Abstract
Recently Brass and Dix introduced the semantics D-WFS for general disjunctive logic programs. The interesting feature of this approach is that it is both semantically and proof-theoretically founded. Semantically, D-WFS is invariant under some natural declarative principles. Proof-theoretically, any program Φ is associated a normalform Φ, called the residual program, by a nontrivial bottom-up construction using least fixpoints of two monotonic operators.
We show in this paper that the original calculus, consisting of some simple transformations, has a very strong and appealing property: it is confluent and terminating. This means that all the transformations can be applied in any order: whenever we arrive at an irreducible program (no more transformation is applicable), this program is already uniquely determined and coincides with the normalform res(Φ) Moreover, for fair sequences it is also strongly terminating: every fair sequence of transformations leads to normalform res(Φ). Another feature of our approach is that D-WFS can be read off from res(Φ) immediately in a very simple way. No proper subset of the calculus has these properties – only when we restrict to certain subclasses of programs.
We also give an equivalent characterization of D-WFS in terms of iterated minimal model reasoning with respect to positive programs. This construction is a generalization of a description of the well-founded semantics: we introduce a very simple and neat construction of a sequence D i that eventually stops and represents the set of derivable disjunctions.
Both characterizations open the way for efficient implementations. The first does so because the ordering of the transformations does not matter: we are free to choose always the “best” transformation, which maximally reduces the program. The second does so because special methods from circumscription, in particular a sophisticated minimal model reasoner for positive programs, might be useful.
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References
Apt, K. R. and Bol, R. N.: Logic programming and negation: A Survey, J. Logic Programming 19–20 (1994), 9–71.
Baral, Ch. and Gelfond, M.: Logic programming and knowledge representation, J. Logic Programming 19–20 (1994).
Baral, Ch., Lobo, J. and Minker, J.: Generalized disjunctive well-founded semantics for logic programs: Declarative semantics, in Z. W. Ras, M. Zemankova and M. L. Emrich (eds), Proc. 5th Int. Symp. on Methodologies for Intelligent Systems, Knoxville, TN, October 1990, North-Holland, 1990, pp. 465–473.
Baral, Ch., Lobo, J. and Minker, J.: WF3: A semantics for negation in normal disjunctive logic programs, in Z. W. Ras and M. Zemankova (eds), Methodologies for Intelligent Systems, LNAI 542, Springer, Berlin, 1991, pp. 459–468.
Baumgartner, P., Furbach, U. and Niemelä, I.: Hyper tableaux, in L. M. Pereira, J. J. Alferes and E. Orlowska (eds), Logics in Artificial Intelligence (JELIA' 96), LNCS 1126, Springer, 1996, pp. 1–17.
Bidoit, B. and Hull, R.: Positivism vs. minimalism in deductive databases, in Proc. 5th ACM Symp. on Principles of Database Systems (PODS'86), 1986, pp. 123–132.
Brass, S. and Dix, J.: Computing disjunctive stable semantics based on Clark's completed database, in Proc. 6th GI-Workshop 'Grundlagen von Datenbanken', Bad Helmstedt, September 1994, 1994, pp. 30–34.
Brass, S. and Dix, J.: A disjunctive semantics based on unfolding and bottom-up evaluation, in Bernd Wolfinger (ed.), Innovationen bei Rechen-und Kommunikationssystemen (IFIP' 94-Congress, Workshop FG2: Disjunctive Logic Programming and Disjunctive Databases), Springer, Berlin, 1994, pp. 83–91.
Brass, S. and Dix, J.: A general approach to bottom-up computation of disjunctive semantics, in J. Dix, L. Pereira, and T. Przymusinski (eds), Nonmonotonic Extensions of Logic Programming, LNAI 927, Springer, Berlin, 1995, pp. 127–155.
Brass, S. and Dix, J.: Characterizations of the disjunctive stable semantics by partial evaluation, J. Logic Programming 32(3) (1997), 207–228. (Extended abstract appeared in: Characterizations of the stable semantics by partial evaluation, LPNMR, Proc. 3rd Int. Conf., Kentucky, LNCS 928, Springer, 1995, pp. 85–98.)
