Abstract
We continue dealing with extensions of first-order logic. We have seen that the expressive power of FO on finite structures is limited in a number of ways: it cannot express counting properties, nor is it capable of expressing properties that require iterative algorithms, as those typically violate locality.
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Bibliographic Notes
N. Immerman and E. Lander. Describing graphs: a first order approach to graph canonization. In Complexity Theory Retrospective, Springer-Verlag, Berlin, 1990.
K. Etessami. Counting quantifiers, successor relations, and logarithmic space. Journal of Computer and System Sciences, 54 (1997), 400–411.
J. Väänänen. Generalized quantifiers. Bulletin of the EATCS, 62 (1997), 115 136.
J. Väänänen. Unary quantifiers on finite models. Journal of Logic, Language and Information, 6 (1997), 275–304.
L. Libkin. On counting logics and local properties. ACM Transactions on Computational Logic, 1 (2000), 33–59.
L. Hella. Logical hierarchies in PTIME. Information and Computation, 129 (1996), 1–19.
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. ACM Transactions on Computational Logic, 1 (2000), 33–59.
D.A.M. Barrington, N. Immerman, and H. Straubing. On uniformity within NC B . Journal of Computer and System Sciences, 41 (1990), 274–306.
H. Vollmer. Introduction to Circuit Complexity. Springer-Verlag, 1999.
L. Hella, L. Libkin, and J. Nurmonen. Notions of locality and their logical characterizations over finite models. Journal of Symbolic Logic, 64 (1999), 1751 1773.
E. Grädel and Y. Gurevich. Metafinite model theory. Information and Computation, 140 (1998), 26–81.
L. Hella, L. Libkin, J. Nurmonen, and L. Wong. Logics with aggregate operators. Journal of the ACM, 48 (2001), 880–907.
L. Libkin. Logics capturing local properties. ACM Transactions on Computational Logic, 2 (2001), 135–153.
L. Libkin and L. Wong. Lower bounds for invariant queries in logics with counting. Theoretical Computer Science, 288 (2002), 153–180.
N. Immerman and E. Lander. Describing graphs: a first order approach to graph canonization. In Complexity Theory Retrospective, Springer-Verlag, Berlin, 1990.
D.A.M. Barrington, N. Immerman, and H. Straubing. On uniformity within NC B . Journal of Computer and System Sciences, 41 (1990), 274–306.
J. Nurmonen. Counting modulo quantifiers on finite structures. Information and Computation, 160 (2000), 62–87.
L. Hella, L. Libkin, J. Nurmonen, and L. Wong. Logics with aggregate operators. Journal of the ACM, 48 (2001), 880–907.
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Libkin, L. (2004). Logics with Counting. In: Elements of Finite Model Theory. Texts in Theoretical Computer Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-07003-1_8
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DOI: https://doi.org/10.1007/978-3-662-07003-1_8
Publisher Name: Springer, Berlin, Heidelberg
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