Boole's expansion theorem

About: Boole's expansion theorem is a research topic. Over the lifetime, 23 publications have been published within this topic receiving 1470 citations. The topic is also known as: Boole's expansion theorem & decomposition.

Logic Minimization Algorithms for VLSI Synthesis

Abstract: 1. Introduction.- 1.1 Design Styles for VLSI Systems.- 1.2 Automatic Logic Synthesis.- 1.3 PLA Implementation.- 1.4 History of Logic Minimization.- 1.5 ESPRESSO-II.- 1.6 Organization of the Book.- 2. Basic Definitions.- 2.1 Operations on Logic Functions.- 2.2 Algebraic Representation of a Logic Function.- 2.3 Cubes and Covers.- 3. Decomposition and Unate Functions.- 3.1 Cofactors and the Shannon Expansion.- 3.2 Merging.- 3.3 Unate Functions.- 3.4 The Choice of the Splitting Variable.- 3.5 Unate Complementation.- 3.6 SIMPLIFY.- 4. The ESPRESSO Minimization Loop and Algorithms.- 4.0 Introduction.- 4.1 Complementation.- 4.2 Tautology.- 4.2.1 Vanilla Recursive Tautology.- 4.2.2 Efficiency Results for Tautology.- 4.2.3 Improving the Efficiency of Tautology.- 4.2.4 Tautology for Multiple-Output Functions.- 4.3 Expand.- 4.3.1 The Blocking Matrix.- 4.3.2 The Covering Matrix.- 4.3.3 Multiple-Output Functions.- 4.3.4 Reduction of the Blocking and Covering Matrices.- 4.3.5 The Raising Set and Maximal Feasible Covering Set.- 4.3.6 The Endgame.- 4.3.7 The Primality of c+.- 4.4 Essential Primes.- 4.5 Irredundant Cover.- 4.6 Reduction.- 4.6.1 The Unate Recursive Paradigm for Reduction.- 4.6.2 Establishing the Recursive Paradigm.- 4.6.3 The Unate Case.- 4.7 Lastgasp.- 4.8 Makesparse.- 4.9 Output Splitting.- 5. Multiple-Valued Minimization.- 6. Experimental Results.- 6.1 Analysis of Raw Data for ESPRESSO-IIAPL.- 6.2 Analysis of Algorithms.- 6.3 Optimality of ESPRESSO-II Results.- 7. Comparisons and Conclusions.- 7.1 Qualitative Evaluation of Algorithms of ESPRESSO-II.- 7.2 Comparison with ESPRESSO-IIC.- 7.3 Comparison of ESPRESSO-II with Other Programs.- 7.4 Other Applications of Logic Minimization.- 7.5 Directions for Future Research.- References.

Three generators for minimal writing-space computations

Abstract: We construct, for each integer n, three functions from {0,1} n to {0,1} such that any boolean mapping from {0,1} n to {0,1} n can be computed with a finite sequence of assignations only using the n input variables and those three functions.

Novel representations for a coherent threshold reliability system: a tale of eight signal ow graphs

Abstract: A threshold system is a reliability system whose success/failure is a threshold switching function in the successes/failures of its components. A coherent system (CS) is one that is causal, monotone, and with relevant components. The coherent threshold system (CTS), typically called the weighted k-out-of-n system, is consequently described by strictly positive weights and threshold. This paper presents recursive relations as well as boundary conditions for eight entities pertaining to a CTS. These are (a) expressions of monoform literals as well as disjoint or probability- ready expressions for either system success or failure, and (b) all-additive formulas as well as inclusion-exclusion ones for either system reliability or unreliability. These entities are obtained according to the best policy of implementing the Boole{Shannon expansion with respect to a higher-weight component before it is made with respect to a lower-weight one. With this best policy, the success and failure expressions with monoform literals are both minimal and shellable. Each of the eight entities considered is represented by an acyclic (loopless) signal ow graph (SFG). The SFG for system success or failure is isomorphic to a reduced ordered binary decision diagram, which is the optimal data structure for a Boolean function. The interrelations between the SFGs demonstrate optimal procedures for implementing (a) the probability (real) transform of a Boolean function, (b) inversion or complementation of a Boolean function, and (c) disjointing or orthogonalization of a sum-of-products expression of a Boolean function. The SFGs discussed herein for a CT can be extended to a general coherent system. They reduce to elegant symmetric regular graphs for the special case of a partially redundant system (k-out-of-n system).

Projection, consistency, and George Boole

Abstract: Although best known for his work in symbolic logic, George Boole made seminal contributions in the logic of probabilities. He solved the probabilistic inference problem with a projection method, leading to the insight that inference (as well as optimization) is essentially a projection problem. This unifying perspective has applications in constraint programming, because consistency maintenance is likewise a form of inference that can be conceived as projection. Viewing consistency in this light suggests a concept of J-consistency, which is achieved by projection onto a subset J of variables. We show how this projection problem can be solved for the satisfiability problem by logic-based Benders decomposition. We also solve it for among, sequence, regular, and all-different constraints. Maintaining J-consistency for global constraints can be more effective than maintaining traditional domain and bounds consistency when propagating through a richer structure than a domain store, such as a relaxed decision diagram. This paper is written in recognition of Boole's 200th birthday.