Marc Zeitoun

Minimization of Counterexamples in SPIN

We propose an algorithm to find a counterexample to some property in a finite state program. This algorithm is derived from SPIN’s one, but it finds a counterexample faster than SPIN does. In particular it still works in linear time. Compared with SPIN’s algorithm, it requires only one additional bit per state stored. We further propose another algorithm to compute a counterexample of minimal size. Again, this algorithm does not use more memory than SPIN does to approximate a minimal counterexample. The cost to find a counterexample of minimal size is that one has to revisit more states than SPIN. We provide an implementation and discuss experimental results.

Infinite-State High-Level MSCs: Model-Checking and Realizability

We consider three natural classes of infinite-state HMSCs: globally-cooperative, locally-cooperative and local-choice HMSCs. We show first that model-checking for globally-cooperative and locally-cooperative HMSCs has the same complexity as for the class of finite-state (bounded) HMSCs. Surprisingly, model-checking local-choice HMSCs turns out to be exponentially more efficient in space than for locally-cooperative HMSCs. We also show that locally-cooperative and local-choice HMSCs can be always implemented by communicating finite states machines, provided we allow some additional (bounded) message data. Moreover, the implementation of local-choice HMSCs is deadlock-free and of linear-size.

Separating Regular Languages by Piecewise Testable and Unambiguous Languages

Separation is a classical problem asking whether, given two sets belonging to some class, it is possible to separate them by a set from another class. We discuss the separation problem for regular languages. We give a Ptime algorithm to check whether two given regular languages are separable by a piecewise testable language, that is, whether a \(\mathcal{B}\Sigma_1(<)\) sentence can witness that the languages are disjoint. The proof refines an algebraic argument from Almeida and the third author. When separation is possible, we also express a separator by saturating one of the original languages by a suitable congruence. Following the same line, we show that one can as well decide whether two regular languages can be separated by an unambiguous language, albeit with a higher complexity.

Infinite-state high-level MSCs: Model-checking and realizability

Message sequence charts (MSC) and High-level MSC (HMSC) is a visual notation for asynchronously communicating processes and a standard of the ITU. They usually represent incomplete specifications of required or forbidden properties of communication protocols. We consider in this paper two basic problems concerning the automated validation of HMSC specifications, namely model-checking and synthesis. We identify natural syntactic restrictions of HMSCs for which we can solve the above questions. We show first that model-checking for globally cooperative (and locally cooperative) HMSCs is decidable within the same complexity as for the restricted class of bounded HMSCs. Furthermore, model-checking local-choice HMSCs turns out to be as efficient as for finite-state (sequential) systems. The study of locally cooperative and local-choice HMSCs is motivated by the synthesis question, i.e., the question of implementing HMSCs through communicating finite-state machines (CFM) with additional message data. We show that locally cooperative and local-choice HMSCs are always implementable. Furthermore, the implementation of a local-choice HMSC is deadlock-free and of linear size.

The pseudovariety J is hyperdecidable

This article defines the notion of hyperdecidability for a class of finite semigroups, which is closely connected to the notion of decidability. It then proves that the pseudovariety J of J-trivial semigroups is hyperdecidable.

Distributed games with causal memory are decidable for series-parallel systems

This paper deals with distributed control problems by means of distributed games played on Mazurkiewicz traces. The main difference with other notions of distributed games recently introduced is that, instead of having a local view, strategies and controllers are able to use a more accurate memory, based on their causal view. Our main result states that using the causal view makes the control synthesis problem decidable for series-parallel systems for all recognizable winning conditions on finite behaviors, while this problem with local view was proved undecidable even for reachability conditions.

Going Higher in the First-Order Quantifier Alternation Hierarchy on Words

We investigate the quantifier alternation hierarchy in first-order logic on finite words. Levels in this hierarchy are defined by counting the number of quantifier alternations in formulas. We prove that one can decide membership of a regular language to the levels \(\mathcal{B}\Sigma_{2}\) (boolean combination of formulas having only 1 alternation) and Σ3 (formulas having only 2 alternations beginning with an existential block). Our proof works by considering a deeper problem, called separation, which, once solved for lower levels, allows us to solve membership for higher levels.

Pebble weighted automata and transitive closure logics

We introduce new classes of weighted automata on words. Equipped with pebbles and a two-way mechanism, they go beyond the class of recognizable formal power series, but capture a weighted version of first-order logic with bounded transitive closure. In contrast to previous work, this logic allows for unrestricted use of universal quantification. Our main result states that pebble weighted automata, nested weighted automata, and this weighted logic are expressively equivalent. We also give new logical characterizations of the recognizable series.

Distributed synthesis for well-connected architectures

We study the synthesis problem for external linear or branching specifications and distributed, synchronous architectures with arbitrary delays on processes. External means that the specification only relates input and output variables. We introduce the subclass of uniformly well-connected (UWC) architectures for which there exists a routing allowing each output process to get the values of all inputs it is connected to, as soon as possible. We prove that the distributed synthesis problem is decidable on UWC architectures if and only if the output variables are totally ordered by their knowledge of input variables. We also show that if we extend this class by letting the routing depend on the output process, then the previous decidability result fails. Finally, we provide a natural restriction on specifications under which the whole class of UWC architectures is decidable.

An automata-theoretic approach to the word problem for ω -terms over R

This paper studies the pseudovariety R of all finite R-trivial semigroups. We give a representation of pseudowords over R by infinite trees, called R-trees. Then we show that a pseudoword is an @w-term if and only if its associated tree is regular (i.e. it can be folded into a finite graph), or equivalently, if the @w-term has a finite number of tails. We give a linear algorithm to compute a compact representation of the R-tree for @w-terms, which yields a linear solution of the word problem for @w-terms over R. We finally exhibit a basis for the @w-variety generated by R and we show that there is no finite basis. Several results can be compared to recent work of Bloom and Choffrut on long words.