On the Sets of Real Numbers Recognized by Finite Automata in Multiple Bases

This article studies the expressive power of finite automata recognizing sets of real numbers encoded in positional notation. We consider Muller automata as well as the restricted class of weak deterministic automata, used as symbolic set representations in actual applications. In previous work, it has been established that the sets of numbers that are recognizable by weak deterministic automata in two bases that do not share the same set of prime factors are exactly those that are definable in the first order additive theory of real and integer numbers. This result extends Cobham's theorem, which characterizes the sets of integer numbers that are recognizable by finite automata in multiple bases. In this article, we first generalize this result to multiplicatively independent bases, which brings it closer to the original statement of Cobham's theorem. Then, we study the sets of reals recognizable by Muller automata in two bases. We show with a counterexample that, in this setting, Cobham's theorem does not generalize to multiplicatively independent bases. Finally, we prove that the sets of reals that are recognizable by Muller automata in two bases that do not share the same set of prime factors are exactly those definable in the first order additive theory of real and integer numbers. These sets are thus also recognizable by weak deterministic automata. This result leads to a precise characterization of the sets of real numbers that are recognizable in multiple bases, and provides a theoretical justification to the use of weak automata as symbolic representations of sets.Comment: 17 page

Convex Hull of Arithmetic Automata

Arithmetic automata recognize infinite words of digits denoting decompositions of real and integer vectors. These automata are known expressive and efficient enough to represent the whole set of solutions of complex linear constraints combining both integral and real variables. In this paper, the closed convex hull of arithmetic automata is proved rational polyhedral. Moreover an algorithm computing the linear constraints defining these convex set is provided. Such an algorithm is useful for effectively extracting geometrical properties of the whole set of solutions of complex constraints symbolically represented by arithmetic automata

Omega-Regular Model Checking

peer reviewed"Regular model checking" is the name of a family of techniques for analyzing infinite-state systems in which states are represented by words or trees, sets of states by finite automata on these objects, and transitions by finite automata operating on pairs of state encodings, i.e. finite-state transducers. In this context, the central problem is then to compute the iterative closure of a finite-state transducer. This paper addresses the use of regular model-checking like techniques for systems whose states are represented by infinite (omega) words. Its main motivation is to show the feasibility and usefulness of this approach through a combination of the necessary theoretical developments, implementation, and experimentation. The iteration technique that is used is adapted from recent work of the authors on the iteration of finite-word transducers. It proceeds by comparing successive elements of a sequence of approximations of the iteration, detecting an "increment" that is added to move from one approximation to the next, and extrapolating the sequence by allowing arbitrary repetitions of this increment. By restricting oneself to weak deterministic Buchi automata, and using a number of implementation optimizations, examples of significant size can be handled. The proposed transducer iteration technique can just as well be exploited to compute the closure of a given set of states by the transducer iteration, which has proven to be a very effective way of using the technique. Examples such as a leaking gas burner in which time is modeled by real variables have been handled completely within the automata-theoretic setting

Enhancing Approximations for Regular Reachability Analysis

This paper introduces two mechanisms for computing over-approximations of sets of reachable states, with the aim of ensuring termination of state-space exploration. The first mechanism consists in over-approximating the automata representing reachable sets by merging some of their states with respect to simple syntactic criteria, or a combination of such criteria. The second approximation mechanism consists in manipulating an auxiliary automaton when applying a transducer representing the transition relation to an automaton encoding the initial states. In addition, for the second mechanism we propose a new approach to refine the approximations depending on a property of interest. The proposals are evaluated on examples of mutual exclusion protocols

Accelerated Data-Flow Analysis

Acceleration in symbolic verification consists in computing the exact effect of some control-flow loops in order to speed up the iterative fix-point computation of reachable states. Even if no termination guarantee is provided in theory, successful results were obtained in practice by different tools implementing this framework. In this paper, the acceleration framework is extended to data-flow analysis. Compared to a classical widening/narrowing-based abstract interpretation, the loss of precision is controlled here by the choice of the abstract domain and does not depend on the way the abstract value is computed. Our approach is geared towards precision, but we don't loose efficiency on the way. Indeed, we provide a cubic-time acceleration-based algorithm for solving interval constraints with full multiplication

Efficient Symbolic Representation of Convex Polyhedra in High-Dimensional Spaces

peer reviewedThis work is aimed at developing an efficient data structure for representing symbolically convex polyhedra. We introduce an original data structure, the Decomposed Convex Polyhedron (DCP), that is closed under intersection and linear transformations, and allows to check inclusion, equality, and emptiness. The main feature of DCPs lies in their ability to represent concisely polyhedra that can be expressed as combinations of simpler sets, which can overcome combinatorial explosion in high dimensional spaces. DCPs also have the advantage of being reducible into a canonical form, which makes them efficient for representing simple sets constructed by long sequences of manipulations, such as those handled by state-space exploration tools. Their practical efficiency has been evaluated with the help of a prototype implementation, with promising results

Bottom-Up Shape Analysis

In this paper we present a new shape analysis algorithm. The key distinguishing aspect of our algorithm is that it is completely compositional, bottom-up and non-iterative. We present our algorithm as an inference system for computing Hoare triples summarizing heap manipulating programs. Our inference rules are compositional: Hoare triples for a compound statement are computed from the Hoare triples of its component statements. These inference rules are used as the basis for a bottom-up shape analysis of programs. Specifically, we present a logic of iterated separation formula (LISF) which uses the iterated separating conjunct of Reynolds [17] to represent program states. A key ingredient of our inference rules is a strong biabduction operation between two logical formulas. We describe sound strong bi-abduction and satisfiability decision procedures for LISF. We have built a prototype tool that implements these inference rules and have evaluated it on standard shape analysis benchmark programs. Preliminary results show that our tool can generate expressive summaries, which are complete functional specifications in many cases

Using Flow Specifications of Parameterized Cache Coherence Protocols for Verifying Deadlock Freedom

We consider the problem of verifying deadlock freedom for symmetric cache coherence protocols. In particular, we focus on a specific form of deadlock which is useful for the cache coherence protocol domain and consistent with the internal definition of deadlock in the Murphi model checker: we refer to this deadlock as a system- wide deadlock (s-deadlock). In s-deadlock, the entire system gets blocked and is unable to make any transition. Cache coherence protocols consist of N symmetric cache agents, where N is an unbounded parameter; thus the verification of s-deadlock freedom is naturally a parameterized verification problem. Parametrized verification techniques work by using sound abstractions to reduce the unbounded model to a bounded model. Efficient abstractions which work well for industrial scale protocols typically bound the model by replacing the state of most of the agents by an abstract environment, while keeping just one or two agents as is. However, leveraging such efficient abstractions becomes a challenge for s-deadlock: a violation of s-deadlock is a state in which the transitions of all of the unbounded number of agents cannot occur and so a simple abstraction like the one above will not preserve this violation. In this work we address this challenge by presenting a technique which leverages high-level information about the protocols, in the form of message sequence dia- grams referred to as flows, for constructing invariants that are collectively stronger than s-deadlock. Efficient abstractions can be constructed to verify these invariants. We successfully verify the German and Flash protocols using our technique