Class Of Transition Systems Research Articles

ON COUNTER MACHINES, REACHABILITY PROBLEMS, AND DIOPHANTINE EQUATIONS

We give a brief survey of results that use “reversal-bounded counters” in studying reachability problems for various classes of transition systems. We also discuss the connection between the decidability of reachability in counter machines and the solvability of certain Diophantine equations.

Checking Simple Properties of Transition Systems Defined by Thue Specifications

In (possibly infinite) deterministic labeled transition systems defined by Thue congruences, labels are considered as functions of states into states. This paper provides a method for computing domains of such functions for a large class of transition systems. The latter are related to model checking of transition systems defined by Thue congruences.

Bisimilarity is decidable in the union of normed BPA and normed BPP processes

Abstract We compare the classes of behaviours (transition systems) which can be generated by normed BPA˙τ and normed BPP˙τ processes. We exactly classify the intersection of these two classes, i.e., the class of transition systems which can be equivalently (up to bisimilarity) described by the syntax of normed BPA˙τ and normed BPP˙τ processes. We provide such a characterization for classes of normed BPA and normed BPP processes as well. Next we show that it is decidable in polynomial time whether for a given normed BPA˙τ (or BPP˙τ) process Δ there is some normed BPP˙τ (or BPA˙τ) process Δ such that Δ is bisimilar to Δ. Moreover, if the answer is positive then the process Δ can be effectively constructed. Simplified versions of the algorithms mentioned above for normed BPA and normed BPP are given too. As an immediate (but important) consequence we also obtain decidability of bisimilarity in the union of normed BPA˙τ and normed BPP˙τ processes.

Hierarchy of Asynchronous Automata

We study a natural notion of communication structure associated with asynchronous automata: we characterize which transition systems are isomorphic to an asynchronous automaton w.r.t. a given communication structure. For that, we present an algorithm to split global states into local states of communicating processes, similar to the regional technique for the synthesis problem of Petri nets. Our main result is an axiomatic criterion for the communication structures which decompose the same class of transition systems; this allows us to characterize and compare several particular classes of asynchronous automata. An immediate corollary of this study is a generic extension of Zielonka's theorem. We finally apply this method to asynchronous automata which describe systems of processes that communicate through shared memories.

Coalgebraic logic

We present a generalization of modal logic to logics which are interpreted on coalgebras of functors on sets. The leading idea is that infinitary modal logic contains characterizing formulas. That is, every model-world pair is characterized up to bisimulation by an infinitary formula. The point of our generalization is to understand this on a deeper level. We do this by studying a fragment of infinitary modal logic which contains the characterizing formulas and is closed under infinitary conjunction and an operation called Δ. This fragment generalizes to a wide range of coalgebraic logics. Each coalgebraic logic is determined by a functor on sets satisfying a few properties, and the formulas of each logic are interpreted on coalgebras of that functor. Among the logics obtained are the fragment of infinitary modal logic mentioned above as well as versions of natural logics associated with various classes of transition systems, including probabilistic transition systems. For most of the interesting cases, there is a characterization result for the coalgebraic logic determined by a given functor. We then apply the characterization result to get representation theorems for final coalgebras in terms of maximal elements of ordered algebras. The end result is that the formulas of coalgebraic logics can be viewed as approximations to the elements of a final coalgebra.

Bisimilarity is decidable in the union of normed BPA and normed BPP processes

We compare the classes of behaviours (transition systems) which can be generated by normed BPA˙τ and normed BPP˙τ processes. We exactly classify the intersection of these two classes, i.e., the class of transition systems which can be equivalently (up to bisimilarity) described by the syntax of normed BPA˙τ and normed BPP˙τ processes. We provide such a characterization for classes of normed BPA and normed BPP processes as well.Next we show that it is decidable in polynomial time whether for a given normed BPA˙τ (or BPP˙τ) process Δ there is some normed BPP˙τ (or BPA˙τ) process Δ such that Δ is bisimilar to Δ. Moreover, if the answer is positive then the process Δ can be effectively constructed. Simplified versions of the algorithms mentioned above for normed BPA and normed BPP are given too.As an immediate (but important) consequence we also obtain decidability of bisimilarity in the union of normed BPA˙τ and normed BPP˙τ processes.

A note on transition systems

This paper introduces a mathematical structure called transition systems. The notion of transition systems has been developed as a result of the study of Petri nets and vector addition systems. The intended use of transition systems is to model concurrent and asynchronous events. The concept of information flow in a complex system and communication between parts of a complex system can be formulated in this formal structure. This paper is concerned with the mathematical properties rather than the applications of transition systems. Patterns of activities in complex systems are defined in terms of termination and finiteness properties of transition systems. Concepts of conservation and repetitivity have been introduced. Structural properties of restricted classes of transition systems have been studied.