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

Functions for the General Solution of Parametric Word Equations

: In this article we introduce the functions Fi (x 1 , x 2 ) l1,..., ls and Th (x 1 , x 2 , x 3 ) i l1,..., l2s (i = 1, 2, 3), of the word variables x i and of the natural number variables li, where s ³ 0. By means of these functions, we give exactly the general solution (i.e. the set of all the solutions) of the first basic parametric equation: x 1 x 2 x 3 x 4 = x 3 x 1 l x 2 x 5 , in a free monoid. 1. Introduction The following four parametric equations: x 1 x 2 x 3 x 4 = x 3 x 1 l x 2 x 5 , x 1 x 2 x 3 x 4 = x 2 x 3 l x 1 x 5 , x 1 x 2 2 x 3 x 4 = x 3 x 1 2 x 2 x 5 , x 1 x 2 l+1 x 3 x 4 = x 3 x 2 µ+1 x 1 x 5 , in a free monoid, are called basic equations. They arise in the graph of the prefixeequations in free monoid (cf. [2], [3]) and play an important role in the hierarchy of the parametric equations, in reason of the structures of their solutions. In particular, the general solution of any equation in a free monoid of the form F(x 1 , x 2 , x 3 ) x 4 = Y(x 1 , x 2 , ..

Towards parametrizing word equations

Classically, in order to resolve an equation u ≈ v over a free monoid X*, we reduce it by a suitable family $\cal F$ of substitutions to a family of equations uf ≈ vf, $f\in\cal F$, each involving less variables than u ≈ v, and then combine solutions of uf ≈ vf into solutions of u ≈ v. The problem is to get $\cal F$ in a handy parametrized form. The method we propose consists in parametrizing the path traces in the so called graph of prime equations associated to u ≈ v. We carry out such a parametrization in the case the prime equations in the graph involve at most three variables