Positive/negative conditional rewriting

  • Stéphane Kaplan
Part 1 Research Articles
Part of the Lecture Notes in Computer Science book series (LNCS, volume 308)


This paper introduces positive/negative conditional term rewriting systems, with rules of the generic form:
$$u = v \wedge \bar u \ne \bar v \Rightarrow \lambda \to \rho$$
as 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:\(\lambda > \rho , u, v, \bar u, \bar v\)(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.


Normal Form Operational Semantic Ground Term Algebraic Semantic Ground Instance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Stéphane Kaplan
    • 1
    • 2
    • 3
  1. 1.Computer Science DepartmentHebrew UniversityGivat Ram JerusalemIsrael
  2. 2.L.R.I., Bât. 490Université des SciencesOrsay CedexFrance
  3. 3.Computer Science DepartmentBar-Ilan UniversityRamat-GanIsrael

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