# On the computational cost of disjunctive logic programming: Propositional case

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## Abstract

This paper addresses complexity issues for important problems arising with disjunctive logic programming. In particular, the complexity of deciding whether a disjunctive logic program is consistent is investigated for a variety of well-known semantics, as well as the complexity of deciding whether a propositional formula is satisfied by all models according to a given semantics. We concentrate on finite propositional disjunctive programs with as well as without integrity constraints, i.e., clauses with empty heads; the problems are located in appropriate slots of the polynomial hierarchy. In particular, we show that the consistency check is Σ _{2} ^{ p } -complete for the disjunctive stable model semantics (in the total as well as partial version), the iterated closed world assumption, and the perfect model semantics, and we show that the inference problem for these semantics is Π _{2} ^{ p } -complete; analogous results are derived for the answer sets semantics of extended disjunctive logic programs. Besides, we generalize previously derived complexity results for the generalized closed world assumption and other more sophisticated variants of the closed world assumption. Furthermore, we use the close ties between the logic programming framework and other nonmonotonic formalisms to provide new complexity results for disjunctive default theories and disjunctive autoepistemic literal theories.

## Keywords

Model Semantic Integrity Constraint Propositional Formula Default Theory Disjunctive Program## Preview

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## References

- [1]K. Apt and H. Blair, Arithmetic classification of perfect models of stratified programs,
*Proc. ICLP/SLP-5*(MIT Press, 1988) pp. 766–779.Google Scholar - [2]K. Apt, H. Blair and A. Walker, Towards a theory of declarative knowledge, in Minker [44](, pp. 89–148.Google Scholar
- [3]C. Baral, J. Lobo and J. Minker, Generalized disjunctive well-founded semantics for logic programs: Declarative semantics,
*Proc. ISMIS-4*, 1990, pp. 465–473.Google Scholar - [4]R. Ben-Eliyahu and R. Dechter, Propositional semantics for disjunctive logic programs,
*Proc. ICLP/SLP-7*, 1992, pp. 813–827. Full paper in Ann. of Math. and Artificial Intelligence 12 (1994) 53–87.Google Scholar - [5]N. Bidoit and C. Froidevaux, General logic databases and programs: Default semantics and stratification, Inf. and Comput. 19 (1991) 15–54.Google Scholar
- [6]N. Bidoit and C. Froidevaux, Negation by default and unstratifiable programs, Theor. Comp. Sci. 78 (1991) 85–112.Google Scholar
- [7]H. Blair and C. Cholak, The complexity of local stratification, Technical Report, School of Computer and Information Sciences, Syracuse University (1992). To appear in Fund. Informaticae.Google Scholar
- [8]H. Blair, W. Marek, A. Nerode, and J. Remmel (editors),
*Informal Proceedings of the Workshop on Structural Complexity and Recursion-Theoretic Methods in Logic Programming*, Washington DC, November 1992 (Cornell University, Mathematical Sciences Institute).Google Scholar - [9]H. Blair, W. Marek and J. Schlipf, The expressiveness of locally stratified programs, Technical Report 92-8, Mathematical Sciences Institute, Cornell University (1992). Ann. of Math. and Artificial Intelligence 15 (1995) 209–229.Google Scholar
- [10]M. Cadoli, The complexity of model checking for circumscriptive formulae, Inf. Processing Lett. 44 (1992) 113–118.Google Scholar
- [11]M. Cadoli and M. Lenzerini, The complexity of closed world reasoning and circumscription, J. Comp. and Syst. Sci. 43 (1994) 165–211.Google Scholar
- [12]M. Cadoli and M. Schaerf, A survey of complexity results for non-monotonic logics, J. Logic Programming 17 (1993) 127–160.Google Scholar
