4,547 research outputs found

    Discrete Function Representations Utilizing Decision Diagrams and Spectral Techniques

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    All discrete function representations become exponential in size in the worst case. Binary decision diagrams have become a common method of representing discrete functions in computer-aided design applications. For many functions, binary decision diagrams do provide compact representations. This work presents a way to represent large decision diagrams as multiple smaller partial binary decision diagrams. In the Boolean domain, each truth table entry consisting of a Boolean value only provides local information about a function at that point in the Boolean space. Partial binary decision diagrams thus result in the loss of information for a portion of the Boolean space. If the function were represented in the spectral domain however, each integer-valued coefficient would contain some global information about the function. This work also explores spectral representations of discrete functions, including the implementation of a method for transforming circuits from netlist representations directly into spectral decision diagrams

    Minimize Logic Synthesis FPGA – Extraction And Substitution Problems

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    The objective of multi-level logic synthesis of FPGA is to find the “best” multi-level structure, where “best” in this case means an equivalent presentation that is optimal with respect to various parameters such as size, speed or power consumption... Five basic operations are used in order to reach this goal: decomposition, extraction, factoring, substitution and collapsing. In this paper we propose a novel application of Walsh spectral transformation to the evaluation of Boolean function correlation. In particular, we present an algorithm with approach to solve the problems of extraction and substitution based on the use of Walsh spectral presentation. The method, operating in the transform domain, has appeared to be more advantageous than traditional approaches, using operations in the Boolean domain, concerning both memory occupation and execution time on some classes of functions

    Conditions of the Affine Extension of an Incompletely Defined Boolean Function

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    The paper presents conditions of extension of the weakly defined Boolean functions to their full affine form. The main goal of the analysis is a fast estimation whether a given incompletely defined function can be extended to a full affine form. If it is possible a simple algorithm of the states completion has been proposed. In such a case undefined points are clearly replaced by 0, 1 values. Spectral coefficients of a Boolean function allow to determine whether a partially defined function can be realised as affine

    R-Functions and WA-Systems of Functions in Modern Information Technologies

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    The review report consists of five parts. It describes the main physical applications of atomic, WA-systems and R-functions

    Categorical invariance and structural complexity in human concept learning

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    An alternative account of human concept learning based on an invariance measure of the categorical\ud stimulus is proposed. The categorical invariance model (CIM) characterizes the degree of structural\ud complexity of a Boolean category as a function of its inherent degree of invariance and its cardinality or\ud size. To do this we introduce a mathematical framework based on the notion of a Boolean differential\ud operator on Boolean categories that generates the degrees of invariance (i.e., logical manifold) of the\ud category in respect to its dimensions. Using this framework, we propose that the structural complexity\ud of a Boolean category is indirectly proportional to its degree of categorical invariance and directly\ud proportional to its cardinality or size. Consequently, complexity and invariance notions are formally\ud unified to account for concept learning difficulty. Beyond developing the above unifying mathematical\ud framework, the CIM is significant in that: (1) it precisely predicts the key learning difficulty ordering of\ud the SHJ [Shepard, R. N., Hovland, C. L.,&Jenkins, H. M. (1961). Learning and memorization of classifications.\ud Psychological Monographs: General and Applied, 75(13), 1-42] Boolean category types consisting of three\ud binary dimensions and four positive examples; (2) it is, in general, a good quantitative predictor of the\ud degree of learning difficulty of a large class of categories (in particular, the 41 category types studied\ud by Feldman [Feldman, J. (2000). Minimization of Boolean complexity in human concept learning. Nature,\ud 407, 630-633]); (3) it is, in general, a good quantitative predictor of parity effects for this large class of\ud categories; (4) it does all of the above without free parameters; and (5) it is cognitively plausible (e.g.,\ud cognitively tractable)
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