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Alternating Weighted Automata

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Fundamentals of Computation Theory (FCT 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5699))

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Abstract

Weighted automata are finite automata with numerical weights on transitions. Nondeterministic weighted automata define quantitative languages L that assign to each word w a real number L(w) computed as the maximal value of all runs over w, and the value of a run r is a function of the sequence of weights that appear along r. There are several natural functions to consider such as Sup, LimSup, LimInf, limit average, and discounted sum of transition weights.

We introduce alternating weighted automata in which the transitions of the runs are chosen by two players in a turn-based fashion. Each word is assigned the maximal value of a run that the first player can enforce regardless of the choices made by the second player. We survey the results about closure properties, expressiveness, and decision problems for nondeterministic weighted automata, and we extend these results to alternating weighted automata.

For quantitative languages L 1 and L 2, we consider the pointwise operations max(L 1,L 2), min(L 1,L 2), 1 − L 1, and the sum L 1 + L 2. We establish the closure properties of all classes of alternating weighted automata with respect to these four operations.

We next compare the expressive power of the various classes of alternating and nondeterministic weighted automata over infinite words. In particular, for limit average and discounted sum, we show that alternation brings more expressive power than nondeterminism.

Finally, we present decidability results and open questions for the quantitative extension of the classical decision problems in automata theory: emptiness, universality, language inclusion, and language equivalence.

This research was supported in part by the Swiss National Science Foundation under the Indo-Swiss Joint Research Programme, by the European Network of Excellence on Embedded Systems Design (ArtistDesign), by the European Combest, Quasimodo, and Gasics projects, by the PAI program Moves funded by the Belgian Federal Government, and by the CFV (Federated Center in Verification) funded by the F.R.S.-FNRS.

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Author information

Authors and Affiliations

  1. Institute of Science and Technology, Austria

    Krishnendu Chatterjee

  2. Université Libre de Bruxelles (ULB), Belgium

    Laurent Doyen

  3. École Polytechnique Fédérale de Lausanne (EPFL), Switzerland

    Thomas A. Henzinger

Authors
  1. Krishnendu Chatterjee
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  2. Laurent Doyen
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  3. Thomas A. Henzinger
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Editor information

Editors and Affiliations

  1. Institute of Mathematics and Computer Science, Wrocław University of Technology, ul. Wybrzeże Wyspiańskiego 27, 50-370, Wrocław, Poland

    Mirosław Kutyłowski  & Maciej Gębala  & 

  2. Institute of Computer Science, University of Wrocław, ul. Joliot-Curie 15, 50-383, Wrocław, Poland

    Witold Charatonik

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Chatterjee, K., Doyen, L., Henzinger, T.A. (2009). Alternating Weighted Automata. In: Kutyłowski, M., Charatonik, W., Gębala, M. (eds) Fundamentals of Computation Theory. FCT 2009. Lecture Notes in Computer Science, vol 5699. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03409-1_2

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  • DOI: https://doi.org/10.1007/978-3-642-03409-1_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03408-4

  • Online ISBN: 978-3-642-03409-1

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