Abstract
This paper studies expressivity bounds for extensions of first-order logic with counting and unary quantifiers in the presence of relations of large degree. There are several motivations for this work. First, it is known that first-order logic with counting quantifiers captures uniform TC° over ordered structures. Thus, proving expressivity bounds for first-order with counting can be seen as an attempt to show TC° ≠⊂ DLOG using techniques of descriptive complexity. Second, the presence of auxiliary built-in relations (e.g., order, successor) is known to make a big impact on expressivity results in finite-model theory and database theory. Our goal is to extend techniques from “pure” setting to that of auxiliary relations.
Until now, all known results on the limitations of expressive power of the counting and unary-quantifier extensions of first-order logic dealt with auxiliary relations of “small” degree. For example, it is known that these logics fail to express some DLOG-queries in the presence of a successor relation. Our main result is that these extensions cannot define the deterministic transitive closure (a DLOG-complete problem) in the presence of auxiliary relations of “large” degree, in particular, those which are “almost linear orders.” They are obtained from linear orders by replacing them by “very thin” preorders on arbitrarily small number of elements. We show that the technique of the proof (in a precise sense) cannot be extended to provide the proof of separation of TC° from DLOG. We also discuss a general impact of having built-in (pre)orders, and give some expressivity statements in the pure setting that would imply separation results for the ordered case.
Part of this work was done when the first author was visiting Institute of Systems Science.
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M. Agrawal, E. Allender and S. Datta. On TC°, AC°, and arithmetic circuits. In Proc. 12th IEEE Conf. on Computational Complexity, 1997.
E. Allender. Circuit complexity before the dawn of the new millennium. In FST&TCS'96, Springer LNCS vol. 1180, 1996, 1–18.
S. Abiteboul, R. Hull, V. Vianu, Foundations of Databases, Addison Wesley, 1995.
D.A. Barrington, N. Immerman, H. Straubing. On uniformity within NC 1. JCSS, 41:274–306,1990.
M. Benedikt, H.J. Keisler. On expressive power of unary counters. ICDT'97, Springer LNCS 1186, 1997, pages 291–305.
J. Cai, M. Fürer and N. Immerman. An optimal lower bound on the number of variables for graph identification. Combinatorica 12 (1992), 389–410.
G. Dong, L. Libkin, L. Wong. Local properties of query languages. Proc. Int. Conf. on Database Theory, Springer LNCS 1186, 1997, pages 140–154.
H.-D. Ebbinghaus. Extended logics: the general framework. In J. Barwise and S. Feferman, editors, Model-Theoretic Logics, Springer-Verlag, 1985, pages 25–76.
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, M. Vardi, On monadic NP vs monadic co-NP, Information and Computation, 120 (1994), 78–92.
H. Gaifman, On local and non-local properties, in “Proceedings of the Herbrand Symposium, Logic Colloquium '81,” North Holland, 1982.
L. Hella. Logical hierarchies in PTIME. Inform.& Comput., 129 (1996), 1–19.
N. Immerman. Languages that capture complexity classes. SIAM Journal of Computing 16 (1987), 760–778.
N. Immerman and E. Lander. Describing graphs: A first order approach to graph canonization. In “Complexity Theory Retrospective”, Springer Verlag, Berlin, 1990.
L. Libkin. On the forms of locality over finite models. In LICS'97, pages 204–215. Full paper “Notions of locality and their logical characterization over finite models” by L. Hella, L. Libkin and J. Nurmonen is available as Bell Labs Technical Memo.
J. Nurmonen. On winning strategies with unary quantifiers. J. Logic and Computation, 6 (1996), 779–798.
M. Otto. Private communication. DIMACS, July 1997.
I. Parberry and G. Schnitger. Parallel computation and threshold functions. JCSS 36 (1988), 278–302.
A. Razborov and S. Rudich. Natural proofs. JCSS 55 (1997), 24–35.
T. Schwentick. Graph connectivity and monadic NP. FOCS'94, pages 614–622.
T. Schwentick. Graph connectivity, monadic NP and built-in relations of moderate degree. In ICALP'95, Springer LNCS 944, 1995, pages 405–416.
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Libkin, L., Wong, L. (1998). Unary quantifiers, transitive closure, and relations of large degree. In: Morvan, M., Meinel, C., Krob, D. (eds) STACS 98. STACS 1998. Lecture Notes in Computer Science, vol 1373. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0028560
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DOI: https://doi.org/10.1007/BFb0028560
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