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Decidable Weighted Expressions with Presburger Combinators

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Fundamentals of Computation Theory (FCT 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10472))

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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. 1.

    \(\#_\sigma (u)\) is the number of occurrences of \(\sigma \) in u.

  2. 2.

    Full proofs are given in the full paper version at http://arxiv.org/abs/1706.08855.

  3. 3.

    Also called formal series in [6].

  4. 4.

    Sometimes, initial and final weight functions are considered in the literature [6], so that non-zero values can be assigned to \(\epsilon \).

  5. 5.

    Chatterjee et al. studied quantitative expressions on infinite words and the automata that they consider are deterministic mean-payoff automata.

References

  1. Alur, R., Freilich, A., Raghothaman, M.: Regular combinators for string transformations. In: CSL, pp. 9:1–9:10 (2014)

    Google Scholar 

  2. Chatterjee, K., Doyen, L., Edelsbrunner, H., Henzinger, T.A., Rannou, P.: Mean-payoff automaton expressions. In: Gastin, P., Laroussinie, F. (eds.) CONCUR 2010. LNCS, vol. 6269, pp. 269–283. Springer, Heidelberg (2010). doi:10.1007/978-3-642-15375-4_19

    Chapter  Google Scholar 

  3. Chatterjee, K., Doyen, L., Henzinger, T.A.: Quantitative languages. ACM Trans. Comput. Log. 11(4) (2010)

    Google Scholar 

  4. Chatterjee, K., Henzinger, T.A., Otop, J.: Nested weighted automata. In: LICS (2015)

    Google Scholar 

  5. Daviaud, L., Guillon, P., Merlet, G.: Comparison of max-plus automata and joint spectral radius of tropical matrices. CoRR, abs/1612.02647 (2016)

    Google Scholar 

  6. Droste, M., Kuich, W., Vogler, H.: Handbook of Weighted Automata. Springer, Heidelberg (2009)

    Book  MATH  Google Scholar 

  7. Eilenberg, S., Schützenberger, M.P.: Rational sets in commutative monoids. J. Algebra 13, 173–191 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  8. Filiot, E., Gentilini, R., Raskin, J.-F.: Finite-valued weighted automata. In: FSTTCS, pp. 133–145 (2014)

    Google Scholar 

  9. Filiot, E., Gentilini, R., Raskin, J.-F.: Quantitative languages defined by functional automata. LMCS 11(3) (2015)

    Google Scholar 

  10. Gurari, E.M., Ibarra, O.H.: The complexity of decision problems for finite-turn multicounter machines. In: Even, S., Kariv, O. (eds.) ICALP 1981. LNCS, vol. 115, pp. 495–505. Springer, Heidelberg (1981). doi:10.1007/3-540-10843-2_39

    Chapter  Google Scholar 

  11. Klimann, I., Lombardy, S., Mairesse, J., Prieur, C.: Deciding unambiguity and sequentiality from a finitely ambiguous max-plus automaton. TCS 327(3) (2004)

    Google Scholar 

  12. Krob, D.: The equality problem for rational series with multiplicities in the tropical semiring is undecidable. Int. J. Algebra Comput. 4(3), 405–425 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Velner, Y.: The complexity of mean-payoff automaton expression. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012. LNCS, vol. 7392, pp. 390–402. Springer, Heidelberg (2012). doi:10.1007/978-3-642-31585-5_36

    Chapter  Google Scholar 

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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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Correspondence to Nicolas Mazzocchi .

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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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