Genetic Algorithms and the Satisfiability of Large-Scale Boolean Expressions.

Abstract

The two new genetic methods overpopulation and bitwise expected value are introduced. In overpopulation a temporary population of size Mn (M $\u3e$ 1) is created using genetic operators and the n children with the highest estimated fitness values are selected as the next generation. The rest are discarded. Bitwise expected value (bev) is the fitness estimation function used. Overpopulation and bitwise expected value are applied to the NP-complete problem 3SAT (a special form of Satisfiability in which the boolean expression consists of the conjunction of an arbitrary number of clauses where each clause consists of the disjunction of 3 boolean variables) with excellent empirical results when compared to the performance of the standard genetic algorithm. Overpopulation increases the cost of producing each generation due to the overhead required to maintain the larger temporary population but results in many fewer generations to solution. Using bitwise expected value as a fitness estimator causes the algorithm to take slightly more generations to solution but is much faster to calculate than the fitness function, leading to a decrease in wall-clock time to solution. Theoretical justification for the success of overpopulation is seen as a result of the generalization of the schema growth equation. Bitwise expected value is viewed as an analogy to the Building Block Hypothesis. Empirical evidence of high correlation between bev and the fitness function is presented. We also introduce the target problem concept, in which a difficult problem is transformed into a well-known problem for which a good genetic method of solution is known. As an example of the target problem concept a transformation from the Traveling Salesman Problem to Satisfiability is demonstrated. Overpopulation and bitwise expected value are applied to the resulting boolean expression, with good results. An interesting convergence property is observed