Infinite-state Systems Research Articles

Informed Swarm Verification of Infinite-State Systems

Informed Swarm Verification of Infinite-State Systems

Proceedings 13th International Workshop on Verification of Infinite-State Systems

Proceedings 13th International Workshop on Verification of Infinite-State Systems

Regular model checking for LTL(MSO)

Regular model checking is a form of symbolic model checking for parameterized and infinite-state systems whose states can be represented as words of arbitrary length over a finite alphabet, in which regular sets of words are used to represent sets of states. We present LTL(MSO), a combination of the logics monadic second-order logic (MSO) and LTL as a natural logic for expressing the temporal properties to be verified in regular model checking. In other words, LTL(MSO) is a natural specification language for both the system and the property under consideration. LTL(MSO) is a two-dimensional modal logic, where MSO is used for specifying properties of system states and transitions, and LTL is used for specifying temporal properties. In addition, the first-order quantification in MSO can be used to express properties parameterized on a position or process. We give a technique for model checking LTL(MSO), which is adapted from the automata-theoretic approach: a formula is translated to a buchi regular transition system with a regular set of accepting states, and regular model checking techniques are used to search for models. We have implemented the technique, and show its application to a number of parameterized algorithms from the literature.

Decentralized Control of Infinite Systems

We propose algorithms for the synthesis of decentralized state-feedback controllers with partial observation of infinite state systems, which are modeled by Symbolic Transition Systems. We first consider the computation of safe controllers ensuring the avoidance of a set of forbidden states and then extend this result to the deadlock free case. The termination of the algorithms solving these problems is ensured by the use of abstract interpretation techniques, but at the price of overapproximations, in particular, in the computation of the states which must be avoided. We then extend our algorithms to the case where the system to be controlled is given by a collection of subsystems (modules). This structure is exploited to locally compute a controller for each module. Our tool SMACS gives an empirical evaluation of our methods by showing their feasibility, usability and efficiency.

On the Decidability of Non Interference over Unbounded Petri Nets

Non-interference, in transitive or intransitive form, is defined here over unbounded (Place/Transition) Petri nets. The definitions are adaptations of similar, well-accepted definitions introduced earlier in the framework of labelled transition systems. The interpretation of intransitive non-interference which we propose for Petri nets is as follows. A Petri net represents the composition of a controlled and a controller systems, possibly sharing places and transitions. Low transitions represent local actions of the controlled system, high transitions represent local decisions of the controller, and downgrading transitions represent synchronized actions of both components. Intransitive non-interference means the impossibility for the controlled system to follow any local strategy that would force or dodge synchronized actions depending upon the decisions taken by the controller after the last synchronized action. The fact that both language equivalence and bisimulation equivalence are undecidable for unbounded labelled Petri nets might be seen as an indication that non-interference properties based on these equivalences cannot be decided. We prove the opposite, providing results of decidability of non-interference over a representative class of infinite state systems.

On Almost-Sure Properties of Probabilistic Discrete Event Systems

Although randomization often increases the degree of flexibility in system design, analyzing system properties in the probabilistic framework introduces additional difficulties and challenges in comparison with their nonprobabilistic counterparts. In this paper, we focus on probabilistic versions of two problems frequently encountered in discrete event systems, namely, the reachability and forbidden-state problems. Our main concern is to see whether there exists a (or for every) non-blocking or fair control policy under which a given finite- or infinite-state system can be guided to reach (or avoid) a set of goal states with probability one. For finite-state systems, we devise algorithmic approaches which result in polynomial time solutions to the two problems. For infinite-state systems modelled as Petri nets, the problems are undecidable in general. For the class of persistent Petri nets, we establish a valuation approach through which the convergence behavior of a system is characterized, which in turn yields solutions to the reachability and forbidden-state problems.

Backward Reachability of Array-based Systems by SMT solving: Termination and Invariant Synthesis

The safety of infinite state systems can be checked by a backward reachability procedure. For certain classes of systems, it is possible to prove the termination of the procedure and hence conclude the decidability of the safety problem. Although backward reachability is property-directed, it can unnecessarily explore (large) portions of the state space of a system which are not required to verify the safety property under consideration. To avoid this, invariants can be used to dramatically prune the search space. Indeed, the problem is to guess such appropriate invariants. In this paper, we present a fully declarative and symbolic approach to the mechanization of backward reachability of infinite state systems manipulating arrays by Satisfiability Modulo Theories solving. Theories are used to specify the topology and the data manipulated by the system. We identify sufficient conditions on the theories to ensure the termination of backward reachability and we show the completeness of a method for invariant synthesis (obtained as the dual of backward reachability), again, under suitable hypotheses on the theories. We also present a pragmatic approach to interleave invariant synthesis and backward reachability so that a fix-point for the set of backward reachable states is more easily obtained. Finally, we discuss heuristics that allow us to derive an implementation of the techniques in the model checker MCMT, showing remarkable speed-ups on a significant set of safety problems extracted from a variety of sources.

