Undecidability of the State Complexity of Composed Regular Operations

Abstract

AbstractWe consider the regularity-preserving operations of intersection and marked catenation and construct an infinite sequence C i , i = 1, 2, …, of compositions formed from the two operations. We construct also an infinite sequence of polynomials S i , i = 1, 2, …, with positive integer coefficients. As a main result we prove that it is undecidable whether or not S i is a state complexity function of C i . All languages needed are over a fixed alphabet with at most 50 letters. We also consider some implications and generalizations, as well as present some open problems.Keywordsfinite deterministic automatonstate complexityundecidabilityregularity-preserving operationscomposition of operations