guarded formula. In our completeness proof we will need a result from [17] which gives a characterisation of the validity of the calculus formulas by means of infinite tableaux. We will briefly recall the result here. First we introduce the concept of a definition list [21] which will name the fixpoint subformulas of a given formula in order of their nesting. We extend vocabulary of the calculus by a countable set Dcons of fresh symbols that will be referred to as definition constants and usually denoted U; V; These new symbols are now allowed to appear ....

Colin P. Stirling and David J. Walker. Local model checking in the modal mu-calculus. In International Joint Conference in Theory and Practice of Software Development, volume 351 of LNCS, pages 369--382. SpringerVerlag, 1989.

....in the mu calculus and other logics, but the proof system and the 1 So a certifying model checker can be used to certify itself algorithm of this paper appear to be the first to do so for symbolic representations. In [Kic,YL97] algorithms are given to create tableau proofs in the style of [SW89] In parallel with our work, Peled and Zuck [PZ01] have developed an algorithm for automatically generating explicit state proofs for LTL properties. The game playing algorithm of [SS98] implicitly generates a kind of proof. Explicit state proofs are of reasonable size only for programs with ....

C. Stirling and D. Walker. Local model checking in the modal mu-calculus. In TAPSOFT,

....behaviors which have been added to the model due to abstraction. Other problems that can be solved using calculus model checking are LTL model checking, bisimulation equivalence and language containment of regular automata [4] Many algorithms for calculus model checking have been suggested [9, 16, 18, 7, 13]. In this work we parallelize a simple sequential algorithm, as presented in [6] The algorithm works bottom up through the formula, evaluating each subformula based on the values of its own subformulas. A formula is interpreted as the set of states in which it is true. Thus, for each calculus ....

C. Stirling and D. J. Walker. Local model checking in the modal mu-calculus. In J. Diaz and F. Orejas, editors, Proceedings of the

.... The works in the rst category generate the whole transition system, corresponding to the concurrent system, but encode it symbolically to reduce the memory size [6, 18] The works belonging to the second category avoid the generation of the whole transition system; local model checking [9, 23], on the y techniques [12, 14] abstractions [3, 7] and compositional reasoning [1, 8, 13, 15] belong to this category. In our work we follow a compositional approach. Compositional techniques reduce state explosion exploiting the natural decomposition of complex systems into processes. Many ....

C. Stirling, D. Walker. Local Model Checking in the Modal MuCalculus. Theoretical Computer Science, 89, 1991. 161-177.

....the original proof goal is again encountered. It is clear that these proof steps can be repeated indefinitely. A number of conditions for ensuring safe termination of proofs at this point, since a greatest fixed point has been unfolded, have been proposed: a constant scheme in Stirling and Walker [SW91] and a tagging scheme in Winskel [Win91] In this thesis, and earlier in Dam et al. DFG98b] an explicit fixed point induction scheme is used for handling a greatest fixed point on the right hand side of the turnstile. First, the fixed point is approximated; we commit to proving the formula for ....

.... calculus) The chosen specification language is frequently a variant of CCS. An excellent introduction to the area can be found in Stirling [Sti01] Early work in the field include numerous algorithms and proof systems for the verification of finite state systems [Sti85, Lar88, Win91, Cle90, SW91] Larsen [Lar88] introduced the concept of local modal checking, i.e. algorithms that considers only the part of the state space reachable from an agent p which is necessary to determine whether a satisfaction p : # holds. Another common topic is compositional proof systems, in order to ....

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C. Stirling and D. Walker. Local model checking in the modal mucalculus. Theoretical Computer Science, 89:161--177, 1991.

....# co NP. 70 Model checking has also been applied successfully to verification of probabilistic systems [PZ93] PZ86] BCHG 97] which is the topic of the ESSLLI 2000 course o#ered by Mark Ryan and Marta Kwiatkowska. For more on model checking, attend that course, or see e.g. EL86] SW91] Zuc93b] KVW94] Zuc93a] Var98] 7.2 Branching time vs linear time logics The debate on the pros and contras of using linear vs branching time temporal logics has been alive and unabating since the begining of the 80 s. See [Lam80] and [EH86] for the beginning of it. Of course, the linear ....

