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On the phase transitions of (k, q)-SAT

On the phase transitions of (k, q)-SAT Call a sequence of k Boolean variables or their negations a k-tuple. For a set V of n Boolean variables, let T k (V) denote the set of all 2 k n k possible k-tuples on V. Randomly generate a set C of k-tuples by including every k-tuple in T k (V) independently with probability p, and let Q be a given set of q “bad” tuple assignments. An instance I = (C,Q) is called satisfiable if there exists an assignment that does not set any of the k-tuples in C to a bad tuple assignment in Q. Suppose that θ, q > 0 are fixed and ε = ε(n) > 0 be such that εlnn/lnlnn→∞. Let k ≥ (1 + θ) log2 n and let $${p_0} = \frac{{\ln 2}}{{q{n^{k - 1}}}}$$ . We prove that $$\mathop {\lim }\limits_{n \to \infty } P\left[ {I is satisfiable} \right] = \left\{ {\begin{array}{*{20}c} {1,} & {p \leqslant (1 - \varepsilon )p_0 ,} \\ {0,} & {p \geqslant (1 + \varepsilon )p_0 .} \\ \end{array} } \right.$$ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

On the phase transitions of (k, q)-SAT

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References (6)

Publisher
Springer Journals
Copyright
Copyright © 2016 by Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-016-0591-8
Publisher site
See Article on Publisher Site

Abstract

Call a sequence of k Boolean variables or their negations a k-tuple. For a set V of n Boolean variables, let T k (V) denote the set of all 2 k n k possible k-tuples on V. Randomly generate a set C of k-tuples by including every k-tuple in T k (V) independently with probability p, and let Q be a given set of q “bad” tuple assignments. An instance I = (C,Q) is called satisfiable if there exists an assignment that does not set any of the k-tuples in C to a bad tuple assignment in Q. Suppose that θ, q > 0 are fixed and ε = ε(n) > 0 be such that εlnn/lnlnn→∞. Let k ≥ (1 + θ) log2 n and let $${p_0} = \frac{{\ln 2}}{{q{n^{k - 1}}}}$$ . We prove that $$\mathop {\lim }\limits_{n \to \infty } P\left[ {I is satisfiable} \right] = \left\{ {\begin{array}{*{20}c} {1,} & {p \leqslant (1 - \varepsilon )p_0 ,} \\ {0,} & {p \geqslant (1 + \varepsilon )p_0 .} \\ \end{array} } \right.$$

Journal

Acta Mathematicae Applicatae Sinica – Springer Journals

Published: Aug 21, 2016

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