Random Sparse Bit Strings at the Threshold of Adjacency

d Abstract appeared in STACS `98) Joel H. Spencer1 and Katherine St. John2 1 Courant Institute, New York University, New York, NY 10012 2 Department of Mathematics, Santa Clara University, Santa Clara, CA 95053-0290 ct We give a complete characterization for the limit probabilities of first order sentences over sparse random bit strings at the threshold of adjacency. For strings of length n, we let the probability that a bit is "on" be c n , for a real positive number c. For every first order sentence , we show that the limit probability function: f(c) = lim n Pr[Un, c n has the property ] (where Un, c n is the random bit string of length n) is infinitely differentiable. Our methodology for showing this is in itself interesting. We begin with finite models, go to the infinite (via the almost sure theories) and then characterize f(c) as an infinite sum of products of polynomials and exponentials. We further show that if a sentence has limiting probability 1 for some c, then ...

• Type: Conference
• Year: 1998
• Where: STACS
• Authors: Joel Spencer, Katherine St. John
• Keywords: Order Sentence, Random Bit String, Santa Clara, STACS 1998, Theoretical Computer Science