Abstract
Random structures (such as the classical random graphs of Erdős and Rényi) are playing an increasingly large role in the theory of computing, as well as in discrete mathematics. The surprising and useful fact that random structures can have properties not found in known constructions—indeed, that they can “have properties” at all—rests on the phenomenon of 0–1 laws; that is, on the fact that for many properties P the probability that a random structure satisfies P is guaranteed to approach either 0 or 1.
We will make a tour of various spaces of random structures, noting when 0-1 laws hold and when they do not; proofs or sketches of proofs are given for the major results. We will also discuss progress in two directions, characterizing structures where the 0-1 law holds for the first-order logic, and extending the 0-1 law as far as possible to more expressive logics.
Ultimately, we wish to improve both our town and our readers’ intuition about how random structures behave.
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References
M.H. Albert & A.M. Frieze, Random graph orders, Order 6 (1989), 19–30.
A. Blass, Y.Gurevich & D. Kozen, A zero-one law for logic with a fixed point operator, Information and Control 67 (1985), 70–90.
B. Bollobás, Random Graphs, Academic Press, New York, 1985.
C.C. Chang & H.J. Keisler, Model Theory, North-Holland, 1990.
K.J. Compton, A logical approach to asymptotic combinatorics I: First-order properties, Advances in Mathematics 78 (1987), 65–96.
K.J. Compton, The computational complexity of asymptotic problems I: partial orders, Inform. and Comput. 78 (1988), 108–123.
K.J. Compton, 0–1 laws in logic and combinatorics, in Algorithms and Order, I. Rival, ed., NATO ASI Series, Kluwer Academic Publishers, Dordrecht, (1989) 353–383.
B. Dushnik & E.W. Miller, Partially ordered sets, Amer. J. Math. 63 (1941), 600–610.
P. Erdős & A. Rényi, On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci. 5 (1960), 17–61.
R. Fagin, Probabilities on finite models, J. Symbolic Logic 41 (1) (1976), 50–58.
Y.V. Glebskii, D.I. Kogan, M.I. Liogon’kii & V.A. Talanov, Range and degree of realizability of formulas in the restricted predicate calculus, Kibernetika 2 (1969), 17–28.
E. Grandjean, Complexity of the first-order theory of almost all structures, Information and Control 52 (1983), 180–204.
J. Justicz, E. Scheinerman, & P. Winkler, Random intervals, Amer. Math. Monthly, 97 (10) (1989), 881–889.
D.J. Kleitman & B.L. Rothschild, Asymptotic enumeration of partial orders on a finite set, Trans. Amer. Math. Soc. 205 (1975), 205–210.
Ph.G. Kolaitis & M.Y. Vardi, Zero-one laws and decision problems for fragments of second-order logic, Information and Computation 87 (1990), 302–338.
Ph.G. Kolaitis & M.Y. Vardi, Zero-one laws for infinitary logics, Proc. 5th IEEE Symp. on Logic in Computer Science (1990), 156–167.
N. Linial, M. Saks & P. Shor, Largest induced suborders satisfying the chain condition, Order 2 (1985), 265–268.
J. Lynch, Probabilities of first-order sentences about unary functions, Trans. Amer. Math. Soc. 287 (1985), 543–568.
L. Pacholski & W. Szwast, The 0–1 law fails for the class of existential second-order Gödel sentences with equality, Proc. 30th IEEE Symp. on Foundations of Computer Science (1989), 160–163.
R. Rado, Universal graphs and universal functions, Acta Arith. 9 (1964), 331–340.
E.R. Scheinerman, Random interval graphs, Combinatorica 8 (1988), 357–371.
S. Shelah and J. Spencer, Zero-one laws for sparse random graphs, J. Amer. Math. Soc. 1 (1988), 97–115.
J.R. Shoenfield, Mathematical Logic, Addison-Wesley, 1967.
J. Spencer, Nonconvergence in the theory of random orders, Order 7 (1991), 341–348.
P. Winkler, Random orders, Order 1 (1985), 317–331.
P. Winkler, Connectedness and diameter for random orders of fixed dimension, Order 2 (1985), 165–171.
P. Winkler, A counterexample in the theory of random orders, Order 5 (1989), 363–368.
P. Winkler, Random orders of dimension 2, Order 7 (1991), 329–339.
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© 1993 Springer Science+Business Media Dordrecht
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Winkler, P. (1993). Random Structures and Zero-One Laws. In: Sauer, N.W., Woodrow, R.E., Sands, B. (eds) Finite and Infinite Combinatorics in Sets and Logic. NATO ASI Series, vol 411. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2080-7_27
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DOI: https://doi.org/10.1007/978-94-011-2080-7_27
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