Abstract
This article describes the Ehrenfeucht game and its applications in two areas of model theory: definability theory and random model theory. Most of the examples are taken from finite model theory, where the Ehrenfeucht game is one of the most useful tools. Extensions of the Ehrenfeucht game to logics more expressive than first-order logic are described, and applications to definability and random models in these logics are outlined.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Ajtai, M., Fagin, R.: Reachability is harder for directed than for undirected finite graphs. J. Symbolic Logic 55 (1990) 113–150
Barwise, J.: On Moschovakis closure ordinals. J. Symbolic Logic 42 (1977) 292–296
Blass, A., Gurevich, Y., Kozen, D.: A zero-one law for logic with a fixed point operator. Inform. and Control 67 (1985) 70–90
Büchi, J.: Weak second-order arithmetic and finite automata. Z. Math. Logik Grundlagen Math. 6 (1960) 66–92
Compton, K.: A logical approach to asymptotic combinatorics I. first order properties. Adv. Math. 65 (1987) 65–96
Compton, K.: A logical approach to asymptotic combinatorics II: monadic second-order properties. J. Comb. Theory, Ser. A 50 (1989) 110–131
Coppersmith, D.: A left coset composed of n-cycles. Research Report rc 19511, IBM (1994)
de Rougemont, M.: Second-order and inductive definability on finite structures. Z. Math. Logik Grundlagen Math. 33 (1987) 47–63
Ebbinghaus, H.-D., Flum, J.: Finite Model Theory. Springer-Verlag (1995)
Ehrenfeucht, A.: An application of games to the completeness problem for formalized theories. Fund. Math. 49 (1961) 129–141
Elgot, C.: Decision problems of finite-automata design and related arithmetics. Trans. AMS 98 (1961) 21–51
Fagin, R.: Generalized first-order spectra and polynomial-time recognizable sets. Complexity of Computation. R. Karp, ed. SIAM-AMS Proc. 7, Am. Math. Soc., New York (1974) 43–73
Fagin, R.: Monadic generalized spectra. Z. Math. Logik Grundlagen Math. 21 (1975) 89–96
Fagin R.: Probabilities on finite models. J. Symbolic Logic 41 (1976) 50–58
Fraïssé, R.: Sur quelques classifications des systéms de relations. Pub. Sci. Univ. Alger Sér. A 1 (1954) 35–182
Gaifman, H.: On local and nonlocal properties. Logic Colloquium '81. J. Stern, ed. North-Holland (1982) 105–135
Glebskii, Y., Kogan, D., Liogon'kii, M., Talanov, V.: Range and degree of realizability of formulas in the restricted predicate calculus. Kibernetika (Kiev) 2 (1969) 17–28; English translation: Cybernetics 5 (1972) 142–154
Hanf, W.: Model-theoretic methods in the study of elementary logic. J. Addison, L. Henkin, A. Tarski, eds. The Theory of Models. North-Holland (1965) 132–145
Immerman, N.: Relational queries computable in polynomial time. Inform. and Control 68 (1986) 86–104
Kanellakis, P.: Elements of relational database theory. The Handbook of Theoretical Computer Science. A. Meyer, M. Nivat, M. Paterson, D. Perrin, J. van Leeuwen, eds. North-Holland (1990)
Kaufmann, M.: A counterexample to the 0-1 law for existential monadic second-order logic. Internal Note #032, Computational Logic Inc. (1988)
Kaufmann, M., Shelah, S.: On random models of finite power and monadic logic. Discrete Math. 54 (1985) 285–293
Kolaitis, Ph.: On asymptotic probabilities of inductive queries and their decision problem. R. Parikh, ed. Logics of Programs '85. Lecture Notes in Computer Science 193, Springer-Verlag (1985) 153–166
Kolaitis, Ph., Vardi, M.: The decision problem for the probabilities of higher-order properties. Proc. 19th ACM Symp. on Theory of Computing (1987) 425–435
Kolaitis, Ph., Vardi, M.: Infinitary logics and 0–1 laws. Inform. and Computation 98 (1992) 258–294
Ladner, R.: Application of model theoretic games to discrete linear orders and finite automata. Inform. and Control 33 (1977) 281–303
Lynch, J.: Almost sure theories. Ann. Math. Logic 18 (1980) 91–135
Lynch, J.: On sets of relations definable by addition. J. Symbolic Logic 47 (1982) 659–668
Lynch, J.: Probabilities of first-order sentences about unary functions. Trans. AMS 287 (1985) 543–568
Lynch, J.: Probabilities of sentences about very sparse random graphs. Random Struct. Alg. 3 (1992) 33–53
Lynch, J.: The quantifier structure of sentences that characterize nondeterministic time complexity. Comput. Complexity 2 (1992) 40–66
Lynch, J.: Convergence laws for random words. Australasian J. of Combinatorics 7 (1993) 145–156
Lynch, J.: Infinitary logics and very sparse random graphs. Proc. Eighth Ann. IEEE Symp. on Logic in Computer Science (1993) 191–198; J. Symbolic Logic (to appear)
Monk, J.: Mathematical Logic. Springer-Verlag, New York (1976)
McKenzie, R., Mycielski, J., Thompson, D.: On boolean functions and connected sets. Math. Systems Theory 5 (1971) 259–270
Robinson, J.: Definablilty and decision problems in arithmetics. J. Symbolic Logic 14 (1949) 98–114
Schwentick, T.: On winning Ehrenfeucht games and monadic NP. Technical Report 3/95, Institut für Informatik, Johannes Gutenberg-Universität Mainz (1995)
Shelah, S., Spencer, J.: Zero-one laws for sparse random graphs. J. AMS 1 (1988) 97–115
Spencer, J.: Zero-one laws via the Ehrenfeucht game. Discrete Appl. Math. 30 (1991) 235–252
Vardi, M.: The complexity of relational query languages. Proc. 14th ACM Symp. on Theory of Computing (1982) 137–146
Woods, A.: Counting finite models. J. Symbolic Logic (to appear)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Lynch, J.F. (1997). Pebble games in model theory. In: Mycielski, J., Rozenberg, G., Salomaa, A. (eds) Structures in Logic and Computer Science. Lecture Notes in Computer Science, vol 1261. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63246-8_5
Download citation
DOI: https://doi.org/10.1007/3-540-63246-8_5
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63246-7
Online ISBN: 978-3-540-69242-3
eBook Packages: Springer Book Archive