# Every Formula-Based Logic Program Has a Least Infinite-Valued Model

## Abstract

Every definite logic program has as its meaning a least Herbrand model with respect to the program-independent ordering \(\subseteq \). In the case of normal logic programs there do not exist least models in general. However, according to a recent approach by Rondogiannis and Wadge, who consider infinite-valued models, every normal logic program does have a least model with respect to a program-independent ordering. We show that this approach can be extended to formula-based logic programs (i.e., finite sets of rules of the form \(A\leftarrow \phi \) where \(A\) is an atom and \(\phi \) an arbitrary first-order formula). We construct for a given program \(P\) an interpretation \(M_P\) and show that it is the least of all models of \(P\).

## Keywords

Logic programming Semantics of programs Negation-as-failure Infinite-valued logics Set theory## Notes

### Acknowledgements.

This work has been financed through a grant made available by the Carl Zeiss Foundation. The author is grateful to Prof. Dr. Peter Schroeder-Heister, Hans-Joerg Ant, M. Comp. Sc., and three anonymous reviewers for helpful comments and suggestions.

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