Founded Semantics and Constraint Semantics of Logic Rules
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.
KeywordsDatalog Unrestricted negation Existential and universal quantifications Fixed-point semantics Constraints Well-founded semantics Stable model semantics Fitting (Kripke-Kleene) semantics Supported model semantics
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.
- 3.Clark, K.L.: Negation as failure. In: Gallaire, H., Minker, J. (eds.) Logic and Databases, pp. 293–322. Plenum Press, New York (1978)Google Scholar
- 5.Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Proceedings of the 5th International Conference and Symposium on Logic Programming, pp. 1070–1080. MIT Press (1988)Google Scholar
- 6.Apt, K.R., Blair, H.A., Walker, A.: Towards a theory of declarative knowledge. In: Foundations of Deductive Databases and Logic Programming, pp. 89–148. Morgan Kaufman (1988)Google Scholar
- 9.Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases: The Logical Level. Addison-Wesley, Reading (1995)Google Scholar
- 10.Liu, Y.A., Stoller, S.D.: The founded semantics and constraint semantics of logic rules. Computing Research Repository arXiv:1606.06269 [cs.LO] (Revised 2017) (2016)
- 13.Zukowski, U.: Flexible computation of the well-founded semantics of normal logic programs. Ph.D. thesis, Faculty of Computer Science and Mathematics, University of Passau (2001)Google Scholar
- 17.Sato, T., Tamaki, H.: Transformational logic program synthesis. In: Proceedings of the International Conference on Fifth Generation Computer Systems, pp. 195–201 (1984)Google Scholar
- 19.Chan, D.: Constructive negation based on the completed database. In: Proceedings of the 5th International Conference and Symposium on Logic Programming, pp. 111–125. MIT Press (1988)Google Scholar
- 20.Foo, N.Y., Rao, A.S., Taylor, A., Walker, A.: Deduced relevant types and constructive negation. In: Proceedings of the 5th International Conference and Symposium on Logic Programming, pp. 126–139 (1988)Google Scholar
- 21.Stuckey, P.J.: Constructive negation for constraint logic programming. In: Proceedings of the 6th Annual IEEE Symposium on Logic in Computer Science, pp. 328–339 (1991)Google Scholar