# Founded Semantics and Constraint Semantics of Logic Rules

## Abstract

Logic rules and inference are fundamental in computer science and have been studied extensively. However, prior semantics of logic languages can have subtle implications and can disagree significantly.

This paper describes a simple new semantics for logic rules, *founded semantics*, and its straightforward extension to another simple new semantics, *constraint semantics*, that unify the core of different prior semantics. The new semantics support unrestricted negation, as well as unrestricted existential and universal quantifications. They are uniquely expressive and intuitive by allowing assumptions about the predicates and rules to be specified explicitly. They are completely declarative and relate cleanly to prior semantics. In addition, founded semantics can be computed in linear time in the size of the ground program.

## Keywords

Datalog Unrestricted negation Existential and universal quantifications Fixed-point semantics Constraints Well-founded semantics Stable model semantics Fitting (Kripke-Kleene) semantics Supported model semantics## Notes

### Acknowledgment

We thank David S. Warren, Michael Kifer, Anil Nerode, Tuncay Tekle, Molham Aref, Marc Denecker, Cordell Green, Goyal Gupta, Bob Kowalski, Fangzhen Lin, Alberto Pettorossi, Maurizio Proietti, Neng-Fa Zhou, and many others for helpful comments and discussions on logic languages, semantics, and efficient computations.

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