We study the problem of decomposing a Boolean function into a set of functions with fewer arguments. This problem has considerable practical importance in VLSI, for example, for designs using field-programmable gate arrays. The decomposition problem is old, and well understood when the function to be decomposed is specified by a truth table, or has one output only. However, modern design tools handle functions with many outputs and represent them by cubes, for reasons of efficiency. We develop a comprehensive theory of serial decompositions for multiple-output, partially specified, Boolean functions represented by cubes. A function f (x1 , . . . , xn) has a serial decomposition if it can be expressed as h(u1 , . . . , ur, g(v1 , . . . , vs)), where U = {u1 , . . . , ur} and V = {v1 , . . . , vs} are subsets of the set X = {x1 , . . . , xn} of input variables, and g and h have fewer input variables than f. The theory uses generalized set systems (which, in turn, are generalized partitions), which we call blankets.

Decomposition of Boolean Functions Specified by Cubes

The functional decomposition of binary and multi-valued discrete functions and relations has been gaining more and more recognition. It has important applications in many fields of modern digital system engineering, such as combinational and sequential logic synthesis for VLSI systems, pattern analysis, knowledge discovery, machine learning, decision systems, data bases, data mining etc. However, its practical usefulness for very complex systems has been limited by the lack of an effective and efficient method for selecting the appropriate input supports for sub-systems. In this paper, a new effective and efficient functional decomposition method is proposed and discussed. This method is based on applying information relationship measures to input support selection. Using information relationship measures allows us to reduce the search space to a manageable size while retaining high-quality solutions in the reduced space. Experimental results demonstrate that the proposed method is able to construct optimal or near-optimal supports very efficiently, even for large systems. It is many times faster than the systematic support selection method, but delivers results of comparable quality.

Functional decomposition with an efficient input support selection for sub-functions based on information relationship measures

The analysis of information relationships is of primary importance for the analysis and synthesis of digital information systems. The paper aims to introduce and discuss the fundamental apparatus for the analysis and evaluation of information relationships. It defines and explains various relationships between information, measures for the amount and importance of information, and measures for the strength and importance of the information relationships. It demonstrates importance of the introduced relationships and measures for efficient synthesis, and shows how to apply them in the synthesis process. The analysis apparatus makes operational the famous theory of partitions and set systems of Hartmanis (1966). While partitions and set systems enable one to model information, the relationships and measures enable one to analyze and measure information and information relationships. Both together form a complete information modeling and analysis apparatus that can be applied in logic design, decision system design, pattern recognition, knowledge discovery, machine learning and other areas.

Information relationships and measures: an analysis apparatus for efficient information system synthesis

Effective and efficient FPGA synthesis through general functional decomposition

Finding appropriate bound sets of variables is the most important task of functional decomposition. When solving some problems, the bound sets need to be larger, for instance in decomposition to symmetric subfunctions realized in MOPS arrays for submicron technologies, or when no good small bound sets exist. In such cases, the creation of the incompatibility graph, which is necessary to evaluate good variable partitionings, becomes very inefficient. Therefore, an algorithm is proposed that can speed up this process by orders of magnitude without sacrificing the quality of the decomposition, because the same graph coloring algorithms (exact or approximate) are still applied to the created graph.

An efficient approach to decomposition of multi-output Boolean functions with large sets of bound variables

During the last decade, many different approaches have been proposed to solve the multiple-level synthesis problem withdifferent minimum functionally complete systems of primitive logic blocks. The most popular of them is the division-basedapproach. However, modem microelectronic technology provides a large variety of building blocks which considerablydiffer from those typically considered. The traditional methods are therefore not suitable for synthesis with many modembuilding blocks. Furthermore, they often fail to find global optima for complex designs and leave unconsidered someimportant design aspects. Some of their weaknesses can be eliminated without leaving the paradigm they are based on, otherones are more fundamental. A paradigm which enables efficient exploitation of the opportunities created by themicroelectronic technology is the general decomposition paradigm. The aim of this paper is to analyze and compare thegeneral decomposition approach and the division-based approach. The most important advantages of the generaldecomposition approach are its generality (any network of any building blocks can be considered) and totality (all importantdesign aspects can be considered) as well as handling the incompletely specified functions in a natural way. In many cases,the general decomposition approach gives much better results than the traditional approaches.

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Fam Frank Volf

Papers

Division-Based Versus General Decomposition-Based Multiple-Level Logic Synthesis