Abstract
This paper introduces positive/negative conditional term rewriting systems, with rules of the generic form:
as they often appear in algebraic specifications. We consider the algebraic semantics of such systems (viewed as sets of axioms). They do not in general have initial models; however, we show that they admit quasi-initial models, that are in some sense extremal within the class of all models. We then introduce the subclass of reducing rewrite systems, constrained by the condition: λ>ρ,u,v,u',v' (for some reduction ordering >). For such systems, we show that an optimal rewrite relation → may be defined, and constructed as a "limit". We prove the total validity of an interpreter that computes the normal forms of terms for →. It is then shown that when → is confluent, the algebra of normal forms is a quasi-initial model. We state a general result about the converse. Lastly, we present a complete critical-pair criterion à la Knuth-Bendix to check for the confluence of reducing systems.
Preview
Unable to display preview. Download preview PDF.
References
K. Apt, H. Blair, A. Walker,Towards a theory of declarative knowledge, I.B.M. T.J. Watson Research Center Report, submitted for publication (1987)
J.A. Goguen, J.W. Thatcher, E.G. Wagner,An Initial Algebra Approach to the Specification, Correctness, and Implementation of Abstract Data Types, Current Trends in Programming Methodology, Vol. 4, Ed. Yeh R., Prentice-Hall, pp. 80–149 (1978)
J. Baeten, J. Bergstra, J.W. Klop,Term rewriting systems with priority, Proc. of the RTA'87 Conf., LNCS 256, Springer Verlag (1987)
J. Bergstra, M. Broy, M. Wirsing, J. Tucker,On the power of algebraic specifications, Proc. of the MFCS'81 Conference, LNCS 118 (1981)
N. Bidoit, R. Hull,Positivism vs. Minimalism in Deductive Data Bases, Proc. of the ACM SIGACT-SIGMOD Symposium on Principle of Data Base Systems, Cambridge (1986)
N. Dershowitz,Termination, Proc. of the 1st Conf. on Rewriting Techniques and Applications, LNCS 202, Dijon-France (1985)
H. Ehrig, B. Mahr,Fundamentals of algebraic specifications. I: Equations and initial semantics, EATCS monographs on Theoretical Computer Science, Springer Verlag (1985)
H. Ganziger,Ground term confluence in parametric conditional equational specifications, Proc. of the STACS'87 Conf., LNCS 252, Springer Verlag (1987)
G. Huet,Confluent reductions: abstract properties and applications to term rewriting systems, Proc. of the 18th FOCS Conf., Providence (1978)
G. Huet, D.C. Oppen,Equations and rewrite rules: a survey, Formal languages: Perspective and open problems, R. Book Ed., Academic Press (1980)
J.-P. Jouannaud, C. Kirchner,Completion of a set of rules modulo a set of equations, Proc. of the 11th POPL Conf. (1984)
J.-P. Jouannaud, B. Waldmann, Reductive conditional term rewriting systems, Proc. of the 3rd TC2 Working Conf. on the formal Description of Programming Concepts, North-Holland Pub. Company (1986)
S. Kaplan,Specifications of abstract data types: the power of several classes of axioms with semi-decidable congruence, Proc. of the AFCET Mathematics for Computer Science Conf., Paris (1982)
S. Kaplan,Conditional rewrite rules, TCS 33 (1984)
S. Kaplan,Fair conditional term rewrite systems, Report 194, University of Paris-South (1984)
S. Kaplan,Simplifying conditional term rewriting systems, to appear in the Journal of Symbolic Computation (1987)
J.W. Klop,Term rewriting systems: a tutorial, Bulletin of the EATCS, 32, pp. 143–183 (1987)
V. Lifschitz,On the declarative semantics of logic programs with negation, Workshop on the Foundations of Deductive Data Bases and Logic Programming, Washington D.C. (1986)
T.C. Przymusinski,On the semantics of stratified deductive databases, Workshop on the Foundations of Deductive Data Bases and Logic Programming, Washington D.C. (1986)
J.L. Rémy, H. Zhang,REVEUR4: a system for validating conditional algebraic specifications of abstract data types, Proc. of the 6th ECAI Conf. (1984)
J.L. Rémy, H. Zhang,Contextual rewriting, Proc. of the 1st RTA Conf., LNCS 202, Springer Verlag (1985)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1988 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kaplan, S. (1988). Positive/Negative conditional rewriting. In: Chytil, M.P., Koubek, V., Janiga, L. (eds) Mathematical Foundations of Computer Science 1988. MFCS 1988. Lecture Notes in Computer Science, vol 324. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0017161
Download citation
DOI: https://doi.org/10.1007/BFb0017161
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-50110-7
Online ISBN: 978-3-540-45926-2
eBook Packages: Springer Book Archive