Colin Stirling and David Walker. Local model checking in the modal mucalculus. Theoretical Computer Science, 89(1):161--177, 1991.

....only in those states that are reachable from the In Proc CL2000, LNAI 1861 http: www.springer.de comp lncs index.html Copyright 2000 Springer Verlag initial state(s) it is often possible to substantially reduce the state space. This is known as local (or on the fly) model checking (e.g. [19]) Model checking has many characteristics in common with logic programming; a logic program is a symbolic description of a model (the so called least Herbrand model in case of definite programs or the standard model in the case of stratified logic programs) and by means of resolution it is ....

C. Stirling and D. Walker. Local Model Checking in the Modal Mu-calculus. Theoretical Computer Science, 89(1):161--177, 1991.

....congruence and makes use of explicit labels, like o(s 1 , et, s 2 ) that carry informations about the actions performed (o) its source and destination (s 1 and s 2 ) and the transmitted information (et) We equip the logic with a proof system based on tableau. This system is inspired by [6, 18, 19]; but we have to deal with the additional difficulties induced by the richer labels. Moreover, we deal also with infinite state systems and explicit values. The rest of the paper is organized as follows. Section 2 contains a brief introduction to KLAIM and its labeled operational semantics. ....

....and Related Works In this paper we have presented a variant of HML enriched with state formulae and more refined action predicates which permit reasoning about properties of KLAIM systems. The proposed logic has been equipped with a proof system based on tableau. This system has been inspired by [6, 18, 19]. However, in our case, we had to deal with the additional difficulties induced by the richer labels. There are a few papers that have tackled the problem of defining a logic for process calculi with primitives for mobility. More specifically they have considered definitions of logics for # ....

C. Stirling and D. Walker. Local model checking in the modal mu-calculus. Theoretical Computer Science, 89(1):161--177, October 1991.

....language level, so value and parameter passing has begun to receive attention at the level of property specification and verification. Based on a modal logic devised by Milner, Parrow, and Walker , Dam [6] introduced a first order extension to L and showed how the model checking approach of [16] extended in a very conservative fashion to this logic. A closely related account of value passing processes, though only addressing modal properties (no temporal connectives) was obtained by Hennessy and Liu [9] Later work includes [2] that addresses both global and compositional approaches to ....

C. Stirling and D. Walker. Local model checking in the modal mu-calculus. Theoretical Computer Science, 89:161--177, 1991.

....two processes are bisimilar iff they satisfy the same closed formulae [Sti91] 1.3.1 Tableau techniques An important problem in the context of program logics is that of model checking. Model checking consists in determining whether a process state p satisfies a formula F written p j= F . In [SW89] Stirling and Walker have given a model checker for the modal mucalculus and finite transition graphs in the form of a tableau system, a goal directed proof 14 Chapter 1. Introduction system for the relation p j= F . An advantage of the tableau based approach to model checking is that it is local ....

....we can find a natural and convenient method of determining whether or not two normed BPA processes are bisimulation 1.5. Layout of the Thesis 19 equivalent. In this thesis we show that it is indeed possible; our method is a tableau technique related to the local model checking systems of [SW89,BS90]. This method is also closely related to the branching algorithms for equivalence problems on grammars introduced by Korenjak and Hopcroft in [KH66] We use the same tableau method to prove that the branching bisimulation equivalence of Weijland and van Glabbeek [vGW89a,vGW89b] is decidable for a ....

[Article contains additional citation context not shown here]

C. Stirling and D. Walker. Local model checking in the modal mu-calculus. In LNCS 351, pages 369--383. Springer-Verlag, 1989.

....Engineering] Testing and Debugging debugging aids 1. INTRODUCTION Tableau based proof systems are used for deductive reasoning in a variety of computing applications, including automated theorem proving [14] and in speci cation and veri cation of temporal properties of concurrent systems [4, 29, 33]. Such systems are typically presented as a set of proof rules. Given a set of proof rules and a goal (which is a proof Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for ....

