THE EXISTENTIAL THEORY OF CONCATENATION OVER A FINITE ALPHABET

STEVEN ORVILLE SENGER, Purdue University

Abstract

This paper investigates several problems dealing with the existential theory of concatenation. In chapter 1 we study several partial orders on words which are of fundamental importance. We also show that the full existential theory of concatenation is equivalent to the solvability of equations in concatenation. That is, we can replace with a single equation the conjunction, disjunction and negation of equations. While the result on disjunction is not of importance for the question of solvability we use it to show that the solutions of an equation over an arbitrary alphabet may be encoded by an equation over a two letter alphabet. Further solvability of the original equation is equivalent to solvability of the encoded equation. In chapter 2 we examine the algorithm given by Makanin for deciding whether an equation in concatenation is solvable and obtain information on which sets are definable by formulas of the existential theory of concatenation. In particular we show that no set or operation giving a notion of length is definable from concatenation alone. We end by developing a conjecture which would simplify Makanin's algorithm and show that the two letter submonoid of a three letter alphabet is not existentially definable from concatenation. In chapter 3 we examine several notions of length and related operators, proving that the existential theory of concatenation augmented by several of these operators is undecidable. We also examine embeddings of concatenation in fragments of arithmetic, obtaining a particular fragment whose existential theory is undecidable.

Degree

Ph.D.

Subject Area

Mathematics

COinS