Grammars for terms and automata

On a book by the late J. Richard Büchi
  • D. Siefkes
Part of the Lecture Notes in Computer Science book series (LNCS, volume 270)


J. Richard Büchi left behind an incomplete manuscript for a book on "Finite Automata, their Algebras and Grammars". He presents classical finite automata theory by treating automata as unary algebras, and then generalizes this approach to general algebras and their terms. He uses grammars (semi-Thue, or Post systems) to produce, recognize, and evaluate terms. Taking terms as trees, he thus deals with tree automata and regular tree systems. Writing terms in linear notation, he thus manipulates terms through contextfree grammars (including LRK-grammars) and pushdown automata. In this way he analyzes the mathematical properties of terms. For Polish notation the book is fairly complete, for the general case the treatment remains sketchy. Therefore I undertake to present here the main ideas, to make better accessible this fascinating approach to a theory of terms.


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  1. Büchi JR (1962) Mathematische Theorie des Verhaltens endlicher Automaten. Zeitschr. Ang. Math. Mech. 42, T9–T16Google Scholar
  2. Büchi JR (1964) Regular canonical systems and finite automata. Archiv Math. Logik Grundl. 6, 91–111Google Scholar
  3. Büchi JR (1966) Algebraic theory of feedback in discrete systems. In: "Automata Theory" Academic Press, 70–101Google Scholar
  4. Büchi JR, Hosken WH (1970) Canonical systems which produce periodic sets. Math. Syst. Th. 4, 81–90Google Scholar
  5. Büchi JR, Wright JB (1960) Mathematical Theory of Automata. Notes Communication Sciences 403, University of MichiganGoogle Scholar
  6. Deussen P (1978) A Unified Approach to the Generation and the Acceptation of Formal Languages. Acta Inf. 9, 377–390Google Scholar
  7. Doner JE (1970) Tree acceptors and some of their applications. J. Comp. Syst. Sci. 4, 406–451Google Scholar
  8. Ehrenfeucht A, Hoogeboom HJ, Rozenberg G (1985) On coordinated rewriting. Fundamentals Computation Theory, Lect. Notes CS 199, 100–111Google Scholar
  9. Eilenberg S, Wright JB (1967) Automata in general algebras. Inf. Control 11, 452–470Google Scholar
  10. Gécseg F, Steinby M (1984) Tree Automata. Akadémiai KiadóGoogle Scholar
  11. Goguen JA, Thatcher JW, Wagner EG, Wright JB (1977) Initial Algebra Semantics and Continuous Algebras. J. ACM 24, 68–95Google Scholar
  12. Harrison MA (1965) Introduction to Switching and Automata Theory. McGraw-HillGoogle Scholar
  13. Huet G, Oppen DC (1980) Equations and Rewrite Rules — A Survey. In: R.V. Book "Formal Language Theory". Academic Press, 349–406Google Scholar
  14. Irons ET (1961) A syntax directed compiler for ALGOL 60. Comm. ACM 4, 51–55Google Scholar
  15. Knuth DE (1965) On the translation from left to right. Inf. Control 8, 607–639Google Scholar
  16. Knuth DE (1968) Semantics of context-free languages. Math. Syst. Th. 2, 127–145Google Scholar
  17. Langmaack H (1971) Application of Regular Canonical Systems to Grammars Translatable from Left to Right. Acta Inf. 1, 111–114Google Scholar
  18. Mezei J, Wright JB (1967) Algebraic automata and context-free sets. Inf. Control 11, 3–29Google Scholar
  19. Post E (1921) Introduction to a general theory of elementary propositions. Amer. J. Math. 43, 163–185Google Scholar
  20. Post E (1936) Finite Combinatory Processes — Formulation I. J. Symb. Logic 1, 103–105Google Scholar
  21. Post E (1943) Formal reductions of the general combinatorial decision problem. Amer. J. Math. 65, 197–215Google Scholar
  22. Thatcher JW (1973) Tree automata — an informal survey. In: A.V. Aho (ed.) "Currents in the Theory of Computing". Prentice-Hall, 143–172Google Scholar
  23. Thatcher JW, Wright JB (1968) Generalized finite automata theory with an application to a decision problem of second order logic. Math. Syst. Th. 2, 57–81Google Scholar
  24. Thue A (1910) Die Lösung eines Spezialfalles eines generellen logischen Problems. Vid.-Sels. Skrifter Christiania, Math.-Nat.Kl. No. 8Google Scholar
  25. Thue A (1914) Probleme über Veränderungen von Zeichenreihen nach gegebenen Regeln. Vid.-Sels. Skrifter Christiania, Math.-Nat. Kl. No. 10Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • D. Siefkes
    • 1
  1. 1.Fachbereich Informatik, FR 6–2Technische Universität BerlinBerlin 10West-Germany

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