A Structural Entropy Measurement Principle of Propositional Formulas in Conjunctive Normal Form

Abstract

The satisfiability (SAT) problem is a core problem in computer science. Existing studies have shown that most industrial SAT instances can be effectively solved by modern SAT solvers while random SAT instances cannot. It is believed that the structural characteristics of different SAT formula classes are the reasons behind this difference. In this paper, we study the structural properties of propositional formulas in conjunctive normal form (CNF) by the principle of structural entropy of formulas. First, we used structural entropy to measure the complex structure of a formula and found that the difficulty solving the formula is related to the structural entropy of the formula. The smaller the compressing information of a formula, the more difficult it is to solve the formula. Secondly, we proposed a λ-approximation strategy to approximate the structural entropy of large formulas. The experimental results showed that the proposed strategy can effectively approximate the structural entropy of the original formula and that the approximation ratio is more than 92%. Finally, we analyzed the structural properties of a formula in the solution process and found that a local search solver tends to select variables in different communities to perform the next round of searches during a search and that the structural entropy of a variable affects the probability of the variable being flipped. By using these conclusions, we also proposed an initial candidate solution generation strategy for a local search for SAT, and the experimental results showed that this strategy effectively improves the performance of the solvers CCAsat and Sparrow2011 when incorporated into these two solvers.

Initial Solution Generation and Diversified Variable Picking in Local Search for (Weighted) Partial MaxSAT

“…In local search, an initial solution (starting point), which lies near a high-quality one, may cost significantly less steps (flips) to achieve that satisfactory solution. In our previous works [ 39 , 40 ], we confirmed that a variable’s structural entropy significantly influences the probability that it will be flipped later in local search. Based on this result, we developed strategies for initiating solutions and such strategies greatly improve two state-of-the-art solvers, and .…”

“…The motivation of VSEI is as follows. The smaller a variable’s structural entropy is, the more stable its truth value is, hence the smaller the probability that it will be flipped later [ 40 ]. That is, a variable with smaller structural entropy should probably be assigned earlier, compared to those variables with greater ones.…”

Complexity Evaluation of an Environmental Control and Life-Support System Based on Directed and Undirected Structural Entropy Methods

“…Since Shannon introduced the theory of information entropy in 1948 [ 36 ], it has been widely used in complexity measurement [ 37 , 38 , 39 , 40 , 41 ]. Entropy is also used to quantitatively describe the uncertainty and order degree of system structure, which can facilitate complexity evaluation [ 42 , 43 , 44 , 45 ].…”