String matching problems over free partially commutative monoids

AbstractThis paper studies two string matching problems over free partially commutative monoids. We analyze these two problems in detail, and present two efficient polynomial time algorithms for solving them

Rational, recognizable, and aperiodic sets in the partially lossy queue monoid

Partially lossy queue monoids (or plq monoids) model the behavior of queues that can forget arbitrary parts of their content. While many decision problems on recognizable subsets in the plq monoid are decidable, most of them are undecidable if the sets are rational. In particular, in this monoid the classes of rational and recognizable subsets do not coincide. By restricting multiplication and iteration in the construction of rational sets and by allowing complementation we obtain precisely the class of recognizable sets. From these special rational expressions we can obtain an MSO logic describing the recognizable subsets. Moreover, we provide similar results for the class of aperiodic subsets in the plq monoid

On the decidability of some problems about rational subsets of free partially commutative monoids

Let I=A+B be a partially commutative alphabet such that two letters commute if one of them belongs to A and the other one belongs to B. Let M=A* B* denote the free partially commutative monoid generated by I. We consider the following six problems for rational (given by regular expressions) subsets X, Y of M: Q1:X ∩ Y= O? Q2: X ⊆ Y? DI X = Y? Q4: X = M? L25: M — X finite? X is recognizable? It was proved by Choffrut (see [2]) that all these problems are undecidable if Card A > 1 and Card B > 1, and they are decidable if Card A = Card B = 1 (Card U denotes the cardinality of U). It was conjectured (see [2], p 79) that these problems are decidable in the remaining cases, where Card A = 1 and Card B > 1. In this paper we show that if Card A = 1 and Card B > 1 then the problem 01 is decidable, and problems Q2-06 are undecidable. Our paper is an application of results concerning reversal-bounded nondeterministic multicounter machines and nondeterministic general sequential machines