Word problems over traces which are solvable in linear time

Abstract

Finite complete replacement systems over free partially commutative monoids have a decidable word problem, and if these systems are weight-reducing, then the word problem is solvable in square time. It is unknown whether this time bound is optimal, but the restriction to semi-Thus systems or vector replacement systems yields linear time. In direction to close this gap, we first give a global condition for trace replacement systems which also ensures a linear time bound: and we prove that this condition is decidable. Then we give a stronger but local condition which leads to the notion of cones and blocks which are certain traces. Trivial examples of cones and blocks are words and vectors, respectively. We show that under a mild additional assumption we may decide the confluence of a noetherian trace replacement system if all left-hand sides are cones or blocks. (In general, confluence is undecidable even for length-reducing systems). Altogether we obtain a sufficient and computable condition such that the word problem of a trace replacement system is solvable in linear time.