# Operators on lattices of ω-Herbrand interpretations

## Abstract

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 for**L* 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.

## Keywords

logic programming Herbrand models fixed point semantics## Preview

Unable to display preview. Download preview PDF.

## References

- [1]K. R. Apt, Introduction to Logic Programming, in J. Van Leeuwen, editor,
*Handbook of Theoretical Computer Science*, North Holland, 1989.Google Scholar - [2]K. R. Apt, H. A. Blair and A. Walker, Towards a Theory of Declarative knowledge, in J. Minker (ed.)
*Foundations of Deductive Databases and Logic Programming*, Morgan, Kaufmann Publishers, Los Altos, 1988, pp. 89–148.Google Scholar - [3]K. R. Apt and M. H. van Emden, Contributions to the Theory of Logic Programming,
*J. ACM*29, 3, July 1982, pp. 841–862.CrossRefGoogle Scholar - [4]K. L. Clark, Negation as Failure, in
*Logic and Databases*, H. Gallaire and J. Minker (eds.), Plenum Press, New York, 1978, 193–322.Google Scholar - [5]K. L. Clark, Predicate Logic as a Computational Formalism,
*Research Report DOC 79/59*, Dept. of Computing, Imperial College, 1979.Google Scholar - [6]J.-L. Lassez, V. L. Nguyen and E. A. Sonnenberg, Fixed Point Theorems and Semantics: A Folk Tale,
*Inf. Proc. Letters*14, 3 (1982), pp. 112–116.CrossRefGoogle Scholar - [7]J. W. Lloyd,
*Foundations of Logic Programming*, Second extended edition, Springer Verlag, 1987.Google Scholar - [8]J. W. Lloyd, E. A. Soneneberg and R. W. Topor, Integrity Constraint Checking in Stratified Databases, Technical Report 86/5, department of Computer Science, University of Melbourne, 1986.Google Scholar
- [9]J. W. Lloyd and R. W. Topor, A Basis for Deductive Database Systems II,
*J. Logic Programming*, vol 3, No. 1, 1986, pp. 55–67.CrossRefGoogle Scholar - [10]J. A. Plaza,
*Fully Declarative Programming with Logic — Mathematical Foundations*, Doctoral Dissertation, City University of New York, July 1990.Google Scholar - [11]J. A. Plaza, An Extension of Clark's Equality Theory for the Foundations of Logic Programming, submitted for publication.Google Scholar
- [12]H. Rasiowa and R. Sikorski,
*The Mathematics of Metamathematics*, third edition, PWN — Polish Scientific Publishers, 1970.Google Scholar - [13]H. Rasiowa and R. Sikorski, A proof of Completeness Theorem of Gödel,
*Fundamenta Mathematicae 37*(1950), pp. 193–200.Google Scholar - [14]H. Rasiowa and R. Sikorski, Algebraic treatment of the notion of satisfiability,
*Fundamenta Mathematicae 40*(1953), pp. 62–95.Google Scholar - [15]J. C. Shepherdson, Negationa as Failure II,
*J. Logic Programming*vol. 2, No. 3, 1985, pp. 185–202.CrossRefGoogle Scholar - [16]J. R. Shoenfield,
*Mathematical Logic*, Addison-Wesley, 1967.Google Scholar - [17]M. H. van Emden and R. A. Kowalski, The Semantics of Predicate Logic as a Programming Language,
*J. ACM*23, 4 (Oct. 1976), pp. 733–742.CrossRefGoogle Scholar - [18]A. Van Gelder, Negation as Failure using Tight Derivations for General Logic Programs,
*Proc. 3rd IEEE Symp. on Logic Programming*, Salt Lake City, 1986, pp. 127–138.Google Scholar