Reducing Inductive Definitions to Propositional Satisfiability
The FO(ID) logic is an extension of classical first-order logic with a uniform representation of various forms of inductive definitions. The definitions are represented as sets of rules and they are interpreted by two-valued well-founded models. For a large class of combinatorial and search problems, knowledge representation in FO(ID) offers a viable alternative to the paradigm of Answer Set Programming. The main reasons are that (i) the logic is an extension of classical logic and (ii) the semantics of the language is based on well-understood principles of mathematical induction.
In this paper, we define a reduction from the propositional fragment of FO(ID) to SAT. The reduction is based on a novel characterization of two-valued well-founded models using a set of inequality constraints on level mappings associated with the atoms. We also show how the reduction to SAT can be adapted for logic programs under the stable model semantics. Our experiments show that when using a state of the art SAT solver both reductions are competitive with other answer set programming systems — both direct implementations and SAT based.
Unable to display preview. Download preview PDF.
- 3.Brachman, R.J., Levesque, H.: Competence in knowledge representation. Proc. of the National Conference on Artificial Intelligence, 189–192 (1982)Google Scholar
- 5.Denecker, M., Ternovska, E.: Inductive situation calculus. In: Principles of Knowledge Representation and Reasoning: Proc. of the 9th International Conference, pp. 545–553. AAAI Press, Menlo Park (2004)Google Scholar
- 9.Fages, F.: Consistency of Clark’s completion and existence of stable models. Journal of Methods of Logic in Computer Science 1, 51–60 (1994)Google Scholar
- 10.Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Logic Programming, 5th International Conference and Symposium, pp. 1070–1080 (1988)Google Scholar
- 11.Janhunen, T.: Representing normal programs with clauses. In: Proc. of the 16th European Conference on Artificial Intelligence, pp. 358–362 (2004)Google Scholar
- 13.Lin, F., Zhao, J.: On tight logic programs and yet another translation from normal logic programs to propositional logic. In: International Joint Conference on Artificial Intelligence, pp. 853–858. Morgan Kaufmann, San Francisco (2003)Google Scholar
- 17.Mitchell, D., Ternovska, E.: A framework for representing and solving NP-search problems. In: Proc. of the National Conference on Artificial Intelligence, pp. 430–435 (2005)Google Scholar
- 19.Ryan, L.: Efficient algorithms for clause-learning SAT solvers. Master’s thesis, Simon Fraser University, Burnaby, Canada (2004)Google Scholar
- 21.Ternovskaia, E.: Causality via inductive definitions. In: Working Notes of ”Prospects for a Commonsense Theory of Causation”. AAAI Spring Symposium Series, pp. 94–100 (1998)Google Scholar
- 22.Ternovskaia, E.: ID-logic and the ramification problem for the situation calculus. In: Proc. of the 14th European Conference on Artificial Intelligence, pp. 563–567 (2000)Google Scholar