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Finite Variable Logics in Descriptive Complexity Theory

Published online by Cambridge University Press:  15 January 2014

Martin Grohe
Martin Grohe*
Affiliation:
Institut für Mathematische Logik, Eckerstrasse 1, 79104 Freiburg, GermanyE-mail: [email protected]
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Extract

Throughout the development of finite model theory, the fragments of first-order logic with only finitely many variables have played a central role. This survey gives an introduction to the theory of finite variable logics and reports on recent progress in the area.

For each k ≥ 1 we let Lk be the fragment of first-order logic consisting of all formulas with at most k (free or bound) variables. The logics Lk are the simplest finite-variable logics. Later, we are going to consider infinitary variants and extensions by so-called counting quantifiers.

Finite variable logics have mostly been studied on finite structures. Like the whole area of finite model theory, they have interesting model theoretic, complexity theoretic, and combinatorial aspects. For finite structures, first-order logic is often too expressive, since each finite structure can be characterized up to isomorphism by a single first-order sentence, and each class of finite structures that is closed under isomorphism can be characterized by a first-order theory. The finite variable fragments seem to be promising candidates with the right balance between expressive power and weakness for a model theory of finite structures. This may have motivated Poizat [67] to collect some basic model theoretic properties of the Lk. Around the same time Immerman [45] showed that important complexity classes such as polynomial time (PTIME) or polynomial space (PSPACE) can be characterized as collections of all classes of (ordered) finite structures definable by uniform sequences of first-order formulas with a fixed number of variables and varying quantifier-depth.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 4 , Issue 4 , December 1998 , pp. 345 - 398
Copyright
Copyright © Association for Symbolic Logic 1998

