André Arnold

Games for synthesis of controllers with partial observation

The synthesis of controllers for discrete event systems, as introduced by Ramadge and Wonham, amounts to computing winning strategies in parity games. We show that in this framework it is possible to extend the specifications of the supervised systems as well as the constraints on the controllers by expressing them in the modal µ-calculus.In order to express unobservability constraints, we propose an extension of the modal µ-calculus in which one can specify whether an edge of a graph is a loop. This extended µ-calculus still has the interesting properties of the classical one. In particular it is equivalent to automata with loop testing. The problems such as emptiness testing and elimination of alternation are solvable for such automata.The method proposed in this paper to solve a control problem consists in transforming this problem into a problem of satisfiability of a µ-calculus formula so that the set of models of this formula is exactly the set of controllers that solve the problem. This transformation relies on a simple construction of the quotient of automata with loop testing by a deterministic transition system. This is enough to deal with centralized control problems. The solution of decentralized control problems uses a more involved construction of the quotient of two automata.This work extends the framework of Ramadge and Wonham in two directions. We consider infinite behaviours and arbitrary regular specifications, while the standard framework deals only with specifications on the set of finite paths of processes. We also allow dynamic changes of the sets of observable and controllable events.

A Log (N) Distributed Mutual Exclusion Algorithm Based on Path Reversal

In this paper, we present a distributed algorithm for mutual exclusion based on path reversal. The algorithm does not use logical clocks to serialize the concurrent events, and all the variables are bounded. When a process invokes a critical section, it sends a request to the tail of a queue. A dynamical rooted tree gives the path to this tail. The algorithm requires onlyO(log(n)) messages on average, wherenis the number of processes in the network. The performance analysis of the algorithm is based on generating formal power series.

The AltaRica formalism for describing concurrent systems

The AltaRica formalism is designed for describing complex systems consisting of a number of interacting components. Its semantics is expressed in terms of transition systems so that a system described in this language can be analysed by any technique or tool applicable to transition systems. The components of a system have two kinds of interactions • event synchronisation, like in the synchronized product of transition systems of Arnold and Nivat, • interface coordination: with each component are associated interfaces whose values depend on the state of the component as well as on the values of interfaces of other components of the system. Another feature of AltaRica is the possibility of defining hierarchical systems: some subsystems can be encapsulated and their mutual interactions as well as their interactions with the rest of the system are supervised by a controller.

A linear algorithm to solve fixed-point equations on transition systems

Abstract This paper gives an algorithm to compute the least fixed-point of a system of equations over a transition system. This algorithm has a time complexity linear in the size of the transition system, thus improving the known algorithms which are quadratic.

MEC: A System for Constructing and Analysing Transition Systems

MEC is a tool for constructing and analysing transition systems modelizing processes and systems of communicating processes.

A relative of the Thue-Morse sequence

Abstract We study a sequence, c, which encodes the lengths of blocks in the Thue-Morse sequence. In particular, we show that the generating function for c is a simple product.

An algebraic characterization of transition system equivalences

Abstract I. Castellani (1987 , J. Comput. System Sci. 34, 210–235) has shown that observation equivalence of transition systems could be characterized by particular reductions: systems are equivalent if, and only if, they can be reduced to the same form. Moreover, every transition system has a minimal reduced form. We extend these results to logical equivalence, by an algebraic interpretation of temporal logics: we characterize logical equivalence of transition systems by particular reductions (saturating quasi-homomorphisms) or their power algebras of sets of states and paths and prove that every power algebra has a minimal reduced form. We then offer alternative proofs for logical characterizations of observation equivalence: in particular we apply our method to prove M. Hennessy and C. Stirlings (1984 , “Lecture Notes in Comput. Sci. Vol. 176,” pp. 301–311, Springer-Verlag, New York/Berlin) result that “Future Perfect” logic characterizes observation equivalence of generalized transition systems, i.e., systems whose infinite behaviours are restricted by arbitrary fairness constraints.

Nivat's processes and their synchronization

This short paper retraces how the notion of synchronization of processes introduced by Maurice Nivat in 1979 has evolved over more than 20 years.

Ambiguous classes in µ-calculi hierarchies

A classical result by Rabin states that if a set of trees and its complement are both Buchi definable in the monadic second order logic, then these sets are weakly definable. In the language of µ-calculi, this theorem asserts the equality between the complexity classes Σ2 ∩ Π2 and Comp(Σ1, Π1) of the fixed-point alternation-depth hierarchy of the µ-calculus of tree languages. It is natural to ask whether at higher levels of the hierarchy the ambiguous classes Σn+1 ∩ Πn+1 and the composition classes Comp(Σn, Πn) are equal, and for which µ-calculi.The first result of this paper is that the alternation-depth hierarchy of the games µ-calculus--whose canonical interpretation is the class of all complete lattices--enjoys this property. More explicitly, every parity game which is equivalent both to a game in Σn+1 and to a game in Πn+1 is also equivalent to a game obtained by composing games in Σn and Πn.The second result is that the alternation-depth hierarchy of the µ-calculus of tree languages does not enjoy the property. Taking into account that any Buchi definable set is recognized by a nondeterministic Buchi automaton, we generalize Rabins result in terms of the following separation theorem: if two disjoint languages are recognized by nondeterministic Πn+1 automata, then there exists a third language recognized by an alternating automaton in Comp(Σn, Πn) containing one and disjoint from the other.Finally, we lift the results obtained for the µ-calculus of tree languages to the propositional modal µ-calculus: ambiguous classes do not coincide with composition classes, but a separation theorem is established for disjunctive formulas.

Sémantique des processus communicants

— In order to study the behaviour ofa set of processes endlessly exchanging informations, we consider a process as a transducer of infinité words, which aîlows to define operational semantics ofa net of processes equivalent to Kahns denotational semantics.