Abstract
In this paper, we define and investigate the complexity of several nonmonotonic logics with quantified Boolean formulas as constraints. We give quantified constraint versions of the constraint programming formalism of Marek, Nerode, and Remmel [15] and of the natural extension of their theory to default logic. We also introduce a new formalism which adds constraints to circumscription. We show that standard complexity results for each of these formalisms generalize in the quantified constraint case. Gogic, Kautz, Papadimitriou, and Selman [8] have introduced a new method for measuring the strengths of reasoning formalisms based on succinctness of model representation. We show a natural hierarchy based on this measure exists between our versions of logic programming, circumscription, and default logic. Finally, we discuss some results about the relative succinctness of our reasoning formalisms versus any formalism for which model checking can be done somewhere in the polynomial time hierarchy.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
T. Baker, J. Gill, and R. Solovay. “Relativizations of the P=? NP question”. SIAM Journal of Computing., 1993. 431–442.
M. Cadoli and M. Schaerf. ‘A survey on complexity results for non-monotoniclogic” Journal of Logic Progrmming., Nov. 1993, vol. 17,(no.2–4):127–60.
W.F. Dowling and J.H. Gallier. “Linear-time algorithms for testing the satisfiability of propositional Horn formulae”. Journal of Logic Programming, 3:267–284, 1984
T. Eiter and G. Gottlob. “Propositional circumscription and extended closed-world reasoning are Π 2complete”. Theoretical Computer Science, 114:231–245, 1993.
T. Eiter and J. Lu and V.S.Subrahmanian. “Computing non-ground representations of stable models”. In: Logic Programming and Nonmonotonic Reasoning, LPNMR'97 This volume
M. Furst and J.B. Saxe and M. Sipser. “Parity, circuits and the polynomial hierarchy”. Math Systems Theory, 17:13–27, 1984.
M. R. Garey and D. S. Johnson. Computers and Intractibility: A Guide to the theory of NP-completeness. W.H. Freeman and Company, 1979
G. Gogic, H. Kautz, C. Papadimitriou, and B. Selman. “The comparative linguistics of knowledge representation”. In Proceedings of International Joint Conference on Artificial Intelligence, IJCAI-95. Springer Lecture notes in Artificial Intelligence; 1042. pages 862–869, 1995.
G. Gogic. Complexity Aspects of Knowledge Representation. Ph.D. Thesis. Computer Science Department, U.C. San Diego. 1996.
J. Goldsmith and D. Joseph. Three results on the polynomial isomorphism of complete sets. In Foundations of Computer Science, volume 27, 1986.
G. Gottlob. “Complexity results for nonmonotonic logics”. Journal of Logic and Computation, 2:397–425, 1992
J. Jaffar and J.-L. Lassez. Constraint Logic Programming. In: Proceedings of the 14th ACM Symposium on Principles of Programming Languages, pages 111–119, Münich, 1987.
J. Jaffar and M. Maher Constraint Logic Programming: A Survey. Journal of Logic Programming 19–20, pages 503–581. 1994.
V.W. Marek and M. Truszczyński. Nonmonotonic Logic. Springer Verlag. 1993.
V.W. Marek, A. Nerode and J.B. Remmel. “On logical constraints in Logic Programming”. In: Logic Programming and Nonmonotonic Reasoning, LPNMR'95, Springer Lecture Notes in Artificial Intelligence 928, pages 43–56, 1995.
A.R. Meyer and L. J. Stockmeyer. “The equivalence problem for regular expressions with squaring requires exponential time”. In Proceedings of the 13th annual symposium on switching and automata theory, pages 125–129. New York, NY: IEEE Computer Society, 1972.
C.H. Papadimitriou. “The complexity of knowledge representation”. In Proceedingsof the 11th Annual IEEE Conference on Computational Complexity, pages 244–248. IEEE Computer Society, 1996.
U. Schöning. “A note on complete sets for the polynomial hierarchy”. ACM SIGACT News 13 pages 30–34. 1981
Author information
Authors and Affiliations
Corresponding author
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pollett, C., Remmel, J.B. (1997). Nonmonotonic reasoning with quantified boolean constraints. In: Dix, J., Furbach, U., Nerode, A. (eds) Logic Programming And Nonmonotonic Reasoning. LPNMR 1997. Lecture Notes in Computer Science, vol 1265. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63255-7_3
Download citation
DOI: https://doi.org/10.1007/3-540-63255-7_3
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63255-9
Online ISBN: 978-3-540-69249-2
eBook Packages: Springer Book Archive