Abstract
The goal of this chapter is to study the complexity of queries expressible in FO. We start with the general definition of different ways of measuring the complexity of a logic over finite structures: these are data, expression, and combined complexity. We then connect FO with Boolean circuits and establish some bounds on the data complexity. We also consider the issue of uniformity for a circuit model, and study it via logical definability. We then move to the combined complexity of FO, and show that it is much higher than the data complexity. Finally, we investigate an important subclass of FO queries — conjunctive queries — which play a central role in database theory.
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Bibliographic Notes
M.Y. Vardi. The complexity of relational query languages. In Proc. ACM Symap. on Theory of Computing, 1982, 137–146.
S. Abiteboul, R. Hull, and V. Vianu. Foundations of Databases, Addison-Wesley, 1995.
N. Immerman. Descriptive Complexity. Springer-Verlag, 1998.
N. Immerman and Ph. Kolaitis, eds. Descriptive Complexity and Finite Models, Proc. of a DIMACS workshop. AMS, 1997.
H. Vollmer. Introduction to Circuit Complexity. Springer-Verlag, 1999.
M.Y. Vardi. On the complexity of bounded-variable queries. In ACM Symp. on Principles of Database Systems ACM Press, 105, pages 266–276.
M. Furst, J. Saxe, and M. Sipser. Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory, 17 (1984), 13–27.
M. Ajtai. Zi formulae on finite structures. Annals of Pure and Applied Logic, 24 (1983), 1–48.
L. Denenberg, Y. Gurevich, and S. Shelah. Definability by constant-depth polynomial-size circuits. Information and Control, 70 (1986), 216–240.
H. Vollmer. Introduction to Circuit Complexity. Springer-Verlag, 1999.
A.J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. Journal of the American Mathematical Society, 9 (1996), 1051–1094.
S. R. Buss. First-order proof theory of arithmetic. In Handbook of Proof Theory, Elsevier, Amsterdam, 1998, pages 79–147.
S.A. Cook. Proof complexity and bounded arithmetic. Manuscript, Univ. of Toronto, 2002.
L. Stockmeyer. The complexity of decision problems in automata and logic. PhD Thesis, MIT, 1974.
M. Yannakakis. Perspectives on database theory. In IEEE Symp. on Foundations of Computer Science, 1995, pages 224–246.
C. Papadimitriou and M. Yannakakis. On the complexity of database queries. Journal of Computer and System Sciences, 58 (1999), 407–427.
R. Downey and M. Fellows. Parameterized Complexity. Springer-Verlag, 1999.
M. Grohe. The parameterized complexity of database queries. In ACM Symp. on Principles of Database Systems,2001, ACM Press, pages 82–92.
M. Grohe. Parameterized complexity for the database theorist. SIGMOD Record, 31 (2002), 86–96.
D. Seese. Linear time computable problems and first-order descriptions. Mathematical Structures in Computer Science, 6 (1996), 505 526.
J. Flum and M. Grohe. Fixed-parameter tractability, definability, and model-checking. SIAM Journal on Computing 31 (2001), 113–145.
A. Chandra and P. Merlin. Optimal implementation of conjunctive queries in relational data bases. In ACM Symp. on Theory of Computing, 1977, pages 77–90.
M. Yannakakis. Algorithms for acyclic database schemes. In Proc. Conf. on Very Large Databases, 1981, pages 82–94.
J. Flum, M. Frick, and M. Grohe. Query evaluation via tree-decompositions. Journal of the ACM, 49 (2002), 716–752.
R. Tarjan and M. Yannakakis. Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hyper-graphs. SIAM Journal on Computing, 13 (1984), 566–579.
M. Grohe, T. Schwentick, and L. Segoufin. When is the evaluation of conjunctive queries tractable? In ACM Symp. ore Theory of Computing, 2001, pages 657–666.
G. Gottlob, N. Leone, and F. Scarcello. The complexity of acyclic conjunctive queries. Journal of the ACM, 48 (2001), 431–498.
A. Dawar, K. Doets, S. Lindell, and S. Weinstein. Elementary properties of finite ranks. Mathematical Logic Quarterly, 44 (1998), 349–353.
H. Vollmer. Introduction to Circuit Complexity. Springer-Verlag, 1999.
C. Papadimitriou and M. Yannakakis. On the complexity of database queries. Journal of Computer and System Sciences, 58 (1999), 407–427.
J. Flum and M. Grohe. Fixed-parameter tractability, definability, and model-checking. SIAM Journal on Computing 31 (2001), 113–145.
G. Gottlob, N. Leone, and F. Scarcello. The complexity of acyclic conjunctive queries. Journal of the ACM, 48 (2001), 431–498.
A. Chandra and P. Merlin. Optimal implementation of conjunctive queries in relational data bases. In ACM Symp. on Theory of Computing, 1977, pages 77–90.
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Libkin, L. (2004). Complexity of First-Order Logic. In: Elements of Finite Model Theory. Texts in Theoretical Computer Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-07003-1_6
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DOI: https://doi.org/10.1007/978-3-662-07003-1_6
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