Publication detail

# What else is decidable about integer arrays?

HABERMEHL, P. IOSIF, R. VOJNAR, T.

Original Title

What else is decidable about integer arrays?

English Title

What else is decidable about integer arrays?

Type

conference paper

Language

en

Original Abstract

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.

English abstract

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.

Keywords

mathematical logic, arrays, decidability, decision procedure, formal verification, automata

RIV year

2008

Released

10.03.2008

Publisher

Springer Verlag

Location

Berlin

ISBN

978-3-540-78497-5

Book

Foundations of Software Science and Computation Structures

Edition

Lecture Notes in Computer Science

Edition number

NEUVEDEN

Pages from

475

Pages to

490

Pages count

16

Documents

BibTex


@inproceedings{BUT30752,
author="Peter {Habermehl} and Iosif {Radu} and Tomáš {Vojnar}",
title="What else is decidable about integer arrays?",
annote="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.",
}