Publication detail

What else is decidable about integer arrays?

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

Original Title

What else is decidable about integer arrays?

Type

report

Language

English

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.

Keywords

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

Authors

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

Released

3. 12. 2008

Publisher

VERIMAG

Location

TR-2007-8, Grenoble

Pages count

36

URL

http://www.fit.vutbr.cz/~vojnar/Publications/hiv-arrays-tr-07.pdf

BibTex

@techreport{BUT63915,
  author="Peter {Habermehl} and Iosif {Radu} and Tomáš {Vojnar}",
  title="What else is decidable about integer arrays?",
  year="2008",
  publisher="VERIMAG",
  address="TR-2007-8, Grenoble",
  pages="36",
  url="http://www.fit.vutbr.cz/~vojnar/Publications/hiv-arrays-tr-07.pdf"
}