Formal Aspects of Computing

, Volume 22, Issue 5, pp 537–545 | Cite as

Solutions of equations in languages

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Original Article
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Abstract

A context-free grammar corresponds to a system of equations in languages. The language generated by the grammar is the smallest solution of the system. We give a necessary and sufficient condition for an arbitrary solution to be the smallest one. We revive an old criterion to decide that a grammar has a unique solution. All this fits in an approach to search for a grammar for an arbitrary language that is given by other means. The approach is illustrated by the derivation of a grammar for a certain set of bit strings. The approach is used to give an elegant derivation of the grammar for a language accepted by a pushdown automaton.

Keywords

Language Grammar Context-free Pushdown automaton Unguarded recursion 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution,and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. Bod65a.
    Bodnarchuk VG (1965) The metrical space of events, part I. Cybernet Syst Anal 1(1): 20–24CrossRefGoogle Scholar
  2. Bod65b.
    Bodnarchuk VG (1965) The metrical space of events, part II. Cybernet Syst Anal 1(4): 27–36MathSciNetGoogle Scholar
  3. Cho62.
    Chomsky N (1962) Context-free grammar and pushdown storage. Quarterly Progress Report, MIT Research Lab. in Electronics 65: 187–194Google Scholar
  4. HMU01.
    Hopcroft JE, Motwani R, Ullman JD (2001) Introduction to automata theory, languages and computation. Addison-Wesley, ReadingMATHGoogle Scholar
  5. Kup97.
    Kupka I (1997) Unique fixpoints in complete lattices with applications to formal languages and semantics. In: Freksa C, Jantzen M, Valk R (eds) Foundations of Computer Science: potential-theory-cognition, vol 1337 of LNCS. Springer, New York, pp 107–115Google Scholar
  6. McW71.
    McWhirter IP (1971) Substitution expressions. J Comput Syst Sci 5(6): 629–637MATHMathSciNetGoogle Scholar
  7. MaG87.
    Mandrioli D, Ghezzi C (1987) Theoretical foundations of Computer Science. Wiley, New YorkMATHGoogle Scholar
  8. Red65.
    Red’ko VN (1965) Some aspects of the theory of languages. Cybernet Syst Anal 1(4): 15–26CrossRefGoogle Scholar
  9. Sud06.
    Sudkamp TA (2006) Languages and machines, 3rd edn. Pearson, BostonGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Department of Computing ScienceUniversity of GroningenGroningenThe Netherlands

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