Brass, S. and Dix, J.: Semantics of disjunctive logic programs based on partial evaluation, J. Logic Programming, accepted for publication, 1998. (Extended abstract appeared in: Disjunctive semantics based upon partial and bottom-up evaluation, Proc. 12th Int. Logic Programming Conference, Tokyo, MIT Press, 1995, pp. 199–213.)
Brass, S., Dix, J., Niemelä, I. and Przymusinski, T. C.: Comparison and efficient computation of the static and the disjunctive WFS, in G. Brewka, E. Weydert, and C. Witteveen (eds), Proc. 3rd Dutch–German Workshop on Nonmonotonic Reasoning and Its Applications, 1997, pp. 37–42.
Brass, S., Dix, J. and Przymusinski, T. C.: Super logic programs, in L. C. Aiello, J. Doyle, and S. C. Shapiro (eds), Principles of Knowledge Representation and Reasoning: Proc. 5th Int. Conf. (KR' 96), Morgan Kaufmann, San Francisco, CA, 1996, pp. 529–541.
Brass, S., Zukowski, U. and Freitag, B.: Transformation based bottom-up computation of the well-founded model, in J. Dix, L. Pereira, and T. Przymusinski (eds), Nonmonotonic Extensions of Logic Programming, LNAI 1216, Springer, Berlin, 1997, pp. 171–201.
Brewka, G., Dix, J. and Konolige, K.: Nonmonotonic Reasoning: An Overview, CSLI Lecture Notes 73, CSLI Publications, Stanford, CA, 1997.
Brewka, G. and Dix, J.: Knowledge representation with logic programs, Technical report, Tutorial Notes of the 12th European Conference on Artificial Intelligence (ECAI' 96), 1996. Also appeared as Technical Report 15/96, Dept. of CS of the University of Koblenz-Landau. Will appear as Chapter 6 in Handbook of Philosophical Logic, 2nd edition (1998), Volume 6, Methodologies.
Clark, K. L.: Negation as failure, in H. Gallaire and J. Minker (eds), Logic and Data-Bases, Plenum, New York, 1978, pp. 293–322.
Dix, J., Pereira, L. and Przymusinski, T.: Non-Monotonic Extensions of Logic Programming, LNAI 927, Springer, Berlin, 1995.
Dix, J., Pereira, L. and Przymusinski, T.: Non-Monotonic Extensions of Logic Programming, LNAI 1216, Springer, Berlin, 1997.
Dix, J.: A framework for representing and characterizing semantics of logic programs, in B. Nebel, C. Rich, and W. Swartout (eds), Principles of Knowledge Representation and Reasoning: Proceedings of the Third International Conference (KR' 92), Morgan Kaufmann, San Mateo, CA, 1992, pp. 591–602.
Dix, J.: A classification-theory of semantics of normal logic programs: I. Strong properties, Fundamenta Informaticae 22(3) (1995), 227–255.
Dix, J.: A classification-theory of semantics of normal logic programs: II. Weak properties, Fundamenta Informaticae 22(3) (1995), 257–288.
Dix, J.: Semantics of logic programs, their intuitions and formal properties: An overview, in A. Fuhrmann and H. Rott (eds), Logic, Action and Information – Essays on Logic in Philosophy and Artificial Intelligence, DeGruyter, 1995, pp. 241–327.
Dix, J. and Stolzenburg, F.: Computation of nonground disjunctive well-founded semantics with constraint logic programming (preliminary report), in J. Dix, L. Pereira, and T. Przymusinski (eds), Nonmonotonic Extensions of Logic Programming, LNAI 1216, Springer, Berlin, 1997, pp. 202–226.
Dix, J. and Stolzenburg, F.: A framework to incorporate nonmonotonic reasoning into constraint logic programming, J. Logic Programming (1998). Special Issue on Constraint Logic Programming, Guest Editors: Kim Marriott and Peter Stuckey, to appear.
Eiter, Th. and Gottlob, G.: Propositional circumscription and extended closed world reasoning are Π P2 -complete, Theoretical Computer Science 144(2) (1993), 231–245, Addendum: 118 (1993), 315.
Eiter, Th., Gottlob, G. and Mannila, H.: Expressive power and complexity of disjunctive DATALOG, in Proc. Workshop on Logic Programming with Incomplete Information, Vancouver Oct. 1993, following ILPS' 93, 1993, pp. 59–79.
