Efficient testing of equivalence of words in a free idempotent semigroup

Abstract

We present an automata-theoretic approach to a simple Burnside-type problem for semigroups. For two words of total length n over an alphabet ∑, we give an algorithm with time complexity and space complexity O(n) which tests their equivalence under the idempotency relation x 2≈x. The algorithm verifies whether one word can be transformed to another one by repetitively replacing any factor x 2 by x or z by z 2. We show that the problem can be reduced to equivalence of acyclic deterministic automata of size O(n.|∑|). An interesting feature of our algorithm is small space complexity - equivalence of introduced automata is checked in space O(n), which is significantly less than the sizes of the automata. This is achieved by processing the acyclic automata layer by layer, each layer only of size O(n), hence only small part of a large virtual automaton is kept in the memory. © 2010 Springer-Verlag Berlin Heidelberg.