J. F. Lynch, Probabilities of First-order Sentences about Unary Functions, Trans. AMS 287:2 (1985), 543--567.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the T # is given in Spencer [22] An examination of the countable models of T # is given in Spencer [20] The Alon, Spencer text [1] also includes some of this material. In this brief paper we have only examined a few examples of random structures. Among the many others we mention Lynch [14] on unary functions; Shelah and Spencer [18] and StJohn and Spencer [24] on random unary predicates with order (considerably di#erent from 1 ) # Luczak [11] on random partially ordered sets. # Luczak and Shelah [12] consider an interesting random graph model on vertex set 1, n where the ....

J. Lynch, Probabilities of first-order sentences about unary functions, Trans. Amer. Math. Soc. 287 (1985), 543-568

....almost fixed percentage p of the vertices. This is similar to the theorem of Clarke ( 7] that a random vertex in a random tree is of degree k with probability 1= k Gamma 1) e. In order to do this, we use a characterization of trees developed in [11] as unary functions, and thus suggestive of [19]. The following construction is also in Chapter 2 of [18] We are going to construct a tree on the set of vertices [n] Let f : n Gamma 1] 7 [n] be a unary function. Let f(1) be the root of the tree (if we are constructing a free tree, use a function f : n Gamma 1] Gamma f1g) 7 [n] ....

....If n is very much bigger than jj, the digraph of a random f : n Gamma 1] 7 [n] a.s. has p n Sigma o(n) vertices of kind , and so the corresponding tree has at least that many vertices of the corresponding kind . As p = 1, o(n) of f is in f s cycles (a fact proven by different means in [19]) and that leaves o(n) of the tree s vertices to account for, and we are done. Again, the proof for free trees is the same, and again p = 1= e a ) Xi Equationophiles will notice that if jj is the number of vertices in kind , then we have the formula = 1. Observe that this method ....

J. F. Lynch, Probabilities of first-order sentences about unary functions, Trans. AMS 287:2 (1985), 543--567.

....function symbols were allowed into the language, then there would not be a 0 1 law. For example, if c is a constant symbol and U a unary relation symbol, then Uc has probability 1 2 . Similarly, if f is a unary function symbol, then 8x(f(x) 6= x) has asymptotic probability 1 e [Fag76] Lynch [Lyn85] has shown that if the language contains only unary function symbols, then there is an asymptotic probability, even though, as we just saw, it need not be 0 or 1. Compton, Henson, and Shelah [CHS87] have shown that if the language consists of a single binary function symbol, then there is not ....

J. Lynch. Probabilities of first-order sentences about unary functions. Trans. American Mathematical Society, 287:543--568, 1985.

....MSO query , there was a p 2 [0; 1] such that for large n, the probability that a random rooted n tree satisfies is p . This was conjectured for labelled and unlabelled trees, and proven by very sophisticated generating function techniques in [Wo95] similar results had been obtained in [Ly85]; trees are very close to unary functions: see [H60] In [Mc b, Mc c] more elementary probabilistic methods (Poisson processes and the second moment method) were used to prove that for free trees, these asymptotic probabilities were 0 or 1, as follows. Recall that every rooted tree T is a ....

J. F. Lynch, Probabilities of First-order Sentences about Unary Functions, Trans. AMS 287:2 (1985), 543--567.

....fixed percentage p of the vertices. This is similar to the theorem of Clarke ( 7] that a random vertex in a random tree is of degree k with probability 1= k Gamma 1) e. In order to do this, we use a characterization of trees developed in [11] as unary functions, and thus suggestive of [19]. The following construction is also in Chapter 2 of [18] We are going to construct a tree on the set of vertices [n] Let f : n Gamma 1] 7 [n] be a unary function. Let f(1) be the root of the tree (if we are constructing a free tree, use a function f : n Gamma 1] Gamma f1g) 7 [n] ....

....n is very much bigger than jj, the digraph of a random f : n Gamma 1] 7 [n] a.s. has p n Sigma o(n) vertices of kind , and so the corresponding tree has at least that many vertices of the corresponding kind . As P p = 1, o(n) of f is in f s cycles (a fact proven by different means in [19]) and that leaves o(n) of the tree s vertices to account for, and we are done. Again, the proof for free trees is the same, and again p = 1= e k a ) Xi Equationophiles will notice that if jj is the number of vertices in kind , then we have the formula P e Gammajj a Gamma1 = 1. ....

J. F. Lynch, Probabilities of first-order sentences about unary functions, Trans. AMS 287:2 (1985), 543--567.

.... shown in [10] In a related context, that of so called Zero One laws and asymptotic laws, large classes of enumerative problems in logic are known to have asymptotic distributions in the limit (the limits are often from the set f0; 1g, whence the name) We refer the reader to the works of Lynch [21] regarding random mappings or Compton [6] regarding general logical frameworks. The classification of distributions that arise in recursive structures represents an appreciably more difficult problem. For instance, path length in planar trees and in binary search trees are described by the two ....

James F. Lynch. Probabilities of first-order sentences about unary functions. Transactions of the American Mathematical Society, 287(2):543--568, February 1985.

....above, En is the expected size of the automorphism group of table based on n elements. Now using this fact and the arguments of Theorem 1 and of (5) and (6) one can show that Murskii s result cited above and his result mentioned in the introduction also hold for unlabeled algebras as well. Lynch [5] has shown that for unary algebras (the limit defining) the labeled probability of any property expressible in first order logic must exist. On the other hand, Compton, Henson, and Shelah have given an example showing that this is not the case for algebras with a binary operation. The results of ....

J. F, Lynch, Probabilities of first-order sentences about unary functions, Trans. Amer. Math. Soc. 287 (1986), 543---568.

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J. F. Lynch, Probabilities of First-order Sentences about Unary Functions, Trans. AMS 287:2 (1985), 543--567.

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J. Lynch, Probabilities of first-order sentences about unary functions, Trans. American Mathematical Society, 287 (1985), pp. 543--568.

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J. Lynch. Probabilities of first-order sentences about unary functions. Trans. American Mathematical Society, 287:543--568, 1985.

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J. F. Lynch, Probabilities of First-order Sentences about Unary Functions, Trans. AMS 287:2 (1985), 543--567. 32

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J. F. Lynch, Probabilities of First-order Sentences about Unary Functions, Trans. AMS 287:2 (1985), 543--567.

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J. F. Lynch, Probabilities of First-order Sentences about Unary Functions, Trans. AMS 287:2 (1985), 543--567. 44

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J. Lynch. Probabilities of first-order sentences about unary functions. Trans. American Mathematical Society, 287:543--568, 1985.

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