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Equations Over Languages and Finite Automata

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Abstract

An alphabet is a finite set of symbols. The set of all finite strings over a fixed alphabet X is denoted by X ⋆. X ⋆ includes the empty string ε. A subset L⊆X ⋆ is called a language over alphabet X.

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Notes

  1. 1.

    By the given definition in agreement with standard textbooks (see  [63]), a DFA must be a complete FA, but a complete FA does not need to be deterministic.

  2. 2.

    This definition is an abstraction to languages of the most common definition of Moore automata/finite state machines.

  3. 3.

    Use the same order I ×U ×O in the languages (L 1) ↑O and (L 2) ↑I .

  4. 4.

    It is not uncommon in practice to find that \(\vert \tilde{S}\vert \leq \vert S\vert \).

  5. 5.

    Apply the closure construction to obtain an equivalent deterministic finite automaton without ε-moves.

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Authors and Affiliations

  1. Dipartimento D’Informatica, Università di Verona, Strada le Grazie, 15, 37134, Verona, Italy

    Tiziano Villa

  2. Department of EECS, Tomsk State University, Tomsk, Russia

    Nina Yevtushenko

  3. Department of Electrical Engineering and Computer Science, University of California, Berkeley, California, USA

    Robert K. Brayton

  4. Department of Electrical Engineering and Computer Science (EECS), University of California, Berkeley, Berkeley, California, USA

    Alan Mishchenko & Alberto Sangiovanni-Vincentelli

  5. Computer Research Institute of Montreal (CRIM), Ogilvy 405, Suite 101, H3N 1M3, Montreal, Quebec, Canada

    Alexandre Petrenko

Authors
  1. Tiziano Villa
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  2. Nina Yevtushenko
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  3. Robert K. Brayton
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  4. Alan Mishchenko
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  5. Alexandre Petrenko
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  6. Alberto Sangiovanni-Vincentelli
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Corresponding author

Correspondence to Tiziano Villa .

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© 2012 Springer Science+Business Media, LLC

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Villa, T., Yevtushenko, N., Brayton, R.K., Mishchenko, A., Petrenko, A., Sangiovanni-Vincentelli, A. (2012). Equations Over Languages and Finite Automata. In: The Unknown Component Problem. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-68759-9_2

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  • DOI: https://doi.org/10.1007/978-0-387-68759-9_2

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  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-0-387-34532-1

  • Online ISBN: 978-0-387-68759-9

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