SELF-SIMILARITY OF SATISFIABLE BOOLEAN EXPRESSIONS DECIPHERED IN TERMS OF GRAPH DIRECTED ITERATED FUNCTION SYSTEMS

Abstract

The decision problem of satisfiability of Boolean expression in k-conjunctive normal form (kSAT) is a typical NP-complete problem. In this paper, by mapping the whole Boolean expressions in k-conjunctive normal form onto a unit hypercube, we visualize the problem space of kSAT. The pattern of kSAT is shown to have self-similarity which can be deciphered in terms of graph directed iterated function system. We provide that the Hausdorff dimension of the pattern of kSAT is equal to the box-counting dimension and increases with k. This suggests that the time complexity of kSAT increases with k.