Abstract
Gire and Hoang introduce a fixed-point logic with a ‘symmetric’ choice operator that makes a nondeterministic choice from a definable set of tuples at each stage in the inductive construction of a relation, as long as the set of tuples is an automorphism class of the structure. We present a clean definition of the syntax and semantics of this logic and investigate its expressive power. We extend the logic of Gire and Hoang with parameterized and nested fixed points and first-order combinations of fixed points. We show that the ability to supply parameters to fixed points strictly increases the power of the logic. Our logic can express the graph isomorphism problem and we show that, on almost all structures, it captures P GI, the class of problems decidable in polynomial time by a deterministic Turing machine with an oracle for graph isomorphism.
Research supported by EPSRC grant GR/S06721.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Arvind, V., Biswas, S.: Expressibility of first order logic with a nondeterministic inductive operator. In: Brandenburg, F.J., Wirsing, M., Vidal-Naquet, G. (eds.) STACS 1987. LNCS, vol. 247, pp. 323–335. Springer, Heidelberg (1987)
Blass, A., Gurevich, Y.: The logic of choice. Journal of Symbolic Logic 65(3), 1264–1310 (2000)
Boppana, R.B., Håstad, J., Zachos, S.: Does co-NP have short interactive proofs? Information Processing Letters 25(2), 127–132 (1987)
Cai, J.-Y., Fürer, M., Immerman, N.: An optimal lower bound on the number of variables for graph identification. Combinatorica 12(4), 389–410 (1992)
Dawar, A.: Generalized quantifiers and logical reducibilities. Journal of Logic and Computation 5(2), 213–226 (1995)
Dawar, A., Richerby, D.M.: Fixed-point logics with nondeterministic choice. Journal of Logic and Computation (to appear)
Ebbinghaus, H.-D.: Extended logics: The general framework. In: Barwise, J., Feferman, S. (eds.) Model-Theoretic Logics, Perspectives in Mathematical Logic, pp. 25–76. Springer, Heidelberg (1985)
Ebbinghaus, H.-D., Flum, J.: Finite Model Theory, 2nd edn. Springer, Heidelberg (1999)
Fagin, R.: Generalized first-order spectra and polynomial-time recognizable sets. In: Karp, R. (ed.) Complexity of Computation. SIAM-AMS Proceedings, vol. 7, pp. 43–73. SIAM-AMS (1974)
Gire, F., Hoang, H.K.: An extension of fixpoint logic with a symmetrybased choice construct. Information and Computation 144, 40–65 (1998)
Grohe, M., Mariño, J.: Definability and descriptive complexity on databases of bounded tree-width. In: Beeri, C., Bruneman, P. (eds.) ICDT 1999. LNCS, vol. 1540, pp. 70–82. Springer, Heidelberg (1998)
Grohe, M.: Fixed-point logics on planar graphs. In: Proceedings of 13th IEEE Annual Symposium on Logic in Computer Science, pp. 6–15. IEEE Computer Society, Los Alamitos (1998)
Grohe, M.: Isomorphism testing for embeddable graphs through definability. In: Proceedings of 32nd ACM Symposium on Theory of Computing, pp. 63–72. ACM, New York (2000)
Gyssens, M., Van den Bussche, J., Van Gucht, D.: Expressiveness of efficient semi-deterministic choice constructs. In: Shamir, E., Abiteboul, S. (eds.) ICALP 1994. LNCS, vol. 820, pp. 106–117. Springer, Heidelberg (1994)
Immerman, N., Lander, E.S.: Describing graphs: A first-order approach to graph canonization. In: Selman, A. (ed.) Complexity Theory Retrospective, pp. 59–81. Springer, Heidelberg (1990)
Immerman, N.: Relational queries computable in polynomial time. Information and Control 68(1–3), 86–104 (1986)
Otto, M.: Bounded Variable Logics and Counting — A Study in Finite Models. Lecture Notes in Logic, vol. 9. Springer, Heidelberg (1997)
Schöning, U.: Graph isomorphism is in the low hierarchy. Journal of Computer and System Sciences 37(3), 312–323 (1988)
Torán, J.: On the hardness of graph isomorphism. In: Proceedings of 41st Annual Symposium on Foundations of Computer Science, pp. 180–186. IEEE Computer Society, Los Alamitos (2000)
Vardi, M.Y.: Complexity of relational query languages. In: Proceedings of 14th ACM Symposium on Theory of Computing, pp. 137–146. ACM, New York (1982)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dawar, A., Richerby, D. (2003). A Fixed-Point Logic with Symmetric Choice. In: Baaz, M., Makowsky, J.A. (eds) Computer Science Logic. CSL 2003. Lecture Notes in Computer Science, vol 2803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45220-1_16
Download citation
DOI: https://doi.org/10.1007/978-3-540-45220-1_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40801-7
Online ISBN: 978-3-540-45220-1
eBook Packages: Springer Book Archive