What's Decidable About Sequences?

We present a first-order theory of sequences with integer elements, Presburger arithmetic, and regular constraints, which can model significant properties of data structures such as arrays and lists. We give a decision procedure for the quantifier-free fragment, based on an encoding into the first-order theory of concatenation; the procedure has PSPACE complexity. The quantifier-free fragment of the theory of sequences can express properties such as sortedness and injectivity, as well as Boolean combinations of periodic and arithmetic facts relating the elements of the sequence and their positions (e.g., "for all even i's, the element at position i has value i+3 or 2i"). The resulting expressive power is orthogonal to that of the most expressive decidable logics for arrays. Some examples demonstrate that the fragment is also suitable to reason about sequence-manipulating programs within the standard framework of axiomatic semantics.Comment: Fixed a few lapses in the Mergesort exampl

Theories of HNN-extensions and amalgamated products

Abstract. It is shown that the existential theory of G with rational constraints, over an HNN-extension G = 〈H, t; t −1 at = ϕ(a)(a ∈ A) 〉 is decidable, provided that the same problem is decidable in the base group H and that A is a finite group. The positive theory of G is decidable, provided that the existential positive theory of G is decidable and that A and ϕ(A) are proper subgroups of the base group H with A ∩ ϕ(A) finite. Analogous results are also shown for amalgamated products. As a corollary, the positive theory and the existential theory with rational constraints of any finitely generated virtually-free group is decidable.

2015 Self-Study

This document is intended to fulfill the self-study requirement associated with ABET’s 2015 accreditation review of the computer engineering program at Kettering University. It has been prepared in accordance with the Engineering Accreditation Commission’s ABET Self-Study Questionnaire: Template for a Self-Study Report (dated August 7, 2014), with reference to ABET’s Criteria for Accrediting Engineering Programs (dated November 7, 2014). vi

Word equations over graph products

For monoids that satisfy a weak cancellation condition, it is shown that the decidability of the existential theory of word equations is preserved under graph products. Furthermore, it is shown that the positive theory of a graph product of groups can be reduced to the positive theories of those factors, which commute with all other factors, and the existential theories of the remaining factors. Both results also include suitable constraints for the variables. Larger classes of constraints lead in many cases to undecidability results