- [13]E. Chan, A possible worlds semantics for disjunctive databases, IEEE Trans. Data and Knowledge Eng. 5(2) (1993) 282–292.Google Scholar
- [14]A. Chandra and D. Harel, Horn clause queries and generalizations, J. Logic Programming 2 (1985) 1–15.Google Scholar
- [15]R. Chang and P. Rohatgi, On unique satisfiability and randomized reductions, Bull. EATCS 47 (1990) 151–159.Google Scholar
- [16]J. Chomicki and V.S. Subrahmanian, Generalized closed world assumption is Π
_{2}^{0}-complete, Inf. Processing Lett. 34 (1990) 289–291.Google Scholar - [17]J. Dix and M. Müller, Abstract properties and computational complexity of semantics for disjunctive logic programs, in Blair et al. [8], pp. 15–28.Google Scholar
- [18]J. Dix and M. Müller, Implementing semantics of disjunctive logic programs using fringes and abstract properties,
*Proc. LPNMR-2*(MIT Press, 1993) pp. 43–59.Google Scholar - [19]T. Eiter and G. Gottlob, Reasoning with parsimonious and moderately grounded expansions, Fund. Informaticae 17(1,2) (1992) 31–53.Google Scholar
- [20]T. Eiter and G. Gottlob, Complexity results for disjunctive logic programming and application to nonmonotonic logics,
*Proc. ILPS-10*(MIT Press, 1993) pp. 266–278.Google Scholar - [21]T. Eiter and G. Gottlob, Propositional circumscription and extended closed world reasoning are Π
_{2}^{p}-complete, Theor. Comp. Sci. 114(2) (1993) 231–245. Addendum 118:315.Google Scholar - [22]J. Fernández and J. Minker, Semantics of disjunctive deductive databases,
*Proc. ICDT-4*(Springer, 1992) pp. 21–50.Google Scholar - [23]M. Garey and D.S. Johnson,
*Computers and Intractability — A Guide to the Theory of NP-Completeness*(Freeman, New York, 1979).Google Scholar - [24]M. Gelfond and V. Lifschitz, The stable model semantics for logic programming,
*Proc. ILPS-5*(MIT Press, 1988) pp. 1070–1080.Google Scholar - [25]M. Gelfond and V. Lifschitz, Classical negation in logic programs and disjunctive databases, New Generation Comput. 9 (1991) 365–385.Google Scholar
- [26]M. Gelfond and H. Przymusinska, Negation as failure: careful closure procedure, Artificial Intelligence 30 (1986) 273–287.Google Scholar
- [27]M. Gelfond, H. Przymusinska, V. Lifschitz and M. Truszczyński, Disjunctive defaults,
*Proc. KR-2*, 1991, pp. 230–237.Google Scholar - [28]M. Gelfond, H. Przymusinska and T. Przymusinski, On the relationship between circumscription and negation as failure, Artificial Intelligence 38 (1989) 75–94.Google Scholar
- [29]G. Gottlob, Complexity results for nonmonotonic logics, J. Logic Comput. 2(3) (1992) 397–425.Google Scholar
- [30]D.S. Johnson, A catalog of complexity classes, in J. van Leeuwen, editor,
*Handbook of Theoretical Computer Science*, Vol. A (Elsevier Science Publ., 1990) chap. 2.Google Scholar - [31]H. Kautz and B. Selman, Hard problems for simple default logics, Artificial Intelligence 49 (1991) 243–279.Google Scholar
- [32]V. Lifschitz, Computing circumscription,
*Proc. IJCAI-9*, 1985, pp. 121–127.Google Scholar - [33]V. Lifschitz and G. Schwarz, Extended logic programs as autoepistemic theories,
*Proc. LPNMR-2*(MIT Press, 1993) pp. 101–114.Google Scholar - [34]J. Lobo, J. Minker and A. Rajasekar,
*Foundations of Disjunctive Logic Programming*(MIT Press, Cambridge, MA, 1992).Google Scholar - [35]W. Marek, A. Nerode and J. Remmel, A theory of nonmonotonic rule systems II, Ann. of Math. and Artificial Intelligence 5 (1992) 229–264.Google Scholar
- [36]W. Marek, A. Nerode and J. Remmel, How complicated is the set of stable models of a recursive logic program? Ann. Pure and Appl. Logic 56 (1992) 119.Google Scholar
- [37]W. Marek, A. Rajasekar and M. Truszczyński, Complexity of computing with extended propositional logic programs, in Blair et al. [8], pp. 93–102.Google Scholar
- [38]W. Marek and M. Truszczyński, Autoepistemic logic, J. ACM 38(3) (1991) 588–619.Google Scholar
- [39]W. Marek and M. Truszczyński, Computing intersection of autoepistemic expansions, in Nerode et al. [46], pp. 37–50.Google Scholar
