J. Spencer and K. StJohn, Random Unary Predicates: Almost Sure Theories and Countable Models, Random Structures & Algorithms 13 (1998), 229-248 Home/Search Document Details and Download
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Random Sparse Bit Strings at the Threshold of Adjacency.. - Spencer, al. (1998) (Correct)
....that the almost sure theories are identical for every c. Theorem 2 Let S = T c S c be the intersection of all the almost sure theories. Then, for every real, positive c, S c = S. To prove these theorems, we look first at the countable models of the almost sure theories (for more on this, see [12]) Let U j= S c be such a model. Each of these models satisfy a set of basic axioms Delta (defined in Section 3) Let a = a 0 ; am Gamma1 ) be a finite sequence of non negative integers, representing the distance pairs of 1 s occur apart. We show, using Ehrenfeucht Fraisse games, that ....
....ff l : ff j fi 1 : fi j Let Delta = f Gamma l ; 1 ; 2 ; j 1 ; j 2 ; V r ff r ; V r ffi r g: Then Delta ae T 1 ; T 2 and for each c 0, Delta ae S c . The set of sentences, Sigma [f V r fi r g, axiomatizes T 1 . This follows from an Ehrenfeucht Fraisse game argument (see [12] for more details) For countable models of T 1 , we cannot have a single infinite chain, since all the 1 s must be isolated. So, we must have infinitely many chains, ordered like the integers (called Z chains) that contain a single 1 with an infinite increasing chain of 0 s at the beginning and ....
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Joel H. Spencer and Katherine St. John. Random unary predicates: Almost sure theories and countable models. Submitted for publication, 1997.
Logic and Random Structures - Spencer Self-citation (Spencer) (Correct)
....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 adjacency probability between i and j depends on i j . While we ....
J. Spencer and K. StJohn, Random Unary Predicates: Almost Sure Theories and Countable Models, Random Structures & Algorithms 13 (1998), 229-248
The Tenacity of Zero-One Laws - Spencer, John Self-citation (Spencer St) (Correct)
....n 1 or n 1=k 1 p(n) n 1= k 1) k 1. the electronic journal of combinatorics 7 (2000) #Rxx 5 3 Very, Very Sparse When the probability generating the random bit strings and circular sequences is small, p n 1 , then, almost surely, ones do not occur, and a Zero One Law holds (see [7]) We will focus on bit strings in this section, but the results hold also for circular sequences. Since the random bit string for these probabilities is just a string of zeros, the game on any structures reduces to counting the number of bits. For n 1 ; n 2 2 k 1, Duplicator wins the k move ....
....game if: 1. The lengths of both structures are greater than 2 k . 2. The number of ones in each structure is greater than 2 k . 3. The distance between any two ones in each structure is greater than 2 k . Also, the distance between any one and an endpoint is greater than 2 k . In [7], these conditions for Duplicator to win the k move game were shown to be necessary (following from the de nition of k types ) Given n, for what k will these conditions hold 1 of the time We have already addressed the rst condition above. We would like the second and third condition to hold ....
Joel H. Spencer and Katherine St. John. Random unary predicates: Almost sure theories and countable models. Random Structures and Algorithms, 13(#3/4), 1998.
Limit Probabilities for Random Sparse Bit Strings - John (1997) (1 citation) Self-citation (St) (Correct)
....the work of Lynch, Spencer, and Thoma: 5] 6] and [9] We achieve a simpler characterization of the limit probabilities than those for random graphs due to our underlying models. To prove these theorems, we look first at the countable models of the almost sure theories (for more on this, see [8] ) Let M =S c be such a model. Each of these models satisfy a set of basic axioms # (defined in Section 3) Further, we show, using Ehrenfeucht Fraisse games, that # # # i is complete, where # i is the firstorder sentence that states there are exactly i elements for which the unary ....
....m , M(#) t 1 , t m . Then, f # (c) 1 e c P i=m i=1 c t i t i 1 for c 0 which contradicts f # (d) 1. So, we must have that M(#) #.This gives that f # is constantly one. So, # # S c for every c, and thus, # # S. Therefore, S = T c S c . # 5 Future Work The work of [7] and [8] characterize the almost sure theories and their countable models for p(n) # n 1 and n 1 k # p(n) # n 1 (k 1) for k # 1. In this paper, we fill the gap between p(n) # n 1 and n 1 # p(n) # n 1 2 by characterizing the almost sure theories of U n, c n and giving the form of ....
Joel H. Spencer and Katherine St. John. Random unary predicates: Almost sure theories and countable models. Submitted for publication, 1997.
Limit Probabilities for Random Sparse Bit Strings - John (1997) (1 citation) Self-citation (St) (Correct)
....the work of Lynch, Spencer, and Thoma: 5] 6] and [9] We achieve a simpler characterization of the limit probabilities than those for random graphs due to our underlying models. To prove these theorems, we look first at the countable models of the almost sure theories (for more on this, see [8]) Let M j= S c be such a model. Each of these models satisfy a set of basic axioms Delta (defined in Section 3) Further, we show, using Ehrenfeucht Fraisse games, that Delta [ foe i g is complete, where oe i is the firstorder sentence that states there are exactly i elements for which the ....
....t mg. Then, f OE (c) 1 Gamma e Gammac P i=m i=1 c t i t i 1 for c 0 which contradicts f OE (d) 1. So, we must have that M( OE) This gives that f OE is constantly one. So, OE 2 S c for every c, and thus, OE 2 S. Therefore, S = T c S c . 2 5 Future Work The work of [7] and [8] characterize the almost sure theories and their countable models for p(n) n Gamma1 and n Gamma1=k p(n) n Gamma1= k 1) for k 1. In this paper, we fill the gap between p(n) n Gamma1 and n Gamma1 p(n) n Gamma1=2 by characterizing the almost sure theories of U n; c ....
Joel H. Spencer and Katherine St. John. Random unary predicates: Almost sure theories and countable models. Submitted for publication, 1997.
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