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References

REFERENCES

[1] Abiteboul, S., Vardi, M. Y., and Vianu, V., Fixpoint logics, relational machines, and computational complexity, Journal of the ACM, vol. 44 (1997), pp. 30–56.Google Scholar
[2] Abiteboul, S. and Vianu, V., Generic computation and its complexity, Proceedings of the 23rd ACM symposium on theory of computing, 1991, pp. 209–219.Google Scholar
[3] Babai, L., Erdös, P., and Selkow, S., Random graph isomorphism, SIAM Journal on Computing, vol. 9 (1980), pp. 628–635.Google Scholar
[4] Babai, L. and Luks, E. M., Canonical labeling of graphs, Proceedings of the 15th ACM symposium on theory of computing, 1983, pp. 171–183.Google Scholar
[5] Barwise, J., Back and forth through infinitary logic, Studies in model theory (Morley, M. D., editor), Studies in Mathematics, Mathematical Association of America, 1973, pp. 5–34.Google Scholar
[6] Barwise, J., On Moschovakis closure ordinals, this Journal, vol. 42 (1977), pp. 292–296.Google Scholar
[7] Blass, A., Gurevich, Y., and Kozen, D., A zero-one law for logic with a fixed-point operator, Information and Control, vol. 67 (1985), pp. 70–90.Google Scholar
[8] Börger, E., Grädel, E., and Gurevich, Y., The classical decision problem, Springer-Verlag, 1997.Google Scholar
[9] Cai, J., Fürer, M., and Immerman, N., An optimal lower bound on the number of variables for graph identification, Combinatorica, vol. 12 (1992), pp. 389–410.Google Scholar
[10] Cameron, P., Combinatorics: Topics, techniques, algorithms, Cambridge University Press, 1994.Google Scholar
[11] Chandra, A. and Harel, D., Structure and complexity of relational queries, Journal of Computer and System Sciences, vol. 25 (1982), pp. 99–128.Google Scholar
[12] Dawar, A., Feasible computation through model theory, Ph.D. thesis , University of Pennsylvania, 1993.Google Scholar
[13] Dawar, A., Generalized quantifiers and logical reducibilities, Journal of Logic and Computation, vol. 5 (1995), pp. 213–226.Google Scholar
[14] Dawar, A., A restricted second order logic for finite structures, Logic and computational complexity: International workshop, LCC '94 (Leivant, D., editor), Lecture Notes in Computer Science, vol. 960, Springer-Verlag, 1995.Google Scholar
[15] Dawar, A., Types and indiscernibles in finite models, Logic colloquium '95 (Makowsky, J. A. and Ravve, E. V., editors), Lecture Notes in Logic, vol. 11, Springer-Verlag, 1998, pp. 51–65.Google Scholar
[16] Dawar, A., Hella, L., and Kolaitis, Ph. G., Implicit definability and infinitary logic in finite model theory, Proceedings of the 22nd international colloquium on automata, languages and programming, 1995, pp. 624–635.Google Scholar
[17] Dawar, A., Lindell, S., and Weinstein, S., Infinitary logic and inductive definability over finite structures, Information and Computation, vol. 119 (1995), pp. 160–175.Google Scholar
[18] Ebbinghaus, H.-D., Extended logics: The general framework, Model-theoretic logics (Barwise, J. and Feferman, S., editors), Springer-Verlag, 1985, pp. 25–76.Google Scholar
[19] Ebbinghaus, H.-D. and Flum, J., Finite model theory, Springer-Verlag, 1995.Google Scholar
[20] Ebbinghaus, H.-D., Flum, J., and Thomas, W., Mathematical logic, 2nd ed., Springer-Verlag, 1994.Google Scholar
[21] Fagin, R., Generalized first-order spectra and polynomial-time recognizable sets, Complexity of computation (Karp, R.M., editor), SIAM-AMS Proceedings, vol. 7,1974, pp. 43–73.Google Scholar
[22] Filotti, I. S. and Mayer, J. N., A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus, Proceedings of the 12th ACM symposium on theory of computing, 1980, pp. 236–243.Google Scholar
[23] Flum, J. and Ziegler, M., private communication.Google Scholar
[24] Grädel, E., Kolaitis, Ph. G., and Vardi, M. Y., On the decision problem for two-variable first-order logic, Bulletin of Symbolic Logic, vol. 3 (1997), pp. 53–69.CrossRefGoogle Scholar
[25] Grädel, E. and Otto, M., Inductive definability with counting on finite structures, Computer science logic, 6th workshop, CSL '92, San Miniato 1992, selected papers (Börger, E., Jäger, G., Büning, H. Kleine, Martini, S., and Richter, M. M., editors), Lecture Notes in Computer Science, vol. 702, Springer-Verlag, 1993, pp. 231–247.Google Scholar
[26] Grädel, E. and Otto, M., On logics with two variables, Theoretical Computer Science, to appear, 1998.Google Scholar
[27] Grädel, E., Otto, M., and Rosen, E., Two-variable logic with counting is decidable, Proceedings of the 12th IEEE symposium on logic in computer science, 1997, pp. 306–317.Google Scholar
[28] Grädel, E., Otto, M., and Rosen, E., Undecidability results on two-variable logics, Proceedings of the 14th annual symposium on theoretical aspects of computer science, 1997, pp. 249–260.Google Scholar
[29] Grohe, M., Equivalence in finite-variable logics is complete for polynomial time, Proceedings of the 37th annual IEEE symposium on foundations of computer science, 1996, pp. 264–273.Google Scholar
[30] Grohe, M., Large finite structures with few L k-types , Proceedings of the 12th IEEE symposium on logic in computer science, 1997, pp. 216–227.Google Scholar
[31] Grohe, M., Canonization for L k-equivalence is hard , Computer science logic, 11th international workshop CSL '97 (Selected papers) (Nielsen, M. and Thomas, W., editors), Lecture Notes in Computer Science, vol. 1414, Springer-Verlag, 1998.Google Scholar
[32] Grohe, M., Fixed-point logics on planar graphs, Proceedings of the 13th IEEE symposium on logic in computer science, 1998, pp. 6–15.Google Scholar
[33] Grohe, M. and Mariño, J., Definability and descriptive complexity on databases of bounded tree-width, Proceedings of the 7th international conference on database theory (Beeri, C., editor), Lecture Notes in Computer Science, Springer-Verlag, 1999, to appear.Google Scholar