Proceedings 12th International Workshop on Verification of Infinite-State Systems

Proceedings 12th International Workshop on Verification of Infinite-State Systems

On (Omega-)regular model checking

Checking infinite-state systems is frequently done by encoding infinite sets of states as regular languages. Computing such a regular representation of, say, the set of reachable states of a system requires acceleration techniques that can finitely compute the effect of an unbounded number of transitions. Among the acceleration techniques that have been proposed, one finds both specific and generic techniques. Specific techniques exploit the particular type of system being analyzed, for example, a system manipulating queues or integers, whereas generic techniques only assume that the transition relation is represented by a finite-state transducer, which has to be iterated. In this article, we investigate the possibility of using generic techniques in cases where only specific techniques have been exploited so far. Finding that existing generic techniques are often not applicable in cases easily handled by specific techniques, we have developed a new approach to iterating transducers. This new approach builds on earlier work, but exploits a number of new conceptual and algorithmic ideas, often induced with the help of experiments, that give it a broad scope, as well as good performances.

A formal approach for the development of reactive systems

Context This paper deals with the development and verification of liveness properties on reactive systems using the Event-B method. By considering the limitation of the Event-B method to invariance properties, we propose to apply the language TLA + to verify liveness properties on Event-B models. Objective This paper deals with the use of two verification approaches: theorem proving and model-checking, in the construction and verification of safe reactive systems. The theorem prover concerned is part of the Click_n_Prove tool associated to the Event-B method and the model checker is TLC for TLA + models. Method To verify liveness properties on Event-B systems, we extend first the expressivity and the semantics of a B model (called temporal B model) to deal with the specification of fairness and eventuality properties. Second, we propose semantics of the extension over traces, in the same spirit as TLA + does. Third, we give verification rules in the axiomatic way of the Event-B method. Finally, we give transformation rules from a temporal B model into a TLA + module. We present in particular, our prototype system called B2TLA +, that we have developed to support this transformation; then we can verify liveness properties thanks to the model checker TLC on finite state systems. For the verification of infinite-state systems, we propose the use of the predicate diagrams and its associated tool DIXIT. As the B refinement preserves invariance properties through refinement steps, we propose some rules to get the preservation of liveness properties by the B refinement. Results The proposed approach is applied for the development of some reactive systems examples and our prototype system B2TLA + is successfully used to transform a temporal B model into a TLA + module. Conclusion The paper successfully defines an approach for the specification and verification of safety and liveness properties for the development of reactive systems using the Event-B method, the language TLA + and the predicate diagrams with their associated tools. The approach is illustrated on a case study of a parcel sorting system.

A Decidability Result for the Model Checking of Infinite-State Systems

We present a decidability result for the model checking of a certain class of properties that can be conveniently expressed as ground formulae of a first-order temporal fragment. The decidability result is obtained by importing into the context of model-checking problems some techniques developed for the combination of decision procedures for the satisfiability of constraints. The general decidability result is then specialized for checking properties of particular interest, such as liveness and safety, and, for the latter case, a more optimized algorithm has been proposed.

Equational approximations for tree automata completion

In this paper we deal with the verification of safety properties of infinite-state systems modeled by term rewriting systems. An over-approximation of the set of reachable terms of a term rewriting system R is obtained by automatically constructing a finite tree automaton. The construction is parameterized by a set E of equations on terms, and we also show that the approximating automata recognize at most the set of R / E -reachable terms. Finally, we present some experiments carried out with the implementation of our algorithm. In particular, we show how some approximations from the literature can be defined using equational approximations.

On Almost-Sure Properties of Probabilistic Discrete Event Systems

Although randomization often increases the degree of flexibility in system design, analyzing system properties in the probabilistic framework introduces additional difficulties and challenges in comparison with their nonprobabilistic counterparts. In this paper, we focus on probabilistic versions of two problems frequently encountered in discrete event systems, namely, the reachability and forbidden-state problems. Our main concern is to see whether there exists a non-blocking or fair control policy under which a given finite- or infinite-state system can be guided to reach (or avoid) a set of goal states with probability one. For finite-state systems, we devise algorithmic approaches which result in polynomial time solutions to the two problems. For infinite-state systems modelled as Petri nets, the problems are undecidable in general. For the class of persistent Petri nets, we establish a valuation approach through which the convergence behavior of a system is characterized, which in turn yields solutions to the reachability and forbidden-state problems.

On external presentations of infinite graphs

The vertices of a finite state system are usually a subset of the natural numbers. Most algorithms relative to these systems only use this fact to select vertices. For infinite state systems, however, the situation is different: in particular, for such systems having a finite description, each state of the system is a configuration of some machine. Then most algorithmic approaches rely on the structure of these configurations. Such characterisations are said internal. In order to apply algorithms detecting a structural property (like identifying connected components) one may have first to transform the system in order to fit the description needed for the algorithm. The problem of internal characterisation is that it hides structural properties, and each solution becomes ad hoc relatively to the form of the configurations. On the contrary, external characterisations avoid explicit naming of the vertices. Such characterisation are mostly defined via graph transformations. In this paper we present two kind of external characterisations: deterministic graph rewriting, which in turn characterise regular graphs, deterministic context-free languages, and rational graphs. Inverse substitution from a generator (like the complete binary tree) provides characterisation for prefix-recognizable graphs, the Caucal Hierarchy and rational graphs. We illustrate how these characterisation provide an efficient tool for the representation of infinite state systems.

