Detail publikace

# What else is decidable about integer arrays?

Originální název

What else is decidable about integer arrays?

Anglický název

What else is decidable about integer arrays?

Jazyk

en

Originální abstrakt

This report is the full version of the corresponding FOSSCAS'08 paper, including full proofs of the achived results. In the work, we introduce a new decidable logic for reasoning about infinite arrays of integers. The logic is in the $\exists^* \forall^*$ first-order fragment and allows (1) Presburger constraints on existentially quantified variables, (2) difference constraints as well as periodicity constraints on universally quantified indices, and (3) difference constraints on values. In particular, using our logic, one can express constraints on consecutive elements of arrays (e.g., $\forall i ~.~ 0 \leq i < n \rightarrow a[i+1]=a[i]-1$) as well as periodic facts (e.g., $\forall i ~.~ i \equiv_2 0 \rightarrow a[i] = 0$). The decision procedure follows the automata-theoretic approach: we translate formulae into a special class of B\"uchi counter automata such that any model of a formula corresponds to an accepting run of an automaton, and vice versa. The emptiness problem for this class of counter automata is shown to be
decidable as a consequence of earlier results on counter automata with a flat control structure and transitions based on
difference constraints.

Anglický abstrakt

This report is the full version of the corresponding FOSSCAS'08 paper, including full proofs of the achived results. In the work, we introduce a new decidable logic for reasoning about infinite arrays of integers. The logic is in the $\exists^* \forall^*$ first-order fragment and allows (1) Presburger constraints on existentially quantified variables, (2) difference constraints as well as periodicity constraints on universally quantified indices, and (3) difference constraints on values. In particular, using our logic, one can express constraints on consecutive elements of arrays (e.g., $\forall i ~.~ 0 \leq i < n \rightarrow a[i+1]=a[i]-1$) as well as periodic facts (e.g., $\forall i ~.~ i \equiv_2 0 \rightarrow a[i] = 0$). The decision procedure follows the automata-theoretic approach: we translate formulae into a special class of B\"uchi counter automata such that any model of a formula corresponds to an accepting run of an automaton, and vice versa. The emptiness problem for this class of counter automata is shown to be
decidable as a consequence of earlier results on counter automata with a flat control structure and transitions based on
difference constraints.

BibTex


@misc{BUT63915,
author="Peter {Habermehl} and Iosif {Radu} and Tomáš {Vojnar}",
title="What else is decidable about integer arrays?",
annote="This report is the full version of the corresponding FOSSCAS'08 paper, including full proofs of the achived results. In the work, we introduce a new decidable logic for reasoning about infinite arrays
of integers. The logic is in the $\exists^* \forall^*$ first-order
fragment and allows (1) Presburger constraints on existentially
quantified variables, (2) difference constraints as well as periodicity
constraints on universally quantified indices, and (3) difference
constraints on values. In particular, using our logic, one can express
constraints on consecutive elements of arrays (e.g., $\forall i ~.~ 0 \leq i < n \rightarrow a[i+1]=a[i]-1$) as well as periodic facts
(e.g., $\forall i ~.~ i \equiv_2 0 \rightarrow a[i] = 0$). The decision
procedure follows the automata-theoretic approach: we translate
formulae into a special class of B\"uchi counter automata such that any
model of a formula corresponds to an accepting run of an automaton, and
vice versa. The emptiness problem for this class of counter automata is
shown to bedecidable as a consequence of earlier results on counter automata with a flat control structure and transitions based ondifference constraints.",
}