Phokion G. Kolaitis
Abstract
The effects of circumscribing first-order formulas are explored from a computational standpoint. First, extending work of V. Lifschitz, it is Shown that the circumscription of any existential first-order formula is equivalent to a first-order formula. After this, it is established that a set of universal Horn clauses has a first-order circumscription if and only if it is bounded (when considered as a logic program); thus it is undecidable to tell whether such formulas have first-order circumscription. Finally, it is shown that there arefirst-order formulas whode circumscription has a coNP-complete model-checking problem.
- 1 AHO, A. V., AND ULLMAN, j.D. Universality of data retrieval languages. In Proceedings of the 6th ACM Symposium on Principles of Programming Languages (San Antonio, Tex., Jan. 29-31). ACM, New York, 1979, pp. 110-117. Google Scholar
- 2 BARWISE, K.J. Admissible Sets and Structures: An Approach to Definability Theory. Springer- Verlag, New York, 1975.Google Scholar
- 3 BARWISE, K. J., AND MOSCHOVAKIS, Y.N. Uniform inductive definability. J. Symb. Logic 43 (1978), 521-534.Google Scholar
- 4 Bossu, G., AND SIEGEL, P. Saturation, nonmonotonic reasoning and the closed world assumption. Artif. Int. 25 (1985), 13-63. Google Scholar
- 5 CHANDRA, A. K., AND HAREL, D. Structure and complexity of relational queries. J. Comput. Syst. Sci. 25 (1982), 99-128.Google Scholar
- 6 CHANG, C. C., AND KEISLER, H.J. Model Theory. North-Holland, Amsterdam, The Netherlands, 1973.Google Scholar
- 7 COSMADAKIS, S. S., GAIFMAN, H., KANELLAKIS, P. C., AND VARDI, M. Decidable optimization problems for database logic programs. In Proceedings of the 20th ACM Symposium on Theory of Computing (Chicago, Ill., May 2-4). ACM, New York, 1988, pp. 477-490. Google Scholar
- 8 DAVIS, M. The mathematics of non-monotonic reasoning. Artif. Int. 13 (1980), 73-80.Google Scholar
- 9 DOYLE, J. Circumscription and implicit definability. J. Automat. Reasoning I (1985), 391-405. Google Scholar
- 10 ETHERINGTON, D., MERCER, R., AND REITER, R. On the adequacy of predicate circumscription for closed-world reasoning. Computat. Intelligence 1 (1985), 1 l- 15.Google Scholar
- 11 FAGIN, R. Generalized first-order spectra and polynomial-time recognizable sets. In Complexity of Computations, R. Karp, ed. Proceedings of the SIAM-AMS, vol. 7. American Mathematical Society, Providence, R.I., 1974, pp. 43-74.Google Scholar
- 12 GAIFMAN, H., MAIRSON, H., SAGIV, Y., AND VARDI, M.Y. Undecidable optimization problems for database logic programs. In Proceedings of the 2nd IEEE Symposium on Logic in Computer Science. IEEE, New York, 1987, pp. 106-115.Google Scholar
- 13 GELFOND, M., AND LIFSCHITZ, V. Compiling circumscriptive theories into logic programs. In Proceedings of the 7th National Conference on Artificial Intelligence (AAAI 88) (Aug.), vol. 2. Morgan-Kaufmann, San Mateo, Calif., 1988, pp. 455-459.Google Scholar
- 14 GELFOND, M., PRZYMUSINSKA, H., AND PRZYMUSINSKI, T. Minimal model semantics vs. negation as failure: A comparison of semantics. In Proceedings of the ACM SIGART Symposium on Methodologies for Intelligence Systems (Torino, Italy). ACM, New York, 1988.Google Scholar
- 15 GENESERETH, M., AND NILSSON, N.J. Logical Foundations of Artificial Intelligence. Morgan- Kaufmann, San Mateo, Calif., 1987. Google Scholar
- 16 IOANNIDES, Y.E. A time bound on the materialization of some recursively defined views. In Proceedings of the 1 l th International Conference on Very Large Data Bases. Morgan-Kaufrnann, San Mateo, Calif., 1985, pp. 219-226.Google Scholar
- 17 KRISHNAPRASAD, T. On the computability of circumscription. Inf. Proc. Lett. 27, 5 (1988), 237-243. Google Scholar
- 18 KUEKER, D.W. Decidability results for universal sentences true on minimal models, preprint. University of Maryland, College Park, Md., 1987.Google Scholar
- 19 LIFSCHITZ, V. Computing circumscription. In Proceedings of the 9th International Joint Conference on Artificial Intelligence. 1985, pp. 121-127.Google Scholar
- 20 LIFSCHITZ, V. On the satisfiability of circumscription. Artif Int. 28 (1986), 17-27. Google Scholar
- 21 LIFSCHITZ, V. Pointwise circumscription. In Readings in Nonmonotonic Reasoning, M. L. Ginsberg, ed. Morgan-Kaufmann, San Mateo, Calif., 1988, pp. 179-193. Google Scholar
