Peter Habermehl, Radu Iosif, Tomas Vojnar

A Logic of Singly Indexed Arrays (2008)

A Logic of Singly Indexed Arrays (2008)

TR-2008-9.ps

**Keywords:**

**Abstract:**We present a logic interpreted over integer arrays, which allows difference bound comparisons between array elements situated within a constant sized window. It is shown that the satisfiability problem for the logic is undecidable for the class of formulae with quantifier prefix $\{\exists,\forall\}^*\forall^*\exists^*\forall^*$. For formulae with quantifier prefixes in the $\exists^*\forall^*$ fragment, decidability is established by an automata-theoretic argument. For each formula in the $\exists^*\forall^*$ fragment we can build a~flat counter automaton with difference bound transition rules (FCADBM) whose traces correspond to the models of the formula. The construction is modular, following the syntax of the formula. Decidability of the $\exists^*\forall^*$ fragment of the logic is a consequence of the fact that reachability of a control state is decidable for FCADBM.