Whats 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.

💡 Research Summary

The paper introduces a first‑order logic called the Theory of Sequences (Seq) that integrates three ingredients: finite sequences of integer elements, Presburger arithmetic, and regular language constraints. The authors argue that existing decidable logics for arrays either lack the ability to relate element values to their positions in a fine‑grained arithmetic way, or they require heavy extensions that quickly become undecidable. By treating a sequence as a primitive object equipped with concatenation, length, and indexed access, Seq can naturally express properties such as sortedness, injectivity, and complex mixed arithmetic‑periodic conditions (e.g., “for every even index i, the element a