What's Decidable About Sequences?

TL;DR

This paper introduces a decidable first-order theory for sequences with integer elements, enabling reasoning about properties like sortedness and injectivity, with applications to data structures and program verification.

Contribution

It presents a decision procedure for the quantifier-free fragment of the theory of sequences, encoding it into the first-order theory of concatenation with PSPACE complexity.

Findings

Decidable quantifier-free fragment with PSPACE complexity

Expressive power includes sortedness, injectivity, and periodic properties

Applicable to reasoning about sequence-manipulating programs

Abstract

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.