Publication detail

# What else is decidable about integer arrays?

HABERMEHL, P., RADU, I., VOJNAR, T.

Original Title

What else is decidable about integer arrays?

English Title

What else is decidable about integer arrays?

Type

presentation

Language

en

Original Abstract

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.

English abstract

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.

Keywords

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

Released

03.12.2008

Publisher

VERIMAG

Location

TR-2007-8, Grenoble

Pages count

36

URL

Documents

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 be
```

decidable as a consequence of earlier results on counter automata with a flat control structure and transitions based on

difference constraints.",
address="VERIMAG",
chapter="63915",
institution="VERIMAG",
year="2008",
month="december",
publisher="VERIMAG",
type="presentation"
}