Least Significant Digit First Presburger Automata

Jérôme Leroux (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

Since 1969 \cite{C-MST69,S-SMJ77}, we know that any Presburger-definable set \cite{P-PCM29} (a set of integer vectors satisfying a formula in the first-order additive theory of the integers) can be represented by a state-based symbolic representation, called in this paper Finite Digit Vector Automata (FDVA). Efficient algorithms for manipulating these sets have been recently developed. However, the problem of deciding if a FDVA represents such a set, is a well-known hard problem first solved by Muchnik in 1991 with a quadruply-exponential time algorithm. In this paper, we show how to determine in polynomial time whether a FDVA represents a Presburger-definable set, and we provide in this positive case a polynomial time algorithm that constructs a Presburger-formula that defines the same set.

Keywords: Presburger; Semi-linear; Arithmetic; Logic; Automaton

Domains: Data Structures and Algorithms [cs.DS]

Jérôme Leroux. Least Significant Digit First Presburger Automata. 2006. ⟨hal-00118748⟩