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Least Significant Digit First Presburger Automata

Jérôme Leroux
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Abstract

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 symmbolic 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.

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Jérôme Leroux. 2006-12-06. Least Significant Digit First Presburger Automata. https://arxiv.org/abs/cs/0612037

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