Heuristics for Million-scale Two-level Logic Minimization

Abstract

Existing two-level logic minimization methods suffer from scalability problems, i.e. they cannot handle the optimization of Boolean functions with more than about 50k or so product terms. However, applications have arisen that produce Boolean functions with hundreds of thousands to millions of minterms. To ameliorate the aforesaid scalability problem, this work presents a suite of heuristics that enables exact or approximate two-level logic minimization of such large Boolean functions by employing a divide and conquer technique. All proposed heuristics first deploy a decision tree to iteratively partition the original specification of a given Boolean function. Next, they apply one of different leaf optimization techniques (e.g., those based on support vector machines or error budgets) to each leaf node of the tree, and, finally, they merge the locally optimized leaves at the root of the tree to perform one round of the global optimization. We show that our support vector machine-based heuristic compresses Boolean functions with 300,000 minterms by a factor of about 100 (i.e. 3,000 cubes in the optimized function), and achieves 98% accuracy. Similarly, our error-budget-driven heuristic compresses a Boolean function with about 3,000,000 minterms by a factor of 1,273, and achieves 95 % accuracy while it only takes 67 seconds to complete the whole optimization process. This is a significant improvement compared to well-known two-level logic minimization tools such as ESPRESSO-II and BOOM, which fail to optimize the same Boolean functions even after running for a few days.