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

What is Decidable about Strings?

We prove several decidability and undecidability results for the satisfiability/validity problem of formulas over a language of finite-length strings and integers (interpreted as lengths of strings). The atomic formulas over this language are equality over string terms (word equations), linear inequality over length function (length constraints), and membership predicate over regularexpressions (r.e.). These decidability questions are important in logic, program analysis and formal verification. Logicians have been attempting to resolve some of these questions for many decades, while practical satisfiability procedures for these formulas are increasingly important in the analysis of string-manipulating programs such as web applications and scripts. We prove three main theorems. First, we consider Boolean combination of quantifier-free formulas constructed out of word equations and length constraints. We show that if word equations can be converted to a solved form, a form relevant in practice, then the satisfiability problem for Boolean combination of word equations and length constraints is decidable. Second, we show that the satisfiability problem for word equations in solved form that areregular, length constraints and r.e. membership predicate is also decidable. Third, we show that the validity problem for the set of sentences written as a forall-exists quantifier alternation applied to positive word equations is undecidable. A corollary of this undecidability result is that this set is undecidable even with sentences with at most two occurrences of a string variable

Chain-Free String Constraints (Technical Report)

We address the satisfiability problem for string constraints that combine relational constraints represented by transducers, word equations, and string length constraints. This problem is undecidable in general. Therefore, we propose a new decidable fragment of string constraints, called weakly chaining string constraints, for which we show that the satisfiability problem is decidable. This fragment pushes the borders of decidability of string constraints by generalising the existing straight-line as well as the acyclic fragment of the string logic. We have developed a prototype implementation of our new decision procedure, and integrated it into in an existing framework that uses CEGAR with under-approximation of string constraints based on flattening. Our experimental results show the competitiveness and accuracy of the new framework