A Method of Producing a Boolean Function Having an Arbitrarily Prescribed Prime Implicant Table

Gimpel, James F.

doi:10.1109/pgec.1965.264175

Cited by 34 publications

(22 citation statements)

References 4 publications

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“…Now it might be the case that when we restrict to the subproblem of MINIMUM COVER consisting of only those instances arising from the Quine-McCluskey procedure, the problem becomes tractable. A couple of papers [10], [20] investigated this question and gave transformations that show that even this subproblem remains NP -complete. We summarize here their main results.…”

Section: Minimum Covermentioning

confidence: 99%

“…The same authors [10], [20] describe how to produce a completely specified function that in a certain sense still represents the original covering table.…”

Section: Minimum Covermentioning

confidence: 99%

“…Therefore, an in-depth investigation of two-level logic minimization has a special place in studying the complexity of logic synthesis problems. The history of classifying the complexity of two-level logic minimization accompanies the field of computational complexity from its beginnings in the 1960s [10] to now. This history extends to a result of only a few years ago settling a conjecture open for more than 20 years, namely, that the following problem is Σ p 2 -complete [30], [31]: "given a sum-of-product (SOP) form of a logic function, is there an equivalent SOP form with at most a given number of terms?"…”

Section: Introductionmentioning

confidence: 99%

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Complexity of two-level logic minimization

Umans

Villa

Sangiovanni-Vincentelli

2006

IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.

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Abstract-The complexity of two-level logic minimization is a topic of interest to both computer-aided design (CAD) specialists and computer science theoreticians. In the logic synthesis community, two-level logic minimization forms the foundation for more complex optimization procedures that have significant real-world impact. At the same time, the computational complexity of two-level logic minimization has posed challenges since the beginning of the field in the 1960s; indeed, some central questions have been resolved only within the last few years, and others remain open. This recent activity has classified some logic optimization problems of high practical relevance, such as finding the minimal sum-of-products (SOP) form and maximal term expansion and reduction. This paper surveys progress in the field with self-contained expositions of fundamental early results, an account of the recent advances, and some new classifications. It includes an introduction to the relevant concepts and terminology from computational complexity, as well a discussion of the major remaining open problems in the complexity of logic minimization.

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Section: Minimum Covermentioning

confidence: 99%

“…The same authors [10], [20] describe how to produce a completely specified function that in a certain sense still represents the original covering table.…”

Section: Minimum Covermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Complexity of two-level logic minimization

Umans

Villa

Sangiovanni-Vincentelli

2006

IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.

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“…Because it i pecific case, it is possible that it is not as complex as the general set covering problem. Howp ever, Gimpel [2] showed that this is not true. He showed that any instance of the set covering roblem occurs as an instance of the sum-of-products problem.…”

Section: Introductionmentioning

confidence: 99%

Complexity Analysis of the Cost-Table Approach to the Design of Multiple-Valued Logic Circuits

Schueller

Butler

1995

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“…In 1965, Gimpel [7] showed that any instance of the set covering problem is an instance of sum-of-products extraction.…”

Section: Introductionmentioning

confidence: 99%

Minimization algorithms for multiple-valued programmable logic arrays

Tirumalai

Butler

1991

IEEE Trans. Comput.

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We show an exact minimization method that is a tree search using backtracking. A considerable reduction in the search space is achieved by considering constrained implicunt sets and by eliminating some implicants altogether. Even with this improvement, the time required for exact minimization is extremely high when compared to all three heuristics.We also examine the case where only prime implicunfs are considered and show that such implicants have marginal value compared to constrained implicant sets. Our basis of comparison is the average number of product terms. We show that the heuristic methods are reasonably close to minimal and produce nearly the same average number of product terms. Interestingly though, there is surprisingly little overlap in the set of functions where the best realization is achieved. Thus, there is a benefit to applying all three heuristics to a given function and then choosing the best realization.Index Terms-Absolute minimization, heuristic minimization, implicant, multiple-valued logic, prime implicant, programmable logic arrays, sum-of-products. I . INTRODUCTIONHE minimization of sum-of-products expressions in binary T logic has received considerable attention for over 30 years.The complexity of the problem has been known for almost as long. In 1965, Gimpel [7] showed that any instance of the set covering problem is an instance of sum-of-products extraction.In 1972, Karp [8] showed that the set covering problem is NP complete; thus, so also is sum-of-products extraction. The best known algorithm then requires exponential time. This is a real barrier; it precludes the exact minimization of functions with even a moderately low number of inputs, e.g., 20. As a result, considerable effort has been devoted to heuristic minimization methods. introduced a method which also seeks the most isolated minterm first, but chooses a product term that tends to introduce the fewest discontinuities when subtracted from the function. There has been little study of the relative merits of available heuristic algorithms even in binary logic. To the credit of Brayton, Hachtel, McMullen,, the realizations produced by ESPRESSO-IIC were compared to the realizations of MINI, PRESTO, and POP [4] over a set of 56 specifically chosen binary functions. To our knowledge, there has been no comparative analysis of minimization algorithms for multiple-valued logic. The justification of the newly introduced heuristics examined by us has rested on an intuitive notion, supported by examples. In this paper, we analyze the following synthesis methods over sets of 7000 random and 7000 random symmetric 4-valued 2-variable functions.1) Random-mintermhandom-implicant 2) Pomper and Armstrong [14] 3) Besslich [3] 4) Dueck and Miller [6] 5) Gold 6) Absolute minimization using prime implicants 7) Absolute minimization using all implicants. Our results show the Besslich heuristic is slightly inferior to the Dueck and Miller heuristic while there is superiority of each over the Pomper and Armstrong heuristic. Our analysis is...

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A Method of Producing a Boolean Function Having an Arbitrarily Prescribed Prime Implicant Table

Cited by 34 publications

References 4 publications

Complexity of two-level logic minimization

Complexity of two-level logic minimization

Complexity Analysis of the Cost-Table Approach to the Design of Multiple-Valued Logic Circuits

Minimization algorithms for multiple-valued programmable logic arrays

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