Daniel Kirsten

Deciding Unambiguity and Sequentiality of Polynomially Ambiguous Min-Plus Automata

This paper solves the unambiguity and the sequentiality problem for polynomially ambiguous min-plus automata. This result is proved through a decidable algebraic characterization involving so-called metatransitions and an application of results from the structure theory of finite semigroups. It is noteworthy that the equivalence problem is known to be undecidable for polynomially ambiguous automata.

Desert Automata and the Finite Substitution Problem

We give a positive solution to the so-called finite substitution problem which was open for more than 10 years [11]: given recognizable languages K and L, decide whether there exists a finite substitution σ such that σ(K)=L. For this, we introduce a new model of weighted automata and show the decidability of its limitedness problem by solving the underlying Burnside problem.

Alternating tree automata and parity games

Since Buchi’s work in 1960 [17], automata play an important role in logic. Numerous different notions of automata provide decision and complexity results in various kinds of logic. Often, one develops a method to translate some given formula ϕ into an appropriate finite automaton A such that L(ϕ) = L(A). Such a translation reduces the model checking problem and the satisfiability problem in some logic to the word problem and the emptiness problem for finite automata. Moreover, such a translation provides algorithms to solve the model checking and the satisfiability problems on a computer. Consequently, one is interested in the decidability and the complexity of the word and emptiness problems of automata.

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

Abstract. In the last decade, research on the star problem in trace monoids (is the iteration of a recognizable language also recognizable?) has pointed out the importance of the finite power property to achieve partial solutions to 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. Intermediate results allow us to give a shorter proof for the decidability of the two previous problems in every trace monoid without a 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.

A Burnside Approach to the Finite Substitution Problem

AbstractWe introduce a new model of weighted automata: the desert automata. We show that their limitedness problem is PSPACE-complete by solving the underlying Burnside problem. As an application of this result, we give a positive solution to the so-called finite substitution problem which was open for more than 10 years: given recognizable languages K and L, decide whether there exists a finite substitution σ such that σ(K) = L.

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 [18], we know that the star problem is decidable in trace monoids which do not contain a C4-submonoid. It is not known whether the star problem is decidable in 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 ∪ L*.

Two Techniques in the Area of the Star Problem

Decidability of the Star Problem, the problem whether the language P* is recognizable for a recognizable language P, remains open. We slightly generalize the problem and show that then its decidability status depends strongly on the assumptions considering the trace monoid and finiteness of P. More precisely, we show that for finite set P ⊂ {A,B}* × {C}* and recognizable R it is decidable whether P* ∩ R is recognizable, but the problem becomes undecidable if we consider recognizable (infinite) P or finite IP ⊂ {A,B}* × {C,D}*.

The Star Problem and the Finite Power Property in Trace Monoids

We deal with the star problem in trace monoids which means to decide whether the iteration of a recognizable trace language is recognizable. We consider trace monoids Kn={a1, b1}*×···×{an, bn}*. Our main result asserts that the star problem is decidable in some trace monoid M iff it is decidable in the biggest Kn submonoid M. Consequently, future research on the star problem can focus on the trace monoids Kn. We develop the main results of the paper for the finite power problem. Then, we establish the link to the star problem by applying the recently shown decidability equivalence between the star problem and the finite power problem (D. Kirsten and G. Richomme, 2001, Theory Comput. Systems34, 193?227).

The Star Problem in Trace Monoids: Reductions Beyond C4

We deal with the star problem in trace monoids which means to decide whether the iteration of a recognizable trace language is recognizable. We consider trace monoids Kn = {a1, b1}ċ ×... × {an, bn}ċ. Our main theorem asserts that the star problem is decidable in a trace monoid M iff it is decidable in the biggest Kn submonoid in M. Thus, future research on the star problem can focus on the trace monoids Kn. The recently shown decidability equivalence between the star problem and the finite power problem [14] plays a crucial role in the paper.

A Connection between the Star Problem and the Finite Power Property in Trace Monoids (Extended Abstract)

This paper deals with a connection between the star problem and the finite power problem in trace monoids. Both problems are decidable in trace monoids without C4 submonoid [21] but remain open in all other trace monoids.