Operators on lattices of ω-Herbrand interpretations
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.
Keywordslogic programming Herbrand models fixed point semantics
Unable to display preview. Download preview PDF.
- K. R. Apt, Introduction to Logic Programming, in J. Van Leeuwen, editor, Handbook of Theoretical Computer Science, North Holland, 1989.Google Scholar
- 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
- 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
- K. L. Clark, Predicate Logic as a Computational Formalism, Research Report DOC 79/59, Dept. of Computing, Imperial College, 1979.Google Scholar
- J. W. Lloyd, Foundations of Logic Programming, Second extended edition, Springer Verlag, 1987.Google Scholar
- 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
- J. A. Plaza, Fully Declarative Programming with Logic — Mathematical Foundations, Doctoral Dissertation, City University of New York, July 1990.Google Scholar
- J. A. Plaza, An Extension of Clark's Equality Theory for the Foundations of Logic Programming, submitted for publication.Google Scholar
- H. Rasiowa and R. Sikorski, The Mathematics of Metamathematics, third edition, PWN — Polish Scientific Publishers, 1970.Google Scholar
- H. Rasiowa and R. Sikorski, A proof of Completeness Theorem of Gödel, Fundamenta Mathematicae 37 (1950), pp. 193–200.Google Scholar
- H. Rasiowa and R. Sikorski, Algebraic treatment of the notion of satisfiability, Fundamenta Mathematicae 40 (1953), pp. 62–95.Google Scholar
- J. R. Shoenfield, Mathematical Logic, Addison-Wesley, 1967.Google Scholar
- 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