Celia Wrathall

Monadic Thue systems

Certain infinite Thue systems over a finite alphabet are studied, in particular, systems S⊆∑∗×(∑∪{e}) such that for each aϵ∑∪{e}, the set {u| (u,a)ϵS} is a context-freelanguage. The syntactic structure of sets of ancestors and sets of descendants is considered, as well as that of unions of congruence classes, taken over (infinite) context-free languages or regular sets. The common descendant problem is shown to be tractable while the common ancestor problem is shown to be undecidable (even for finite systems). The word problem for confluent systems of this type is shown to be tractable. The question of whether an infinite system of this type is confluent is shown to be undecidable as is the question of whether the congruence generated by such a system has a confluent presentation.

The word problem for free partially commutative groups

A linear-time algorithm is given for the word problem for free partially commutative groups. The correctness of the algorithm follows from the fact that certain Thue systems, presenting such groups, are preperfect.

Bounded query machines: On NP() and NPQUERY()

Abstract The NP() and NPQUERY() operators are studied in order to develop necessary and sufficient conditions for the class PSPACE to be equal to the union of the polynomial-time hierarchy.

Efficient solution of some problems in free partially commutative monoids

Abstract Linear-time algorithms are presented for several problems concerning words in a partially commutative monoid, including whether one word is a factor of another and whether two words are conjugate in the monoid.

A note on thue systems with a single defining relation

A combinatorial characterization is given for those one-rule Thue systems of the form {(w1,w2)} with 0≦ |w2|≦|ov(w1)| that are Church-Rosser. Here ov(w1) denotes the longest proper self-overlap ofw1. Further, it is shown that a Thue system of this form is Church-Rosser if and only if there is an equivalent Thue system that is Church-Rosser.

Overlaps in free partially commutative monoids

Abstract It is shown that the set of overlaps of a connected word in a free partially commutative monoid forms a lattice. In addition, a linear-time algorithm is presented for computing the maximum element of that lattice.

On confluence of one-rule trace-rewriting systems

The paper presents combinatorial criteria for confluence of one-rule trace-rewriting systems. The criteria are based on self-overlaps of traces, which are closely related to the notion of conjugacy of traces, and can be tested in linear time. As a special case, we reobtain the corresponding results for strings.

One-Rule Trace-Rewriting Systems and Confluence

Confluence is an undecidable property even for finite noetherian trace-rewriting systems. Here we investigate this property for the special case of trace-rewriting systems with a single rule. For certain classes of one-rule trace-rewriting systems R we present syntactic characterizations that are necessary and sufficient for R to be confluent. Based on these characterizations confluence is easily decidable for these systems. In fact, each system satisfying one of these characterizations is strongly confluent. This raises the question of whether every confluent onerule trace-rewriting system is strongly confluent. We close this paper with an example that answers this question in the negative.

A note on complete sets and transitive closure

It is shown that there is a length-preserving relation that is NP-complete such that the transitive closure of the relation is PSPACE-complete.

The lattices of prefixes and overlaps of traces

Abstract The subject of this paper is the sets of prefixes and the sets of overlaps of traces (elements of free partially commutative monoids), when the prefixes or overlaps are ordered in one natural way. Lattice-theoretic characterizations, and related properties, are developed. Specifically, it is shown that the collection of prefix-sets of traces constitutes precisely the class of finite distributive lattices and that the overlap-set of a trace is a sublattice of its prefix-set, so the overlap-sets of traces also form finite distributive lattices. Several characterizations of the class of overlap-lattices of traces are given; for example, incomparable join-irreducible elements of such lattices must meet at the minimum element of the lattice.