In computer science, the well-founded semantics is a three-valued semantics for logic programming, which gives a precise meaning to general logic programs.

History

The well-founded semantics was defined by Van Gelder, et al. in 1988.[1][2] The Prolog system XSB implements the well-founded semantics since 1997.[3][4]

Three-valued logic

The well-founded semantics assigns a unique model to every general logic program. However, instead of only assigning propositions true or false, it adds a third value unknown for representing ignorance.[1]

A simple example is the logic program that encodes two propositions a and b, and in which a must be true whenever b is not and vice versa:

a :- not(b).
b :- not(a).

neither a nor b are true or false, but both have the truth value unknown. In the two-valued stable model semantics, there are two stable models, one in which a is true and b is false, and one in which b is true and a is false.

Stratified logic programs have a 2-valued well-founded model, in which every proposition is either true or false. This coincides with the unique stable model of the program. The well-founded semantics can be viewed as a three-valued version of the stable model semantics.[5]

Complexity

In 1989, Van Gelder suggested an algorithm to compute the well-founded semantics of a propositional logic program whose time complexity is quadratic in the size of the program.[6] As of 2001, no general subquadratic algorithm for the problem was known.[7]

References

  1. 1 2 Van Gelder, Allen; Ross, Kenneth A.; Schlipf, John S. (July 1991). "The well-founded semantics for general logic programs". Journal of the ACM. 38 (3): 619–649. doi:10.1145/116825.116838. ISSN 0004-5411.
  2. ↑ Van Gelder, Allen; Ross, Kenneth; Schlipf, John S. (1988). "Unfounded sets and well-founded semantics for general logic programs". Proceedings of the Seventh ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems - PODS '88. New York, New York, USA: ACM Press: 221–230. doi:10.1145/308386.308444. ISBN 0897912632.
  3. ↑ Körner, Philipp; Leuschel, Michael; Barbosa, João; Costa, Vítor Santos; Dahl, Verónica; Hermenegildo, Manuel V.; Morales, Jose F.; Wielemaker, Jan; Diaz, Daniel; Abreu, Salvador; Ciatto, Giovanni (November 2022). "Fifty Years of Prolog and Beyond". Theory and Practice of Logic Programming. 22 (6): 776–858. doi:10.1017/S1471068422000102. hdl:10174/33387. ISSN 1471-0684.
  4. ↑ Rao, Prasad; Sagonas, Konstantinos; Swift, Terrance; Warren, David S.; Freire, Juliana (1997), Dix, Jürgen; Furbach, Ulrich; Nerode, Anil (eds.), "XSB: A system for efficiently computing well-founded semantics", Logic Programming And Nonmonotonic Reasoning, Berlin, Heidelberg: Springer Berlin Heidelberg, vol. 1265, pp. 430–440, doi:10.1007/3-540-63255-7_33, ISBN 978-3-540-63255-9, retrieved 2023-11-17
  5. ↑ Przymusinski, Teodor. Well-founded Semantics Coincides with Three-Valued Stable Semantics. Fundamenta Informaticae XIII pp. 445-463, 1990.
  6. ↑ Van Gelder, A. (1989). The alternating fixpoint of logic programs with negation. Proceedings of the eighth ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems. ACM Press. pp. 1–10. doi:10.1145/73721.73722. ISBN 978-0-89791-308-9.
  7. ↑ Lonc, Zbigniew; Truszczyński, Mirosław (2001). "On the problem of computing the well-founded semantics". Theory and Practice of Logic Programming. 1 (5): 591–609. arXiv:cs/0101014. doi:10.1017/S1471068401001053. ISSN 1471-0684.
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