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.
E. Filiot is a research associate of F.R.S.-FNRS. This work has been supported by the following projects: the ARC Project Transform (Federation Wallonie-Brussels), the FNRS CDR project Flare.
N. Mazzocchi is a PhD funded by a FRIA fellowship from the F.R.S.-FNRS.
J.-F. Raskin is supported by an ERC Starting Grant (279499: inVEST), by the ARC project - Non-Zero Sum Game Graphs: Applications to Reactive Synthesis and Beyond - funded by the Fdration Wallonie-Bruxelles, and by a Professeur Francqui de Recherche grant awarded by the Francqui Fondation.
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Notes
- 1.
\(\#_\sigma (u)\) is the number of occurrences of \(\sigma \) in u.
- 2.
Full proofs are given in the full paper version at http://arxiv.org/abs/1706.08855.
- 3.
Also called formal series in [6].
- 4.
Sometimes, initial and final weight functions are considered in the literature [6], so that non-zero values can be assigned to \(\epsilon \).
- 5.
Chatterjee et al. studied quantitative expressions on infinite words and the automata that they consider are deterministic mean-payoff automata.
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Acknowledgements
We are very grateful to Ismaël Jecker and Nathan Lhote for fruitful discussions on this work, and for their help in establishing the undecidability result.
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Filiot, E., Mazzocchi, N., Raskin, JF. (2017). Decidable Weighted Expressions with Presburger Combinators. In: Klasing, R., Zeitoun, M. (eds) Fundamentals of Computation Theory. FCT 2017. Lecture Notes in Computer Science(), vol 10472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55751-8_20
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DOI: https://doi.org/10.1007/978-3-662-55751-8_20
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