Decision Procedures for Sequence Theories (Technical Report)

TL;DR

This paper extends the decidability of string theories with concatenation and regular constraints to sequence theories over Boolean algebra alphabet theories, introduces a new solver SeCo, and explores the limits of automata interpretations.

Contribution

It demonstrates decidability results for sequence theories with parametric automata and develops a new solver, SeCo, for practical applications.

Findings

Decidability holds with parametric automata but not with register automata.

SeCo effectively handles array invariants and symbolic register automata benchmarks.

Adding length constraints makes the problem Turing-equivalent to word equations.

Abstract

Sequence theories are an extension of theories of strings with an infinite alphabet of letters, together with a corresponding alphabet theory (e.g. linear integer arithmetic). Sequences are natural abstractions of extendable arrays, which permit a wealth of operations including append, map, split, and concatenation. In spite of the growing amount of tool support for theories of sequences by leading SMT-solvers, little is known about the decidability of sequence theories, which is in stark contrast to the state of the theories of strings. We show that the decidable theory of strings with concatenation and regular constraints can be extended to the world of sequences over an alphabet theory that forms a Boolean algebra, while preserving decidability. In particular, decidability holds when regular constraints are interpreted as parametric automata (which extend both symbolic automata and variable automata), but fails when interpreted as register automata (even over the alphabet theory of equality). When length constraints are added, the problem is Turing-equivalent to word equations with length (and regular) constraints. Similar investigations are conducted in the presence of symbolic transducers, which naturally model sequence functions like map, split, filter, etc. We have developed a new sequence solver, SeCo, based on parametric automata, and show its efficacy on two classes of benchmarks: (i) invariant checking on array-manipulating programs and parameterized systems, and (ii) benchmarks on symbolic register automata.