Operators on lattices of ω-Herbrand interpretations

Extended abstract
  • Jan A. Plaza
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 620)


Interpretations defined by J. Herbrand in early 1930's have found new applications when logic programming emerged in 1970's. Now they are being used as a major technical tool in the theory underlying construction of languages such as Prolog.

In an Herbrand interpretation of a first-order language L elements of the universe are precisely the closed terms of L. In our research we consider the following generalization: by an ω-Herbrand interpretation forL we understand an Herbrand interpretation for the language resulting from L by adjoining ω new individual constants.

We have shown that the theory of equality determined by such interpretations is decidable. In this paper we consider operators corresponding to formulas A ← B where B is a positive formula with general quantifiers. We prove that such operators on the lattice of ω-Herbrand models reach their least fixed points in ω steps. (This shows a great conceptual difference between conventional Herbrand models and ω-Herbrand models.)

We see the following applications of these results: The decidability algorithm can be used as a computing mechanism in an implementation of a fully declarative programming language that overcomes several deficiencies of Prolog; The result on least fixed points provides a basis for the proof of declarative correctness and completeness of that language.


logic programming Herbrand models fixed point semantics 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Jan A. Plaza
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of MiamiCoral GablesUSA

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