A Statistical Analysis of PLA Folding

Current understanding of the PLA folding problem is limited to simple empirical evidence from studies of heuristic methods. This paper presents a theoretical approach through an analytical and statistical analysis. The problem is first mapped into a set theoretic model. Using a random selection heuristic as a basis, a probability density function (PDF) is derived for the expected number of folds under a set of simplifying assumptions. This PDF is derived in terms of the three fundamental properties of a PLA, r the number of rows, c the number of columns, and d the density. Empirical results obtained from folding thousands of randomly generated PLA's verify the accuracy of the derived probability density function. A technique is developed whereby the PDF can also be used to predict the size of optimal folding sets. A new folding heuristic is introduced which is shown to perform better than other heuristic algorithms in the literature, when applied to a set of randomly generated PLA's. This is the first folding heuristic to have an analytical basis for its expected results, as derived from the PDF function.