- 22 MAIER, D., ULLMAN, J. D., AND VARDI, M. Y. On the foundations of the universal relation model. ACM Trans. Datab. Syst. 9, 2 (June 1984), 283-308. Google Scholar
- 23 MCCARTHY, J. CircumscriptionmA form of non-monotonic reasoning. Artif. Int. 13 (1980), 27-39.Google Scholar
- 24 MCCARTHY, J. Applications of circumscription in formalizing common sense knowledge. Artif. Int. 28 (1986), 89- l 16. Google Scholar
- 25 MOSCHOVAKIS, Y.N. Elementary Induction on Abstract Structures. North-Holland, Amsterdam, The Netherlands, 1974. Google Scholar
- 26 NAUGHTON, J. Data independent recursion in deductive databases. In Proceedings of the 5th ACM SIGACT-SIGMOD Symposium on Principles of Database Systems (Cambridge, Mass., Mar. 24-26). ACM, New York, 1986, pp. 267-279. Google Scholar
- 27 NAUGHTON, J. F., AND SAGIV, Y. A decidable class of bounded recursions. In Proceedings of the 6th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (San Diego, Calif., Mar. 23-25). ACM, New York, 1987, pp. 227-236. Google Scholar
- 28 PAPADIMITRIOU, C. H., AND STEIGLITZ, K. Combinatorial Optimization. Prentice-Hall, New York, 1982. Google Scholar
- 29 PERLIS, D., AND MINKER, J. Completeness results for circumscription. Artif. Int. 28 (1986), 29-42. Google Scholar
- 30 SAGIV, Y. On computing restricted projections of representative instances. In Proceedings of the 4th ACM SIGACT-SIGMOD Symposium on Principles of Database Systems (Portland, Ore., Mar. 25-27). ACM, New York, 1985, pp. 171-180. Google Scholar
- 31 SCHLIPF, J.S. How uncomputable is general circumscription. In Proceedings of the 1st IEEE Conference on Logic in Computer Science. IEEE, New York, 1986, pp. 92-95.Google Scholar
- 32 VARDI, M.Y. Querying Logical Databases. J. Comput. Syst. Sci. 33, 2 (1986), 142-160. Google Scholar
- 33 VARDI, M.Y. Decidability and undecidability results for boundedness of linear recursive queries. In Proceedings of the 7th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (Austin, Tex., Mar. 21-23). ACM, New York, 1988, pp. 341-350. Google Scholar
Index Terms
- Some computational aspects of circumscription
Recommendations
From answer set logic programming to circumscription via logic of GK
We first embed Pearce's equilibrium logic and Ferraris's propositional general logic programs in Lin and Shoham's logic of GK, a nonmonotonic modal logic that has been shown to include as special cases both Reiter's default logic in the propositional ...
Loop formulas for circumscription
Clark's completion is a simple nonmonotonic formalism and a special case of several nonmonotonic logics. Recently there has been work on extending completion with ''loop formulas'' so that general cases of nonmonotonic logics such as logic programs (...
Computing Circumscription Revisited: A Reduction Algorithm
In recent years, a great deal of attention has been devoted to logics of common-sense reasoning. Among the candidates proposed, circumscription has been perceived as an elegant mathematical technique for modeling nonmonotonic reasoning, but difficult to ...
Reviews
Mihai Constantin
The circumscription formalism of commonsense reasoning, first introduced by McCarthy, transforms logical formulas by adding a requirement of minimality. The circumscription of a first-order formula is a second-order formula, implying second-order quantification. This process leads to an increase in logical complexity. In the introductory section, the authors describe some results that identify classes of first-order formulas whose circumscription is equivalent to a first-order formula. Then they demonstrate that the circumscription of any existential first-order formula is equivalent to a first-order formula. The result represents an extension of the work of V. Lifschitz. In the next stage, they show that the circumscription of Horn clauses is first-order if and only if the corresponding program is bounded. Finally, the authors deduce that first-order formulas exist whose circumscription has a coNP-complete model-checking problem. Based on these results, they propose a list of open problems.
Access critical reviews of Computing literature here
Become a reviewer for Computing Reviews.
Comments