Proceedings International Workshop on Verification of Infinite-State Systems

Proceedings International Workshop on Verification of Infinite-State Systems

Contrôle décentralisé de systèmes symboliques infinis sous observation partielle

We propose algorithms for the synthesis of decentralized controllers with partial observation of infinite state systems modelled by Symbolic Transition Systems. We provide models of safe controllers both for potentially deadlock and deadlock free controlled systems. To obtain algorithms for these problems, we use abstract interpretation techniques which provide overapproximations of the transitions set to be disabled. Our tool SMACS allowed us to make an empirical validation of our methods and show their feasibility and eficiency.

Light-Weight SMT-based Model Checking

Recently, the notion of an array-based system has been introduced as an abstraction of infinite state systems (such as mutual exclusion protocols or sorting programs) which allows for model checking of invariant (safety) and recurrence (liveness) properties by Satisfiability Modulo Theories (SMT) techniques. Unfortunately, the use of quantified first-order formulae to describe sets of states makes fix-point checking extremely expensive. In this paper, we show how invariant properties for a sub-class of array-based systems can be model-checked by a backward reachability algorithm where the length of quantifier prefixes is efficiently controlled by suitable heuristics. We also present various refinements of the reachability algorithm that allows it to be easily implemented in a client-server architecture, where a ''light-weight'' algorithm is the client generating proof obligations for safety and fix-point checks and an SMT solver plays the role of the server discharging the proof obligations. We also report on some encouraging preliminary experiments with a prototype implementation of our approach.

Reachability is decidable for weakly extended process rewrite systems

Process rewrite systems (PRS) are widely accepted as a formalism for the description of infinite-state systems. It is known that the reachability problem for PRS is decidable. The problem becomes undecidable when PRS are extended with a finite-state control unit. In this paper, we show that the problem remains decidable when PRS are extended with a weak (i.e. acyclic except for self-loops) finite-state control unit. We also present some applications of this decidability result.

Approximating Probabilistic Behaviors of Biological Systems Using Abstract Interpretation

In this paper we apply the Abstract Interpretation approach [P. Cousot and R. Cousot. Abstract Interpretation: A Unified Lattice Model for Static Analysis of Programs by Construction or Approximation of Fixpoints . Proc. of POPL'77, 238–252, 1977; P. Cousot and R. Cousot. Systematic Design of Program Analysis Frameworks . Proc. of POPL'79, 269–282, 1979] for approximating the behavior of biological systems, modeled specifically using the Chemical Ground Form calculus [L. Cardelli. On Process Rate Semantics . Theoretical Computer Science, 391 190–215, 2008], a new stochastic calculus rich enough to model the dynamics of biochemical reactions. Our analysis computes an Interval Markov Chain (IMC) that safely approximates the Discrete-Time Markov Chain , describing the probabilistic behavior of the system, and reports both lower and upper bounds for probabilistic temporal properties. Our analysis has several advantages: (i) the method is effective (even for infinite state systems) and allows us to systematically derive an IMC from an abstract labeled transition system; (ii) using intervals for abstracting the multiplicity of reagents allows us to achieve conservative bounds for the concrete probabilities of a set of concrete experiments which differs only for initial concentrations.

Model checking race-freeness

With the introduction of highly concurrent systems in standard desktop computers, ensuring correctness of industrial-size concurrent programs is becoming increasingly important. One of the most important standards in use for developing multi-threaded programs is the POSIX Threads standard, commonly known as PThreads. Of particular importance, the analysis of industrial code should, as far as possible, be automatic and not require annotations or other forms of specifications of the code. Model checking has been one of the most successful approaches to program verification during the last two decades. The size and complexity of applications which can be handled have increased rapidly through integration with symbolic techniques. These methods are designed to work on finite (but large) state spaces. This framework fails to deal with several essential aspects of behaviours for multithreaded programs: there is no bound a priori on the number of threads which may arise in a given run of the system; each thread manipulates local variables which often range over unbounded domains; and the system has a dynamic structure in the sense that threads can be created and killed throughout execution of the system. In this paper we concentrate on checking a particular class of properties for concurrent programs, namely safety properties. In particular, we focus on race-freeness, that is, the absence of race conditions (also known as data races) in shared-variable pthreaded programs. We will follow a particular methodology which we have earlier developed for model checking general classes of infinite-state systems [1, 3, 6, 8, 9] and apply a symbolic backward reachability analysis to verify the safety property. Since we construct a model as an over-approximation of the original program, proving the safety property in the model implies that the property also holds in the original system. Surprisingly, it leads to a quite efficient analysis which can be carried out fully automatically.