C.P. Stirling and D.J. Walker. Local model checking in the modal mu-calculus. In Proceedings of TAPSOFT, LNCS 351, pages 369-382, 1989.

....our approach, instead, we require that the standard and the abstract system satisfy exactly the same set of formulae. Other approaches exist avoiding the generation of the whole transition system during verification. In the on the fly methodology [22, 27] and in the local model checking approach [16, 32], the states of the transition system are dynamically generated taking into account the goal of the formula verification. In the automata theoretic approach [35] each temporal logic formula is associated with an automaton accepting exactly all computations that satisfy the (negation of the) ....

C. Stirling, D. Walker. Local Model Checking in the Modal Mu-Calculus. Theoretical Computer Science, 89, (1991). 161-177.

....42707 Lawrence Place, Fremont, CA 94538 armin verysys.com, yunshan verysys.com 2 Computer Science Department, Carnegie Mellon University 5000 Forbes Avenue, Pittsburgh, PA 15213, U.S.A Edmund.Clarke cs.cmu.edu Abstract. The verification process of reactive systems in local model checking [2, 9, 28] and in explicit state model checking [14, 16] is on the fly. Therefore only those states of a system have to be traversed that are necessary to prove a property. In addition, if the property does not hold, than often only a small subset of the state space has to be traversed to produce a ....

....combination with a symbolic representation, such as BDDs, this extension potentially leads to an exponential reduction in tableau size. The idea of handling set of states on the left hand side (LHS) of sequents already occurred in [4] as an extension of local model checking for the modal calculus [28]. The tableau construction in this section extends the LTL model checking algorithm of [2] in a similar way. A multiple state sequent (M Sequent) consists of a set of states S and a list of LTL formulae F= F 1 , F n ) written S # E(F) We use the same symbol # to separate left and right ....

C. Stirling and D. Walker. Local model checking in the modal mu-calculus. Theoretical Computer Science, 89, 1991.

....expressiveness, it turns out that validity is decidable for the modal calculus, and for finite state processes the problem of deciding satisfaction between a process and an assertion is decidable too. A range of algorithms and proof systems for this problem has been given in the literature, e.g. [9, 4, 11, 18, 6, 25, 8, 2, 21, 12, 7, 1]. They mostly rely on globally Appears in: Proceedings of LICS 94, IEEE Computer Society Press. Supported by the Danish Technical Research Council. Basic Research in Computer Science, Centre of the Danish National Research Foundation. or locally computing the underlying transition system. ....

....system can be seen as a result of turning the operational reductions of Larsen and Xinxin and the syntactic reductions of Andersen and Winskel into proof rules. But the match is not exact; apart from the new static rules the treatment of fixed points is closer to the work on local model checking [11, 18, 6, 25]. 2 Languages p a:p a t[rec x:t=x] rec x:t p Theta q ff Thetafi Theta q pf Xig f Xig Xi(ff) fi p ff 2 Figure 1: Operational rules. The process language has a general parallel composition operator called a product, t 0 Theta ....

[Article contains additional citation context not shown here]

Colin Stirling and David Walker. Local model checking in the modal mu-calculus. Theoretical Computer Science, 89(1):161--177, 1991.

....modal calculus, and for nite state processes the problem of deciding satisfaction between a process and an assertion is decidable too. A range of algorithms and proof systems for this problem has been given in the literature, e.g. Emerson and Lei, 1986, Arnold and Crubille, 1988, Larsen, 1988, Stirling and Walker, 1991, Cleaveland, 1990, Winskel, 1989, Cleaveland and Ste en, 1992, Andersen, 1994, Vergauwen and Lewi, 1992, Larsen, 1992, Cleaveland et al. 1992, Andersen, 1993] They mostly rely on globally or locally computing the underlying transition system. However, what we seek here is a method that is ....

