Polynomial-time algorithms for generation of prime implicants

AbstractA notion of a neighborhood cube of a term of a Boolean function represented in the canonical disjunctive normal form is introduced. A relation between neighborhood cubes and prime implicants of a Boolean function is established. Various aspects of the problem of prime implicants generation are identified and neighborhood cube-based algorithms for their solution are developed. The correctness of algorithms is proven and their time complexity is analyzed. It is shown that all presented algorithms are polynomial in the number of minterms occurring in the canonical disjunctive normal form representation of a Boolean function. A summary of the known approaches to the solution of the problem of the generation of prime implicants is also included

Logic Synthesis as an Efficient Means of Minimal Model Discovery from Multivariable Medical Datasets

In this paper we review the application of logic synthesis methods for uncovering minimal structures in observational/medical datasets. Traditionally used in digital circuit design, logic synthesis has taken major strides in the past few decades and forms the foundation of some of the most powerful concepts in computer science and data mining. Here we provide a review of current state of research in application of logic synthesis methods for data analysis and provide a demonstrative example for systematic application and reasoning based on these methods

SOME REMARKS ON APPLICATION OF HIGHER ORDER KARNAUGH MAP TO RELIABILITY ORIENTATION

In this paper the Karnaugh map is performed with some practical contributions from the point of engineering applications. The properties of adjacency of Karnaugh maps are more clearly explained by the equivalence existing between its planar representation and its axonometrical one. Some algorithms and examples are presented for a more easy understanding of the method of this map. A few reversion problems are presented and solved which seem important for system reliability evaluations and circuit designs. To avoid misunderstanding and save time to the reader a short appendix is given

Boolean Functions: Theory, Algorithms, and Applications

This monograph provides the first comprehensive presentation of the theoretical, algorithmic and applied aspects of Boolean functions, i.e., {0,1}-valued functions of a finite number of {0,1}-valued variables. The book focuses on algebraic representations of Boolean functions, especially normal form representations. It presents the fundamental elements of the theory (Boolean equations and satisfiability problems, prime implicants and associated representations, dualization, etc.), an in-depth study of special classes of Boolean functions (quadratic, Horn, shellable, regular, threshold, read-once, etc.), and two fruitful generalizations of the concept of Boolean functions (partially defined and pseudo-Boolean functions). It features a rich bibliography of about one thousand items. Prominent among the disciplines in which Boolean methods play a significant role are propositional logic, combinatorics, graph and hypergraph theory, complexity theory, integer programming, combinatorial optimization, game theory, reliability theory, electrical and computer engineering, artificial intelligence, etc. The book contains applications of Boolean functions in all these areas

BOOM - A Heuristic Boolean Minimizer

This paper presents an algorithm for two-level Boolean minimization (BOOM) based on a new implicant generation paradigm. In contrast to all previous minimization methods, where the implicants are generated bottom-up, the proposed method uses a top-down approach. Thus, instead of increasing the dimensionality of implicants by omitting literals from their terms, the dimension of a term is gradually decreased by adding new literals. The method is advantageous especially for functions with many input variables (up to thousands) and with only few care terms defined, where other minimization tools are not applicable because of the long runtime. The method has been tested on several different kinds of problems and the results were compared with ESPRESSO

Fast Heuristic and Exact Algorithms for Two-Level Hazard-Free Logic Minimization

None of the available minimizers for 2-level hazard-free logic minimization can synthesize very large circuits. This limitation has forced researchers to resort to manual and automated circuit partitioning techniques. This paper introduces two new 2-level logic minimizers:ESPRESSO-HF, a heuristic method which is loosely based on ESPRESSO-II, and IMPYMIN, an exact method based on implicit data structures. Both minimizers can solve all currently available examples, which range up to 32 inputs and 33 outputs.These include examples that have never been solved before.For examples that can be solved by other minimizers our methods are several orders of magnitude faster. As by-products of these algorithms, we also present two additional results. First, we introduce a fast new algorithm to check if a hazard-free covering problem can feasibly be solved. Second, we introduce a novel formulation of the 2-level hazard-free logic minimization problem by capturing hazard-freedom constraints within a synchronous function by adding new variables

Fast Heuristic and Exact Algorithms for Two-Level Hazard-Free Logic Minimization

None of the available minimizers for 2-level hazard-free logic minimization can synthesize very large circuits. This limitation has forced researchers to resort to manual and automated circuit partitioning techniques. This paper introduces two new 2-level logic minimizers:ESPRESSO-HF, a heuristic method which is loosely based on ESPRESSO-II, and IMPYMIN, an exact method based on implicit data structures. Both minimizers can solve all currently available examples, which range up to 32 inputs and 33 outputs.These include examples that have never been solved before.For examples that can be solved by other minimizers our methods are several orders of magnitude faster. As by-products of these algorithms, we also present two additional results. First, we introduce a fast new algorithm to check if a hazard-free covering problem can feasibly be solved. Second, we introduce a novel formulation of the 2-level hazard-free logic minimization problem by capturing hazard-freedom constraints within a synchronous function by adding new variables

Identification of SNP interactions using logic regression

Interactions of single nucleotide polymorphisms (SNPs) are assumed to be responsible for complex diseases such as sporadic breast cancer. Important goals of studies concerned with such genetic data are thus to identify combinations of SNPs that lead to a higher risk of developing a disease and to measure the importance of these interactions. There are many approaches based on classification methods such as CART and Random Forests that allow measuring the importance of single variables. But with none of these methods the importance of combinations of variables can be quantified directly. In this paper, we show how logic regression can be employed to identify SNP interactions explanatory for the disease status in a case- control study and propose two measures for quantifying the importance of these interactions for classification. These approaches are then applied, on the one hand, to simulated data sets, and on the other hand, to the SNP data of the GENICA study, a study dedicated to the identification of genetic and gene-environment interactions associated with sporadic breast cancer. --Single Nucleotide Polymorphism,Feature Selection,Variable Importance Measure,GENICA