[34] Gurevich, Y., Toward logic tailored for computational complexity, Computation and proof theory (Richter, M. M., Börger, E., Oberschelp, W., Schinzel, B., and Thomas, W., editors), Lecture Notes in Mathematics, vol. 1104, Springer-Verlag, 1984, pp. 175–216.Google Scholar
[35] Gurevich, Y., Logic and the challenge of computer science, Current trends in theoretical computer science (Börger, E., editor), Computer Science Press, 1988, pp. 1–57.Google Scholar
[36] Gurevich, Y., From invariants to canonization, Bulletin of the European Association for Theoretical Computer Science, vol. 63 (1997), pp. 115–119.Google Scholar
[37] Gurevich, Y. and Shelah, S., Fixed point extensions of first-order logic, Annals of Pure and Applied Logic, vol. 32 (1986), pp. 265–280.Google Scholar
[38] Gurevich, Y. and Shelah, S., On finite rigid structures, this Journal, vol. 61 (1996), pp. 549–562.Google Scholar
[39] Hella, L., Definability hierarchies of generalized quantifiers, Annals of Pure and Applied Logic, vol. 43 (1989), pp. 235–271.Google Scholar
[40] Hella, L., Logical hierarchies in PTIME, Information and Computation, vol. 129 (1996), pp. 1–19.CrossRefGoogle Scholar
[41] Hella, L., Kolaitis, Ph. G., and Luosto, K., Almost everywhere equivalence of logics in finite model theory, Bulletin of Symbolic Logic, vol. 2 (1996), pp. 422–443.CrossRefGoogle Scholar
[42] Hella, L., Kolaitis, Ph. G., and Luosto, K., How to define a linear order on finite models, Annals of Pure and Applied Logic, vol. 87 (1997), pp. 241–267.Google Scholar
[43] Hodges, W., Model theory, Cambridge University Press, 1993.Google Scholar
[44] Hopcroft, J. E. and Tarjan, R., Efficient planarity testing, Journal of the ACM, vol. ? (1974), pp. 549–568.CrossRefGoogle Scholar
[45] Immerman, N., Upper and lower bounds for first-order expressibility, Journal of Computer and System Sciences, vol. 25 (1982), pp. 76–98.CrossRefGoogle Scholar
[46] Immerman, N., Relational queries computable in polynomial time, Information and Control, vol. 68 (1986), pp. 86–104.Google Scholar
[47] Immerman, N., Expressibility as a complexity measure: results and directions, Proceedings of the 2nd IEEE symposium on structure in complexity theory, 1987, pp. 194–202.Google Scholar
[48] Immerman, N., Languages that capture complexity classes, SIAM Journal on Computing, vol. 16 (1987), pp. 760–778.Google Scholar
[49] Immerman, N. and Lander, E., Describing graphs: A first-order approach to graph canonization, Complexity theory retrospective (Selman, A., editor), Springer-Verlag, 1990, pp. 59–81.Google Scholar
[50] Karp, R. M. and Lipton, R. J., Turing machines that take advice, L'Enseignement Mathématique, vol. 28 (1982), pp. 191–209.Google Scholar
[51] Kolaitis, Ph. G. and Vardi, M. Y., Fixpoint logic vs. infinitary logic in finite-model theory, Proceedings of the 7th IEEE symposium on logic in computer science, 1992, pp. 46–57.Google Scholar
[52] Kolaitis, Ph. G. and Vardi, M. Y., Infinitary logic and 0-1 laws, Information and Computation, vol. 98 (1992), pp. 258–294.CrossRefGoogle Scholar
[53] Kolaitis, Ph. G. and Vardi, M. Y., Infinitary logic for computer science, Proceedings of the 19th international colloquium on automata, languages and programming, Lecture Notes in Computer Science, vol. 623, Springer-Verlag, 1992, pp. 450–473.CrossRefGoogle Scholar
[54] Kolaitis, Ph. G. and Vardi, M. Y., On the expressive power of variable-confined logics, Proceedings of the 11th IEEE symposium on logic in computer science, 1996.Google Scholar
[55] Lindell, S., An analysis of fixed-point queries on binary trees, Theoretical Computer Science, vol. 85 (1991), pp. 75–95.Google Scholar
[56] Livchak, A., The relational model for process control, Automated Documentation and Mathematical Linguistics, vol. 4 (1983), pp. 27–29.Google Scholar
[57] Lovász, L. and Gács, P., Some remarks on generalized spectra, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 23 (1977), pp. 27–144.Google Scholar
[58] Matijasevič, J. V., Enumerable sets are diophantic, Soviet Mathematics Doklady, vol. 11 (1970), pp. 345–357.Google Scholar
[59] Miller, G. L., Isomorphism testing for graphs of bounded genus, Proceedings of the 12th ACM symposium on theory of computing, 1980, pp. 225–235.Google Scholar
[60] Mortimer, M., On languages with two variables, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. ? (1975), pp. 135–140.Google Scholar
[61] Moschovakis, Y. N., On nonmonotone inductive definability, Fundamenta Mathematicae, vol. 82 (1974), pp. 39–83.Google Scholar
[62] Otto, M., Ptime canonization for two variables with counting, Proceedings of the 10th IEEE symposium on logic in computer science, 1995, pp. 342–352.Google Scholar
[63] Otto, M., The expressive power of fixed-point logic with counting, this Journal, vol. 61 (1996), pp. 147–176.Google Scholar
[64] Otto, M., Bounded variable logics and counting, Lecture Notes in Logic, vol. 9, Springer-Verlag, 1997.Google Scholar
[65] Pacholski, L., Szwast, W., and Tendera, L., Complexity of two-variable logic with counting, Proceedings of the 12th IEEE symposium on logic in computer science, 1997, pp. 318–327.Google Scholar
[66] Papadimitriou, C. H., Computational complexity, Addison-Wesley, 1994.Google Scholar
[67] Poizat, B., Deux ou trois choses que je sais de Ln , this Journal, vol. 47 (1982), pp. 641–658.Google Scholar
[68] Scott, D., Logics with denumerably long formulas and finite strings of quantifiers, Theory of models (Addison, J. W., Henkin, L., and Tarski, A., editors), North-Holland, 1965, pp. 329–341.Google Scholar
[69] Seth, A., When do fixed point logics capture complexity classes?, Proceedings of the 10th IEEE symposium on logic in computer science, 1995, pp. 353–363.Google Scholar
[70] Vardi, M. Y., The complexity of relational query languages, Proceedings of the 14th ACM symposium on theory of computing, 1982, pp. 137–146.Google Scholar
[71] Vardi, M. Y., On the complexity of bounded-variable queries, Proceedings of the 14th ACM symposium on principles of database systems, 1995, pp. 266–276.Google Scholar