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Approximate Comparison of Functions Computed by Distance Automata

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Abstract

Distance automata are automata weighted over the semiring \((\mathbb {N}\cup \{\infty \},\min , +)\) (the tropical semiring). Such automata compute functions from words to \(\mathbb {N}\cup \{\infty \}\). It is known from Krob that the problems of deciding ‘ f≤g’ or ‘ f=g’ for f and g computed by distance automata is an undecidable problem. The main contribution of this paper is to show that an approximation of this problem is decidable. We present an algorithm which, given ε>0 and two functions f,g computed by distance automata, answers “yes” if f≤(1−ε)g, “no” if f≦̸g, and may answer “yes” or “no” in all other cases. The core argument behind this quasi-decision procedure is an algorithm which is able to provide an approximated finite presentation of the closure under products of sets of matrices over the tropical semiring. Lastly, our theorem of affine domination gives better bounds on the precision of known decision procedures for cost automata, when restricted to distance automata.

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Notes

  1. Technically, polynomial domination is not stated in [2], but can be derived directly from the proofs which explicitly compute the function α using operations preserving polynomials.

  2. Theorem 2 holds for more general classes of automata, cost automata, for which affine domination does not hold. Affine domination is specific to distance automata.

  3. Modern proofs of this theorem can be found in [3, 7], in particular with the exact bound of N=3|T|−1 (Simon’s original proof only provides N=9|T|).

  4. It corresponds formally to the virtual weighted matrix (I n ,0) (virtual since weight 0 is not allowed).

  5. Technically, here, some entries of the matrices may become negative. Since the arguments in Corollary 1 only involve the order, this does not make any difference.

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Acknowledgments

We thank the anonymous reviewers for their very careful reading and the numerous comments that highly improved the quality of the final document. We are also very grateful to Jean Mairesse for very fruitful discussions on the subject.

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Authors and Affiliations

  1. LIAFA, CNRS UMR 7089, Université Paris Diderot - Paris 7, Case 7014, 75205, Paris Cedex 13, France

    Thomas Colcombet & Laure Daviaud

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  1. Thomas Colcombet
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  2. Laure Daviaud
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Correspondence to Laure Daviaud.

Additional information

The research leading to these results has received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement n∘ 259454, and from the project ANR 2010 BLAN 0202 02 FREC.

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Colcombet, T., Daviaud, L. Approximate Comparison of Functions Computed by Distance Automata. Theory Comput Syst 58, 579–613 (2016). https://doi.org/10.1007/s00224-015-9643-3

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