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Semantics of non-terminating rewrite systems using minimal coverings

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Computer Science Logic (CSL 1995)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1092))

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Abstract

We propose a new semantics for rewrite systems based on interpreting rewrite rules as inequations between terms in an ordered algebra. In particular, we show that the algebra of normal forms in a terminating system is a uniquely minimal covering of the term algebra. In the non-terminating case, the existence of this minimal covering is established in the completion of an ordered algebra formed by rewriting sequences. We thus generalize the properties of normal forms for non-terminating systems to this minimal covering. These include the existence of normal forms for arbitrary rewrite systems, and their uniqueness for confluent systems, in which case the algebra of normal forms is isomorphic to the canonical quotient algebra associated with the rules when seen as equations. This extends the benefits of algebraic semantics to systems with non-deterministic and non-terminating computations. We first study properties of abstract orders, and then instantiate these to term rewriting systems.

Supported in part by JNICT under contracts BD-1102-90/IA and Praxis XXI / BD / 4069 / 94.

The research reported in this paper has been supported in part by the Science and Engineering Research Council, the CEC under ESPRIT-2 BRA Working Groups 6071, IS-CORE (Information Systems Correctness and REusability) and 6112, COMPASS (COMPrehensive Algebraic Approach to System Specification and development), Fujitsu Laboratories Limited, and under the management of the Information Technology Promotion Agency (IPA), Japan, as part of the Industrial Science and Technology Frontier Program “New Models for Software Architectures,” sponsored by NEDO (New Energy and Industrial Technology Development Organization).

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Authors and Affiliations

  1. Departamento de Informática, Universidade do Minho, Braga, Portugal

    José Barros

  2. Programming Research Group, Oxford University, Oxford, UK

    Joseph Goguen

Authors
  1. José Barros
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  2. Joseph Goguen
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Hans Kleine Büning

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© 1996 Springer-Verlag Berlin Heidelberg

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Barros, J., Goguen, J. (1996). Semantics of non-terminating rewrite systems using minimal coverings. In: Kleine Büning, H. (eds) Computer Science Logic. CSL 1995. Lecture Notes in Computer Science, vol 1092. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61377-3_29

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  • DOI: https://doi.org/10.1007/3-540-61377-3_29

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61377-0

  • Online ISBN: 978-3-540-68507-4

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