Abstract. Methods are given for automatically verifying temporal properties of concurrent systems containing an arbitrary number of finite-state processes that communicate using CCS actions. Two models of systems are considered. Systems in the first model consist of a unique contro [ process and an arbitrary number of user processes with identical det-lnitions, For this model, a decision procedure to check whether all the executions of a process satisfy a given specification is presented. This algorithm runs in time double exponential mthe sizes of the control andthe user process definitions. It is also proven that it is decidable whether all the fair executions of a process satisfy a gwen specification. The second model is a special case of the first. In this model, all the processes have identical definitions. For this model, an efficient decision procedure is presented that checks if every execution of a process satisfies a given temporal logic specification. This algorithm runs in time polynomial inthesize of the process definition. Itisshown howtoverify certamglobal properties such as mutual exchrslon and absence of deadlocks. Finally, it is shown how these decision procedures can beusedto reason about certain systems with a communication network,

) E. Allen Emerson and Kedar S. Namjoshi Department of Computer Sciences, The University of Texas at Austin, U.S.A. Abstract. Systems with an arbitrary number of homogeneous processes occur in many applications. The Parameterized Model Checking Problem (PMCP) is to determine whether a temporal property is true of every size instance of the system. We consider systems formed by a synchronous parallel composition of a single control process with an arbitrary number of homogeneous user processes, and show that the PMCP is decidable for properties expressed in an indexed propositional temporal logic. While the problem is in general PSPACE-complete, our initial experimental results indicate that the method is usable in practice. 1 Introduction Systems with an arbitrary number of homogeneous processes occur in many contexts, especially in protocols for data communication, cache coherence, and classical synchronization problems. Current verification work on such systems has focussed mostly...

We identify the computational complexity of the satisfiability problem for FO², the fragment of first-order logic consisting of all relational first-order sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity of its decision problem has not been pinpointed so far. In 1975 Mortimer proved that FO² has the finite-model property, which means that if an FO²-sentence is satisfiable, then it has a finite model. Moreover, Mortimer showed that every satisfiable FO²-sentence has a model whose size is at most doubly exponential in the size of the sentence. In this paper, we improve Mortimer's bound by one exponential and show that every satisfiable FO²-sentence has a model whose size is at most exponential in the size of the sentence. As a consequence, we establish that the satisfiability problem for FO² is NEXPTIME-complete.

We study two-dimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like K4:3, S4:3, GL:3, Grz:3, or the logic determined by the Cartesian square of any infinite linear order. This theorem solves a number of open problems of Gabbay and Shehtman [7]. We also prove a sufficient condition for such products to be not recursively enumerable and give a simple axiomatisation for the square K4:3 K4:3 of the minimal liner logic using non-structural Gabbay-type inference rules.

We describe BDD-based decision procedures for the modal logic K. Our approach is inspired by the automata-theoretic approach, but we avoid explicit automata construction. Instead, we compute certain fixpoints of a set of types---which can be viewed as an on-the-fly emptiness of the automaton. We use BDDs to represent and manipulate such type sets, and investigate different kinds of representations as well as a "level-based" representation scheme. The latter turns out to speed up construction and reduce memory consumption considerably. We also study the effect of formula simplification on our decision procedures. To proof the viability of our approach, we compare our approach with a representative selection of other approaches, including a translation of to QBF. Our results indicate that the BDD-based approach dominates for modally heavy formulae, while search-based approaches dominate for propositionally heavy formulae.

It is known that the tiling technique can be used to give simple proofs of undecidability of various two-dimensional modal and temporal logics. However, up until now, the simplest two-dimensional temporal logic, the compass logic of Venema, has eluded such treatment. We present a new coding of an enumeration of the tiling plane which enables us to show that the compass logic is undecidable.

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Model Checking is well established as a verification technique for finite-state systems. Several important types of systems, such as protocols parameterized by the number of processes, are however inherently infinitestate, hence Model Checking cannot be applied directly to determine correctness of the system.

In this paper we shall introduce a simple temporal logic suitable for reasoning about the temporal aspects of parallel universes, parallel processes, distributed systems or multiple agents. We will use a variant of the mosaic method to prove decidability of this logic. We also present an axiomatization for this logic but show that it does not have the finite model property. This shows that the mosaic method is sometimes a stronger way of establishing decidability. The author would like to thank Valentin Goranko, Alberto Zanardo and the researchers in algebraic logic at London University for their help. This work is supported by EPSRC grant GR/K54946. 1 Introduction The logic, FP\Sigma, investigated here is not new. It has been mentioned in [Thomason, 1984], for example, and is one of the simplest ways of combining temporal and modal operators. It is a propositional logic with formulas built using the two Priorean temporal connectives F and P along with a modal operator \Sigma. The s...

Quantified µ-calculi combine the fix-point and modal operators of temporal logics with (existential and universal) quantifiers, and they allow for reasoning about the possible behaviour of individual components within a software system. In this paper we introduce a novel approach to the semantics of such calculi: we consider a sort of labelled transition systems called counterpart models as semantic domain, where states are algebras and transitions are defined by counterpart relations (a family of partial homomorphisms) between states. Then, formulae are interpreted over sets of state assignments (families of partial substitutions, associating formula variables to state components). Our proposal allows us to model and reason about the creation and deletion of components, as well as the merging of components. Moreover, it avoids the limitations of existing approaches, usually enforcing restrictions of the transition relation: the resulting semantics is a streamlined and intuitively appealing one, yet it is general enough to cover most of the alternative proposals we are aware of. The paper is rounded up with some considerations about expressiveness and decidability aspects.