Decidable and Undecidable Fragments of First-Order Concatenation Theory

Abstract

We identify a number of decidable and undecidable fragments of first-order concatenation theory. We also give a purely universal axiomatization which is complete for the fragments we identify. Furthermore, we prove some normal-form results.

First-order concatenation theory with bounded quantifiers

“…More on the history of concatenation theory can be found in Visser [24]. The present paper is a significantly extended and improved version of the conference paper Kristiansen and Murwanashyaka [13]. We assume that the reader is familiar with the basics of mathematical logic and computability theory.…”

“…In [13] we use the Post's Correspondence Problem (Post [18]) to prove that the fragments Σ D 3,0,2 , Σ D 4,1,1 , Σ B 1,2,1 and Σ B 1,0,2 are undecidable. We will improve these results considerably in the next section.…”

First-Order Concatenation Theory with Bounded Quantifiers

“…The present paper is a significantly extended an improved version of the conference paper Kristiansen & Murwanashyaka [13]. We assume that the reader is familiar with the basics of mathematical logic and computability theory.…”

Weak essentially undecidable theories of concatenation

“…The theory WD is thus the weakest Σ 1 -complete axiomatization of D (modulo closure under logical implication) and the theory D is a natural finitely axiomatizable extension of WD. A variant of D where we have an identity element was introduced in Kristiansen and Murwanashyaka [9] as a Σ 1 -complete axiomatization of the structure D extended with the empty string. In [9,10], we identify a number of decidable and undecidable fragments of D and related structures.…”

“…A variant of D where we have an identity element was introduced in Kristiansen and Murwanashyaka [9] as a Σ 1 -complete axiomatization of the structure D extended with the empty string. In [9,10], we identify a number of decidable and undecidable fragments of D and related structures.…”