- [40]W. Marek and M. Truszczyński, Reflexive autoepistemic logic and logic programming,
*Proc. LPNMR-2*(MIT Press, 1993) pp. 115–131.Google Scholar - [41]J. McCarthy, Circumscription — a form of non-monotonic reasoning, Artificial Intelligence 13 (1980) 27–39.Google Scholar
- [42]J. McCarthy, Applications of circumscription to formalizing common-sense knowledge, Artificial Intelligence 28 (1986) 89–116.Google Scholar
- [43]J. Minker, On indefinite data bases and the closed world assumption,
*Proc. CADE-6*, 1982, pp. 292–308.Google Scholar - [44]J. Minker, (editor)
*Foundations of Deductive Databases and Logic Programming*(Morgan Kaufman, Washington DC, 1988).Google Scholar - [45]R. Moore, Semantical considerations on nonmonotonic logics, Artificial Intelligence 25 (1985) 75–94.Google Scholar
- [46]A. Nerode, W. Marek and V.S. Subrahmanian (editors),
*Proc. LPNMR-1*(MIT Press, 1991).Google Scholar - [47]I. Niemelä, Towards automatic reasoning,
*Proc. Europ. Workshop on Logics in AI*, Amsterdam, September 1990, LNCS # 478 (Springer, 1991).Google Scholar - [48]T. Przymusinski, On the declarative and procedural semantics of stratified deductive databases, in Minker [44](, pp. 193–216.Google Scholar
- [49]T. Przymusinski, Stable semantics for disjunctive programs, New Generation Comput. 9 (1991) 401–424.Google Scholar
- [50]T. Przymusinski, Three-valued nonmonotonic formalisms and semantics of logic programming, Artificial Intelligence 49 (1991) 309–344.Google Scholar
- [51]A. Rajasekar, J. Lobo and J. Minker, Weak generalized closed world assumption, J. Autom. Reasoning 5 (1989) 293–307.Google Scholar
- [52]R. Reiter, On closed-world databases, in H. Gallaire and J. Minker (editors)
*Logic and Data Bases*(Plenum Press, New York, 1978) pp. 55–76.Google Scholar - [53]R. Reiter, A logic for default reasoning, Artificial Intelligence 13 (1980) 81–132.Google Scholar
- [54]K. Ross, The well-founded semantics for disjunctive logic programs,
*Proc. DOOD-1*, 1989, pp. 352–369.Google Scholar - [55]K. Ross and R. Topor, Inferring negative information from disjunctive databases, J. Autom. Reasoning 4(2) (1988) 397–424.Google Scholar
- [56]Ch. Sakama, Possible model semantics for disjunctive databases,
*Proc. DOOD-1*, 1989, pp. 337–351.Google Scholar - [57]M. Schaerf, Logic programming and autoepistemic logics: new relations and complexity results,
*Proc. Italian AI Conference (IA*^{*}AI), 1993, Springer LNCS/AI #728.Google Scholar - [58]J.S. Schlipf, The expressive powers of logic programming semantics, Technical Report CIS-TR-90-3, Computer Science Department, University of Cincinnati (1990). Preliminary version in
*Proc. PODS-90*, pp. 196–204. To appear in J. Comp. and Syst. Sci.Google Scholar - [59]J.S. Schlipf, A survey of complexity and undecidability results in logic programming, in Blair et al. [8], pp. 93–102.Google Scholar
- [60]J.S. Schlipf., When is closed world reasoning tractable?,
*Proc. ISMIS-3*(Elsevier Science Publ., 1998) pp. 485–494.Google Scholar - [61]G. Schwarz, Autoepistemic logic of knowledge, in Nerode et al. [46], pp. 260–274.Google Scholar
- [62]J. Stillman, The complexity of propositional default logic,
*Proc. AAAI-10*, 1992, pp. 794–799.Google Scholar - [63]L. Stockmeyer and A. Meyer, Word problems requiring exponential time,
*Proc. STOC-5*(ACM, 1973) pp. 1–9.Google Scholar - [64]A. Van Gelder, Negation as failure using tight derivations for general logic programs, in Minker [44](, pp. 1149–1176.Google Scholar
- [65]A. Van Gelder, K. Ross and J.S. Schlipf, The well-founded semantics for general logic programs, J. ACM 38(3) (1991) 620–650.Google Scholar
- [66]K. Wagner, Bounded query classes, SIAM J. Comp. 19(5) (1990) 833–846.Google Scholar
- [67]A. Yahya and L. Henschen, Deduction in non-Horn databases, J. Autom. Reasoning 1(2) (1985) 141–160.Google Scholar