First-order concatenation theory with bounded quantifiers

Abstract

We study first-order concatenation theory with bounded quantifiers. We give axiomatizations with interesting properties, and we prove some normal-form results. Finally, we prove a number of decidability and undecidability results.

Weak Essentially Undecidable Theories of Concatenation II

“…By Q 7 ≡ ∀xy [ x × Sy = x × y + x ] 0 × Su = 0u + 0 = 0 + 0 = 0 where the second equality follows from the assumption that u satisfies (6) and the last equality holds by Q 4 . Thus, Su satisfies (6).…”

“…(1) 0 + u = u We verify that 0 ∈ N 0 . We need to show that 0 satisfies (1)- (6). By Q 4 ≡ ∀x [ x + 0 = x ], 0 satisfies (1)- (5).…”

“…By Q 4 ≡ ∀x [ x + 0 = x ], 0 satisfies (1)- (5). By Q 6 ≡ ∀x [ x0 = 0 ], 0 satisfies (6). Thus, 0 ∈ N 0 .…”

“…Murwanashyaka [6]). Σ 1 -formulas are formulas on negation normal form where universal quantifiers occur bounded, i.e., they are of the form ∀x t. Axioms D 4 −D 7 are essential for coding sequences in D since they allow us to work with Σ 0 -formulas, formulas where all quantifiers are of the form ∃ t, ∀x t. In [10], we show that Q is interpretable in D by using especially axioms D 4 −D 7 to restrict the universe of D to a domain K on which the analogue of Q 3 holds, that is, the sentence…”

On Interpretability Between Some Weak Essentially Undecidable Theories

“…More on AST and adjunctive set theory can found in Damnjanovic [2]. For recent results related to the work in the present paper, we refer the reader to Jerabek [5], Cheng [1] and Kristiansen and Murwanashyaka [7].…”