Abstract
Rewrite systems are directed equations used to compute by repeatedly replacing subterms of a given formula with equal terms until the simplest form possible is obtained. This simplest form is what we call a normal form. The theory of rewriting is in essence a theory of normal forms. Most frequently we are interested in ground normal forms (normal forms without any variables). Since we are only looking at ground terms, the set of ground normal forms of a rewrite system R may be viewed as a formal language generated by R.
In this paper we study the language of ground normal forms (GNFR for short) and give effective algorithms for deciding certain problems about it such as finiteness and representation. To solve these problems pumping lemmas are stated and proved for GNFR.
The results we obtain here have a number of direct applications to problems appearing in Inductive theorem proving, Logic programming, Functional programming, Machine Learning, Algebraic specifications, Equational Logic, Communicating processus etc...
Preview
Unable to display preview. Download preview PDF.
5. Bibliography
[Boyer and Moore(1979)]: A Computational Logic. (Academic Press, New York, 1979).
[Burstall (1969)]: "Proving properties of programs by structural induction" Computer Journal 12(1), 41–48, 1969.
[Colmerauer (1984)]: "Equations and Inequations on finite and infinite trees" Proc. FGCS 85–99,1984.
[Comon(1986)]: "SufficientCompleteness,Rewrite Systems and Anti-Unification" Proc.8th CADE 128–140, 1986.
[Dershowitz (1985)]: "Computing with rewrite systems" Information and Control 65(2/3) 122–157, 1985
[Gallier and Book (1985)]: "Reductions in tree replacement systems" TCS 37(2) Nov. 1985 123–150.
[Guttag and Horning (1978)]: "The algebraic specifications of abstract types" Acta Informatica 10, 27–52 1978.
[Hopcroft and Ullman (1979)]: Introduction to Automata Theory, Languages, and Computation. (Addison-Wesley, Reading. Mass. 1979)
[Huet (1980)]: "Confluent reductions: abstract properties and applications to term rewriting systems. JACM 27 (4),1980 797–821.
[Huet and Hullot (1982)]: "Proofs by induction in equational theories with constructors" JCSS 25 (2) 1982
[Jouannaud and Kounalis (1986)]: "Automatic Proofs by induction in equational theories without constructors" Proc. 1st Symp. on Logic in Computer Science, 1986. Full paper in Information and Control Vol 82 (1989) 1–33.
[Kapur, Narendran and Kapur, (1987)]: "On Sufficient Completeness and Related Properties of term rewriting systems" Acta Informatiqua 24, 395–415 (1987)
[Knuth and Bendix (1970)]: "Simple Word Problems in Universal Algebras," In Computational problems in Abstract Algebra (1970) 263–297.
[Kounalis (1985)]: "Validation des Spécifications Algébriques par Complétion Inductive" Thèse Univ. Nancy 1.
[Kounalis (1990)]: "Testing for inductive (co)-reducibility". Proc.15 th Coll. on Trees in Algebra and Programming (CAAP 90), LNCS to appear.
[Kounalis and Rusinowithc (1990)]: "Mechanizing inductive reasoning". Proc of 8th conf. of American Association in Artificial Inteligence (AAAI 90) to appear.
[Kunen 1987]: "Negation in Logic Programming". J. Logic Programming 4(4) 1987 289–308.
[Lassez and Marriott (1987)]: "Explicit Representation of terms Defined by Counterexamples". J. Automated Reasoning 3 1987 301–317.
[Lloyd (1984)]: Foundations of Logic Programming. Springer-Verlag 1984.
[Lugiez (1989)]: "A deduction procedure for first-order logic programs" Proc. 6th Conf. Logic Programming 1989.
[Maher (1988)]: "Complete axiomatizations of the algebras, rational and infinite trees" Proc.3st Symp.on Logic in Computer Science, 1988 348–357
[Michalski (1983)]: "A theory and methodology of Inductive learning" Artificial Intelligence 20, 1983, 111–161.
[Peyton-Jones (1987)]: The Implementation of Functional Programming Languages. Prentice-Hall, 1987.
[Plaisted (1985)]: "Semantic Confluence tests and Completion methods." Information and Control 65 (2/3) 1985
[Raoult (1980)]: "Finiteness results on Term rewriting" RAIRO vol. 15 (4), 1981, 373–391.
[Schnoebelen (1988)]: "Refined compilation of pattern-matching for functional languages". Science of Computer Programming 11, 1988, 133–159.
[Schnoebelen (1987)]: "Rewriting techniques for the temopal analysis of communicating processes". Proc. PARLE in LNCS 259, 1987, 402–419.
[Tarski (1968)] TARSKI, A. "Equational logic and Equational theories of algebras". In K. Schutte ed., Contribution to Mathematical Logic, Horth-Holland, Amsterdam, 1988.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1990 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kounalis, E. (1990). Pumping lemmrs for tree languages generated by rewrite systems. In: Rovan, B. (eds) Mathematical Foundations of Computer Science 1990. MFCS 1990. Lecture Notes in Computer Science, vol 452. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0029631
Download citation
DOI: https://doi.org/10.1007/BFb0029631
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-52953-8
Online ISBN: 978-3-540-47185-1
eBook Packages: Springer Book Archive