Decidable weighted expressions with Presburger combinators

Abstract

In this paper, we investigate the expressive power and the algorithmic properties of weighted expressions, which define functions from finite words to integers. First, we consider a slight extension of an expression formalism, introduced by Chatterjee. et. al. in the context of infinite words, by which to combine values given by unambiguous (max, +)-automata, using Presburger arithmetic. We show that important decision problems such as emptiness, universality and comparison are PSPACE-C for these expressions. We then investigate the extension of these expressions with Kleene star. This allows to iterate an expression over smaller fragments of the input word, and to combine the results by taking their iterated sum. The decision problems turn out to be undecidable, but we introduce the decidable and still expressive class of synchronised expressions.

The boundedness and zero isolation problems for weighted automata over nonnegative rationals

“…The decidability border between the finitely-ambiguous and polynomially-ambiguous classes is not surprising. It is often the case that undecidable problems for weighted automata are decidable for the finitely-ambiguous class [19]; and remain undecidable even for very restricted variants of polynomially-ambiguous automata, e.g. copyless linear CRA [1].…”