....seen as a result of turning the operational reductions of Larsen and Xinxin and the syntactic reductions of Andersen and Winskel into proof rules. But the match is not exact; apart from the new static rules the treatment of xed points is closer to the work on local model checking [Larsen, 1988, Stirling and Walker, 1991, Cleaveland, 1990, Winskel, 1989] 2 Languages p a:p a t[rec x:t=x] rec x:t p q q pf g f g ( p 2 Table 1: Operational rules. The process language has a general parallel composition operator called a product, t 0 t ....

[Article contains additional citation context not shown here]

Stirling, C. and Walker, D. (1991). Local model checking in the modal mu-calculus. Theoretical Computer Science, 89(1):161-177.

....Indeed, the decision procedure for the equivalence of simple grammars [24] may be seen as a special case of our method. The tableau method is also related to the tableau methods used by the second author in the different context of local model 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 ....

C. Stirling and D. Walker. Local model checking in the modal mucalculus. In LNCS 351, pages 369--383. Springer-Verlag, 1989.

....These methods tend to be global : to show if E j= V one constructs the sets fF 2 P(E) F j= V g for each subformula of . Local techniques, in contrast, try and directly solve whether E j= V . Local methods also apply to in nite state systems, and are often presented using tableaux [35, 7]. Another general question is to what extent property checking can be guided by the algebraic structure of the process, for instance see [2] Discovering xed point sets in general is not easy, and is therefore liable to lead to errors. Instead we would like simpler, and consequently safer, ....

Stirling, C. and Walker, D. (1991). Local model checking in the modal mu-calculus. Theoretical Computer Science, 89, 161-177.

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Stirling, C. and D. Walker. "Local Model Checking in the Modal Mu-Calculus." In Proceedings of TAPSOFT '89, Lecture Notes in Computer Science 351. Springer-Verlag, Berlin, 1989.

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C. Stirling and D. J. Walker. Local model checking in the modal mu-calculus. In Theory and Practice of Software Development, LNCS, 1989.

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C. Stirling and D. Walker. Local model checking in the modal mu-calculus. Theoretical Computer Science, 89(1):161--177, Oct. 1991.

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C. Stirling and D. Walker. Local model checking in the modal mu-calculus. Theoretical Computer Science, 89(1):161--177, Oct. 1991.

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Stirling, C. and Walker, D. (1991). Local model checking in the modal mu-calculus. Theoretical Computer Science, 89:161--177.

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C. Stirling and D. Walker. Local model checking in the modal mu-calculus. Theoretical Computer Science, 89, 161--177, 1991.

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C. Stirling and D. Walker, Local model checking in the modal mu-calculus. Theoret. Comput. Sci. 89 (1991) 161--177.

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Colin Stirling and David Walker. Local model checking in the modal mucalculus. Theoretical Computer Science, 89(1):161--177, 1991.

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C. Stirling and D. Walker. Local model checking in the modal mu-calculus. Theoretical Computer Science, 89(1):161--177, Oct. 1991.

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C. Stirling and D. Walker. Local Model Checking in the Modal Mu-Calculus. In TAPSOFT '89, LNCS 351, pages 369--383. Springer, 1989.

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C. Stirling and David Walker. Local Model Checking in the Modal Mu-calculus. Theoretical Computer Science, 89(1):161--177, 1991.

No context found.

C. Stirling and D. Walker. Local model checking in the modal mu-calculus. Theoretical Computer Science, 89(1):161--177, 1991.

No context found.

C. Stirling and David Walker. Local model checking in the modal mu-calculus. Theoretical Computer Science, 89(1):161--177, 1991.

No context found.

C. Stirling and D. Walker, Local model checking in the modal mu-calculus. Theor. Comput. Sci. 89 161-177. (1991).

No context found.

C. Stirling and D. Walker. Local model checking in the modal mu-calculus. Theoretical Computer Science, 89:161--177, 1991.

No context found.

C. Stirling and D. J. Walker. Local Model Checking in the Modal Mu-Calculus, CAAP 1989, LNCS 351

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