This chapter presents both exact and heuristic two-level logic minimization algorithms. For exact logic minimization, it shows various techniques to reduce the complexity of covering problems, discusses branching heuristics, and presents several methods to prune the recursions. For heuristic minimization, it presents the core procedures of the ESPRESSO minimizer. Finally, the chapter surveys various works related to two-level logic minimization.

Two-level logic minimization

Covering problems are widely used as a modeling tool in electronic design automation. Recent years have seen dramatic improvements in algorithms for the unate/binate covering problem (UCP/BCP). Despite these improvements, BCP is a well-known computationally hard problem with many existing real-world instances that currently are hard or even impossible to solve. In this paper we apply search pruning techniques from the Boolean satisfiability domain to branch-and-bound algorithms for BCP. Furthermore, we generalize these techniques, in particular the ability to infer and record new constraints from conflicts and the ability to backtrack nonchronologically, to situations where the branch-and-bound BCP algorithm backtracks due to bounding conditions. Experimental results, obtained on representative real-world instances of the UCP/BCP, indicate that the proposed techniques are effective and can provide significant performance gains for specific classes of instances.

/pdf/search-pruning-techniques-in-sat-based-branch-and-bound-2ccymqs4is.pdf

Search pruning techniques in SAT-based branch-and-bound algorithms for the binate covering problem

Logic synthesis is a fundamental step in hardware design whose goal is to find structural representations of Boolean functions while minimizing delay and area. If the function is completely-specified, the implementation accurately represents the function. If the function is incompletely-specified, the implementation has to be true only on the care set. While most of the algorithms in logic synthesis rely on SAT and Boolean methods to exactly implement the care set, we investigate learning in logic synthesis, attempting to trade exactness for generalization. This work is directly related to machine learning where the care set is the training set and the implementation is expected to generalize on a validation set. We present learning incompletely-specified functions based on the results of a competition conducted at IWLS 2020. The goal of the competition was to implement 100 functions given by a set of care minterms for training, while testing the implementation using a set of validation minterms sampled from the same function. We make this benchmark suite available and offer a detailed comparative analysis of the different approaches to learning.

Logic Synthesis Meets Machine Learning: Trading Exactness for Generalization

We introduce a new technique for solving some discrete optimization problems exactly. The motivation is that when searching the space of solutions by a standard branch-and-bound (B&B) technique, often a good solution is reached quickly and then improved only a few times before the optimum is found: hence, most of the solution space is explored to certify optimality, with no improvement in the cost function. This suggests that more powerful lower bounding would speed up the search dramatically. More radically, it would be desirable to modify the search strategy with the goal of proving that the given subproblem cannot yield a solution better than the current best one (negative thinking), instead of branching further in search for a better solution (positive thinking). For illustration we applied our approach to the unate covering problem. The algorithm starts in the positive-thinking mode by a standard B&B procedure that generates recursively smaller subproblems. If the current subproblem is "deep" enough, the algorithm switches to the negative thinking mode where it tries to prove that solving the subproblem does not improve the solution. The latter is achieved by a new search procedure invoked when the difference between the upper and lower bound is "small". Such a procedure is complete: either it yields a lower bound that matches the current upper bound, or it yields a new solution better than the current one. We implemented our new search procedure on top of ESPRESSO and SCHERZO, two state-of-art covering solvers used for computer-aided design applications, showing that in both cases we obtain new search engines (respectively, AURA and AURA II) much more efficient than the original ones.

/pdf/negative-thinking-in-branch-and-bound-the-case-of-unate-43krz4gl75.pdf

Negative thinking in branch-and-bound: the case of unate covering

Covering problems are widely used as a modeling tool in Electronic Design Automation (EDA). Recent years have seen dramatic improvements in algorithms for the Unate/Binate Covering Problem (UCP/BCP). Despite these improvements, BCP is a well-known computationally hard problem, with many existing real-world instances that currently are hard or even impossible to solve. In this paper we apply search pruning techniques from the Boolean Satisfiability (SAT) domain to BCP. Furthermore, we generalize these techniques, in particular the ability to backtrack non-chronologically to exploit the actual formulation of covering problems. Experimental results, obtained on representative instances of the unate and binate covering problems, indicate that the proposed techniques provide significant performance gains for different classes of instances.

/pdf/on-using-satisfiability-based-pruning-techniques-in-covering-3cziili733.pdf

On using satisfiability-based pruning techniques in covering algorithms

We introduce a new technique to solve exactly a discrete optimization problem, based on the paradigm of "negative" thinking. The motivation is that when searching the space of solutions, often a good solution is reached quickly and then improved only a few times before the optimum is found: hence most of the solution space is explored to certify optimality, but it does not yield any improvement of the cost function. So it is quite natural for an algorithm to be "skeptical" about the chance to improve the current best solution. For illustration we have applied our approach to the unate covering problem. We designed a procedure, raiser, implementing a negative thinking search, which is incorporated into a common branch-and-bound procedure. Experiments show that our program, AURA, outperforms both ESPRESSO and our enhancement of ESPRESSO using Coudert's limit lower bound. It is always faster and in the most difficult examples either has a running time better by up to two orders of magnitude, or the other programs fail to finish due to timeout or spaceout. The package SCHERZO is faster on some examples and loses on others, due to a less powerful pruning strategy of the search space, partially mitigated by a more effective computation of the maximal independent set.

/pdf/negative-thinking-by-incremental-problem-solving-application-22676dxb2t.pdf

Negative thinking by incremental problem solving: application to unate covering

We present a new matrix formulation of the face hypercube embedding problem that motivates the design of an efficient search strategy to find an encoding that satisfies all faces of minimum length. Increasing dimensions of the Boolean space are explored; for a given dimension constraints are satisfied one at a time. The following features help to reduce the nodes of the solution space that must be explored: candidate cubes instead of candidate codes are generated, cubes yielding symmetric solutions are not generated, a smaller sufficient set of solutions (producing basic sections) is explored, necessary conditions help discard unsuitable candidate cubes, early detection that a partial solution cannot be extended to be a global solution prunes infeasible portions of the search tree. We have implemented a prototype package MINSK based on the previous ideas and run experiments to evaluate it. The experiments show that MINSK is faster and solves more problems than any available algorithm. Moreover, MINSK is a robust algorithm, while most of the proposed alternatives are not. Besides most problems of the complete MCNC benchmark suite, other solved examples include an important set of decoder PLAs coming from the design of microprocessor instruction sets.

/pdf/a-fast-and-robust-exact-algorithm-for-face-embedding-3wvzb1iv72.pdf

A fast and robust exact algorithm for face embedding

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Negative thinking by incremental problem solving: application to unate covering

A fast and robust exact algorithm for face embedding