Phase Transitions in Knowledge Compilation: an Experimental Study

TL;DR

This paper empirically investigates phase transitions in knowledge compilation, revealing patterns in compilation sizes and their relation to problem parameters, and explaining the underlying microstructural causes.

Contribution

It provides the first detailed empirical analysis of phase transitions in knowledge compilation, identifying key patterns and microstructural explanations.

Findings

Existence of easy-hard-easy pattern in compilation sizes.

Peak size related to clause-variable ratio, independent of target language.

Transition from polynomial to exponential growth in compilation sizes.

Abstract

Phase transitions in many complex combinational problems have been widely studied in the past decade. In this paper, we investigate phase transitions in the knowledge compilation empirically, where DFA, OBDD and d-DNNF are chosen as the target languages to compile random k-SAT instances. We perform intensive experiments to analyze the sizes of compilation results and draw the following conclusions: there exists an easy-hard-easy pattern in compilations; the peak point of sizes in the pattern is only related to the ratio of the number of clauses to that of variables when k is fixed, regardless of target languages; most sizes of compilation results increase exponentially with the number of variables growing, but there also exists a phase transition that separates a polynomial-increment region from the exponential-increment region; Moreover, we explain why the phase transition in compilations occurs by analyzing microstructures of DFAs, and conclude that a kind of solution interchangeability with more than 2 variables has a sharp transition near the peak point of the easy-hard-easy pattern, and thus it has a great impact on sizes of DFAs.