Abstract
Well-known theorems of Hanf’s and Gaifman’s establishing locality of first-order definable properties have been used in many applications. These theorems were recently generalized to other logics, which led to new applications in descriptive complexity and database theory. However, a logical characterization of local properties that correspond to Hanf’s and Gaifman’s theorems, is still lacking. Such a characterization only exists for structures of bounded valence.
In this paper, we give logical characterizations of local properties behind Hanf’s and Gaifman’s theorems. We first deal with an infinitary logic with counting terms and quantifiers, that is known to capture Hanf-locality on structures of bounded valence. We show that testing isomorphism of neighborhoods can be added to it without violating Hanf-locality, while increasing its expressive power. We then show that adding local second-order quantification to it captures precisely all Hanf-local properties. To capture Gaifman-locality, one must also add a (potentially infinite) case statement. We further show that the hierarchy based on the number of variants in the case statement is strict.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Part of this work was done while visiting INRIA.
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
D.A.M. Barrington, N. Immerman, H. Straubing. On uniformity within NC 1. JCSS, 41:274–306,1990.
M. Benedikt, H.J. Keisler. Expressive power of unary counters. Proc. Int. Conf. on Database Theory (ICDT’97), Springer LNCS 1186, January 1997, pages 291–305.
J. Cai, M. Fürer and N. Immerman. On optimal lower bound on the number of variables for graph identification. Combinatorica, 12 (1992), 389–410.
G. Dong, L. Libkin and L. Wong. Local properties of query languages. Theoretical Computer Science, to appear. Extended abstract in ICDT’97, pages 140–154.
H.-D. Ebbinghaus and J. Flum. Finite Model Theory. Springer Verlag, 1995.
K. Etessami. Counting quantifiers, successor relations, and logarithmic space, JCSS, 54 (1997), 400–411.
R. Fagin, L. Stockmeyer and M. Vardi, On monadic NP vs monadic co-NP, Information and Computation, 120 (1995), 78–92.
H. Gaifman. On local and non-local properties, Proceedings of the Herbrand Symposium, Logic Colloquium’ 81, North Holland, 1982.
E. Grädel. On the restraining power of guards. J. Symb. Logic, to appear.
E. Grädel and Y. Gurevich. Metafinite model theory. Information and Computation 140 (1998), 26–81.
M. Grohe and T. Schwentick. Locality of order-invariant first-order formulas. In MFCS’98, pages 437–445.
W. Hanf. Model-theoretic methods in the study of elementary logic. In J.W. Addison et al, eds, The Theory of Models, North Holland, 1965, pages 132–145.
L. Hella. Logical hierarchies in PTIME. Information and Computation, 129 (1996), 1–19.
L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over finite models. J. Symb. Logic, to appear. Extended abstract in LICS’97, pages 204–215 (paper by the 2nd author).
L. Hella, L. Libkin, J. Nurmonen and L. Wong. Logics with aggregate operators. In LICS’99, pages 35–44.
N. Immerman. Descriptive Complexity. Springer Verlag, 1999.
N. Immerman and E. Lander. Describing graphs: A first order approach to graph canonization. In “Complexity Theory Retrospective”, Springer Verlag, Berlin, 1990.
Ph. Kolaitis and J. Väänänen. Generalized quantifiers and pebble games on finite structures. Annals of Pure and Applied Logic, 74 (1995), 23–75.
L. Libkin. On counting logics and local properties. In LICS’98, pages 501–512.
L. Libkin. Logics capturing local properties. Bell Labs Technical Memo, 1999.
L. Libkin and L. Wong. Unary quantifiers, transitive closure, and relations of large degree. In STACS’98, Springer LNCS 1377, pages 183–193.
J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 779–798.
M. Otto. Bounded Variable Logics and Counting: A Study in Finite Models. Springer Verlag, 1997.
T. Schwentick and K. Barthelmann. Local normal forms for first-order logic with applications to games and automata. In STACS’98, Springer LNCS 1377, 1998, pages 444–454.
M. Vardi. Why is monadic logic so robustly decidable? In Proc. DIMACS Workshop on Descriptive Complexity and Finite Models, AMS 1997.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Libkin, L. (2000). Logics Capturing Local Properties. In: Reichel, H., Tison, S. (eds) STACS 2000. STACS 2000. Lecture Notes in Computer Science, vol 1770. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46541-3_18
Download citation
DOI: https://doi.org/10.1007/3-540-46541-3_18
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67141-1
Online ISBN: 978-3-540-46541-6
eBook Packages: Springer Book Archive