Two Techniques in the Area of the Star Problem

This paper deals with decision problems related to the star problem in trace monoids, which means to determine whether the iteration of a recognizable trace language is recognizable. Due to a theorem by G. Richomme from 1994 [32, 33], we know that the star problem is decidable in trace monoids which do not contain a submonoid of the form {a,c}* x {b,d}*. Here, we consider a more general problem: Is it decidable whether for some recognizable trace language and some recognizable or finite trace language P the intersection R ∩ P* is recognizable? If P is recognizable, then we show that this problem is decidale iff the underlying trace monoid does not contain a submonoid of the form {a,c}* x b*. In the case of finite languages P, we show several decidability and undecidability results

A Connection between the Star Problem and the Finite Power Property in Trace Monoids

This paper deals with a connection between two decision problems for recognizable trace languages: the star problem and the finite power property problem. Due to a theorem by Richomme from 1994 [26, 28], we know that both problems are decidable in trace monoids which do not contain a C4 submonoid. It is not known, whether the star problem or the finite power property are decidable in the C4 or in trace monoids containing a C4. In this paper, we show a new connection between these problems. Assume a trace monoid IM (Σ, I) which is isomorphic to the Cartesian Product of two disjoint trace monoids IM (Σ1, I1) and IM (Σ2, I2). Assume further a recognizable language L in IM (Σ, I) such that every trace in L contains at least one letter in Σ1 and at least in one letter in Σ2. Then, the main theorem of this paper asserts that L* is recognizable iff L has the finite power property

Some Undecidability Results related to the Star Problem in Trace Monoids

This paper deals with decision problems related to the star problem in trace monoids, which means to determine whether the iteration of a recognizable trace language is recognizable. Due to a theorem by Richomme from 1994[30,31], we know that the Star Problem is decidable in trace monoids which do not contain a C4-submonoid. The C4 is (isomorphic to) the Caresian Product of two free monoids over doubleton alphabets. It is not known, whether the Star Problem is decidable in C4 or in trace monoids containing a C4. In this paper, we show undecidability of some related problems: Assume a trace monoid which contains a C4. Then, it is undecidable whether for two given recognizable languages K and L, we have K ⊆ L*, although we can decide K* ⊆ L. Further, we can not decide recognizability of K ∩ L* as well as universality and recognizability of K U L*

Regular Trace Event Structures

We propose trace event structures as a starting point for constructing effective branching time temporal logics in a non-interleaved setting. As a first step towards achieving this goal, we define the notion of a regular trace event structure. We then provide some simple characterizations of this notion of regularity both in terms of recognizable trace languages and in terms of finite 1-safe Petri nets

Decidability Equivalence between the Star Problem and the Finite Power Problem in Trace Monoids

In the last decade, some researches on the star problem in trace monoids (is the iteration of a recognizable language also recognizable?) has pointed out the interest of the finite power property to achieve partial solutions of this problem. We prove that the star problem is decidable in some trace monoid if and only if in the same monoid, it is decidable whether a recognizable language has the finite power property. Intermediary results allow us to give a shorter proof for the decidability of the two previous problems in every trace monoid without C4-submonoid. We also deal with some earlier ideas, conjectures, and questions which have been raised in the research on the star problem and the finite power property, e.g. we show the decidability of these problems for recognizable languages which contain at most one non-connected trace

Index problems for game automata

For a given regular language of infinite trees, one can ask about the minimal number of priorities needed to recognize this language with a non-deterministic, alternating, or weak alternating parity automaton. These questions are known as, respectively, the non-deterministic, alternating, and weak Rabin-Mostowski index problems. Whether they can be answered effectively is a long-standing open problem, solved so far only for languages recognizable by deterministic automata (the alternating variant trivializes). We investigate a wider class of regular languages, recognizable by so-called game automata, which can be seen as the closure of deterministic ones under complementation and composition. Game automata are known to recognize languages arbitrarily high in the alternating Rabin-Mostowski index hierarchy; that is, the alternating index problem does not trivialize any more. Our main contribution is that all three index problems are decidable for languages recognizable by game automata. Additionally, we show that it is decidable whether a given regular language can be recognized by a game automaton