Gelfond, M. and Lifschitz, V.: Classical negation in logic programs and disjunctive databases, New Generation Computing 9 (1991), 365–387. (Extended abstract appeared in: Logic Programs with Classical Negation. Proc. 7th Int. Logic Programming Conference, Jerusalem, MIT Press, 1990, pp. 579–597.)
Lobo, J., Minker, J. and Rajasekar, A.: Foundations of Disjunctive Logic Programming, MIT-Press, 1992.
Minker, J.: On indefinite databases and the closed world assumption, in Proc. 6th Conference on Automated Deduction, New York, Springer, Berlin, 1982, pp. 292–308.
Minker, J.: An overview of nonmonotonic reasoning and logic programming, J. Logic Programming, Special Issue 17 (1993).
Minker, J.: Logic and databases: A 20 year retrospective, in D. Pedreschi and C. Zaniolo (eds), Proc. Int. Workshop on Logic in Databases (LID), LNCS 1154, Springer, Berlin, 1996, pp. 3–58.
Niemelä, I.: Implementing circumscription using a tableau method, in W. Wahlster (ed.), Proc. European Conference on Artificial Intelligence, John Wiley, Budapest, Hungary, 1996, pp. 80–84.
Niemelä, I.: A tableau calculus for minimal model reasoning, in P. Miglioli, U. Moscato, D. Mundici, and M. Ornaghi (eds), Proc. Fifth Workshop on Theorem Proving with Analytic Tableaux and Related Methods, LNAI 1071, Springer-Verlag, Terrasini, Italy, 1996, pp. 278–294.
Niemelä, I. and Simons, P.: Efficient implementation of the well-founded and stable model semantics, in M. Maher (ed.), Proc. Joint International Conference and Symposium on Logic Programming, The MIT Press, Bonn, Germany, 1996, pp. 289–303.
Pzymusinski, T.: Stable semantics for disjunctive programs, New Generation Computing Journal 9 (1991), 401–424. (Extended abstract appeared in: Extended stable semantics for normal and disjunctive logic programs. Proc. 7th Int. Logic Programming Conference, Jerusalem, MIT Press, Cambridge, Mass., 1990, pp. 459–477.)
Przymusinski, T.: Stationary semantics for normal and disjunctive logic programs, in C. Delobel, M. Kifer, and Y. Masunaga (eds), DOOD' 91, Proc. 2nd Int. Conf., LNCS 566, Springer, Muenchen, Berlin, 1991.
Przymusinski, T.: Static semantics for normal and disjunctive logic programs, Ann. Math. Artificial Intelligence 14 (1995), 323–357.
Rajasekar, A., Lobo, J. and Minker, J.: Weak generalized closed world assumption, J. Automated Reasoning 5 (1989), 293–307.
Ross, K. A.: The well-founded semantics for disjunctive logic programs, in Proc.1st Int, Conf. on Deductive and Object Oriented Databases, Kyoto, Japan, 1989, pp. 1–22.
Ross, K. A. and Topor, R. A.: Inferring negative information from disjunctive databases, J. Automated Reasoning 4 (1988), 397–424.
Sakama, Ch. and Seki, H.: Partial deduction of disjunctive logic programs: A declarative approach, in Logic Program Synthesis and Transformation – Meta Programming in Logic, LNCS 883, Springer, Berlin, 1994, pp. 170–182.
Schlipf, J. S.: Formalizing a logic for logic programming, Ann. Math. Artificial Intelligence 5 1992), 279–302.
van Gelder, A., Ross, K. A. and Schlipf, J. S.: The well-founded semantics for general logic programs, J. ACM 38 (1991), 620–650.
You, J.-H. and Yuan, Li-Y.: Autoepistemic circumscription and logic programming, J. Automated Reasoning 10 (1993), 143–160.
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Brass, S., Dix, J. Characterizations of the Disjunctive Well-Founded Semantics: Confluent Calculi and Iterated GCWA. Journal of Automated Reasoning 20, 143–165 (1998). https://doi.org/10.1023/A:1005952908693
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DOI: https://doi.org/10.1023/A:1005952908693