# Faculty of Information Technology, BUT

Publication Details

# What else is decidable about integer arrays?

HABERMEHL Peter, IOSIF Radu and VOJNAR Tomáš. What else is decidable about integer arrays?. TR-2007-8, Grenoble: VERIMAG, 2008.
Czech title
Co dále je rozhodnutelné o polích?
Type
technical report
Language
english
Authors
Habermehl Peter (UPAR7)
Vojnar Tomáš, prof. Ing., Ph.D. (DITS FIT BUT)
URL
Keywords
mathematical logic, arrays, decidability, decision procedure, formal verification, automata
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.
Published
2008
Pages
36
Publisher
VERIMAG
Place
TR-2007-8, Grenoble, FR
BibTeX
@TECHREPORT{FITPUB8819,
author = "Peter Habermehl and Radu Iosif and Tom\'{a}\v{s} Vojnar",
title = "What else is decidable about integer arrays?",
pages = 36,
year = 2008,
location = "TR-2007-8, Grenoble, FR",
publisher = "VERIMAG",
language = "english",
url = "https://www.fit.vut.cz/research/publication/8819"
}