scholarly journals A Generalized If-Then-Else Operator for the Representation of Multi-Output Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
Ilya Levin ◽  
Osnat Keren
Keyword(s):  
Boolean Functions ◽  
Compact Representation ◽  
Decomposition Problem ◽  
New Approach

The paper deals with fundamentals of systems of Boolean functions called multi-output functions (MOFs). A new approach to representing MOFs is introduced based on a Generalized If-Then-Else (GITE) function. It is shown that known operations on MOFs may be expressed by a GITE function. The GITE forms the algebra of MOFs. We use the properties of this algebra to solve an MOF-decomposition problem. The solution provides a compact representation of MOFs.

scholarly journals IDENTIFICATION OF BINARY CELLULAR AUTOMATA FROM SPATIO-TEMPORAL BINARY PATTERNS USING A FOURIER REPRESENTATION

2007 ◽  
Vol 17 (06) ◽  
pp. 1985-1996 ◽  
Author(s):  
L. Z. GUO ◽  
S. A. BILLINGS
Keyword(s):  
Fourier Transform ◽  
Cellular Automata ◽  
Least Squares ◽  
Boolean Functions ◽  
Multilinear Polynomial ◽  
Polynomial Representation ◽  
New Approach ◽  
Spatio Temporal ◽  
The Fourier Transform ◽  
Robust To Noise

The identification of binary cellular automata from spatio-temporal binary patterns is investigated in this paper. Instead of using the usual Boolean or multilinear polynomial representation, the Fourier transform representation of Boolean functions is employed in terms of a Fourier basis. In this way, the orthogonal forward regression least-squares algorithm can be applied directly to detect the significant terms and to estimate the associated parameters. Compared with conventional methods, the new approach is much more robust to noise. Examples are provided to illustrate the effectiveness of the proposed approach.

scholarly journals A New Approach to Simplifying Boolean Functions

2004 ◽  
Vol 1 (1) ◽  
pp. 39
Author(s):  
A. H. M. Ashfak Habib ◽  
Md. Abdus Salam ◽  
Zia Nadir ◽  
Hemen Goswami
Keyword(s):  
Boolean Functions ◽  
New Approach ◽  
Manual Method ◽  
The Cost

There are many benefits to simplifying Boolean functions before they are implemented in hardware. A reduced number of gates decreases considerably the cost of the hardware, reduces the heat generated by the chip and, most importantly, increases the speed. But no method is effective for the simplification of Boolean functions, if it involves more than six variables. This paper presents a new manual method of simplification that can be effectively applied to problems with a large number of variables. 

UNIVERSAL CNN CELLS

1999 ◽  
Vol 09 (01) ◽  
pp. 1-48 ◽  
Author(s):  
RADU DOGARU ◽  
LEON O. CHUA
Keyword(s):  
Linear Function ◽  
Boolean Functions ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Linear Functions ◽  
Compact Representation ◽  
Value Functions ◽  
Linear Discriminant ◽  
Absolute Value ◽  
Parity Function

A cellular neural/nonlinear network (CNN) [Chua, 1998] is a biologically inspired system where computation emerges from a collection of simple nonlinear locally coupled cells. This paper reviews our recent research results beginning from the standard uncoupled CNN cell which can realize only linearly separable local Boolean functions, to a generalized universal CNN cell capable of realizing arbitrary Boolean functions. The key element in this evolutionary process is the replacement of the linear discriminant (offset) function w(σ)=σ in the "standard" CNN cell in [Chua, 1998] by a piecewise-linear function defined in terms of only absolute value functions. As in the case of the standard CNN cells, the excitation σ evaluates the correlation between a given input vector u formed by the outputs of the neighboring cells, and a template vector b, which is interpreted in this paper as an orientation vector. Using the theory of canonical piecewise-linear functions [Chua & Kang, 1977], the discriminant function [Formula: see text] is found to guarantee universality and its parameters can be easily determined. In this case, the number of additional parameters and absolute value functions m is bounded by m<2n-1, where n is the number of all inputs (n=9 for a 3×3 template). An even more compact representation where m<n-1 is also presented which is based on a special form of a piecewise-linear function; namely, a multi-nested discriminant: w (σ) =s (zm +| zm -1 +⋯ | z1 +| z0 +σ |||). Using this formula, the "benchmark" Parity function with an arbitrary number of inputs n is found to have an analytical solution with a complexity of only m =O ( log 2 (n)).

scholarly journals Multi-Level Logic Synthesis Based on Kronecker Decision Diagrams and Boolean Ternary Decision Diagrams for Incompletely Specified Functions

VLSI Design ◽  
1995 ◽  
Vol 3 (3-4) ◽  
pp. 301-313 ◽  
Author(s):  
Marek A. Perkowski ◽  
Malgorzata Chrzanowska-Jeske ◽  
Andisheh Sarabi ◽  
Ingo Schäfer
Keyword(s):  
Boolean Functions ◽  
Logic Synthesis ◽  
General Purpose ◽  
Compact Representation ◽  
Decision Diagrams ◽  
Circuit Realization ◽  
Orthogonal Expansions ◽  
Fine Grain ◽  
Multi Level ◽  
Don't Cares

This paper introduces several new families of decision diagrams for multi-output Boolean functions. The introduced families include several diagrams known from literature (BDDs, FDDs) as subsets. Due to this property, these diagrams can provide a more compact representation of functions than either of the two decision diagrams. Kronecker Decision Diagrams (KDDs) with negated edges are based on three orthogonal expansions (Shannon, Positive Davio, Negative Davio) and are created here for incompletely specified Boolean functions as well. An improved efficient algorithm for the construction of KDD is presented and applied in a mapping program to ATMEL 6000 fine-grain FPGAs. Four other new families of functional decision diagrams are also presented: Pseudo KDDs, Free KDDs, Boolean Ternary DDs, and Boolean Kronecker Ternary DDs. The last two families introduce nodes with three edges and require AND, OR and EXOR gates for circuit realization. There are two variants of each of the last two families: canonical and non-canonical. While the canonical diagrams can be used as efficient general-purpose Boolean function representations, the non-canonical variants are also applicable to incompletely specified functions and create don't cares in the process of the creation of the diagram.. They lead to even more compact circuits in logic synthesis and technology mapping.

scholarly journals Bi-Kronecker Functional Decision Diagrams: A Novel Canonical Representation of Boolean Functions

2019 ◽  
Vol 33 ◽  
pp. 2867-2875 ◽  
Author(s):  
Xuanxiang Huang ◽  
Kehang Fang ◽  
Liangda Fang ◽  
Qingliang Chen ◽  
Zhao-Rong Lai ◽  
...  
Keyword(s):  
Data Structure ◽  
Boolean Functions ◽  
Canonical Representation ◽  
Experimental Results ◽  
Compact Representation ◽  
Decision Diagrams ◽  
Canonical Forms

In this paper, we present a novel data structure for compact representation and effective manipulations of Boolean functions, called Bi-Kronecker Functional Decision Diagrams (BKFDDs). BKFDDs integrate the classical expansions (the Shannon and Davio expansions) and their bi-versions. Thus, BKFDDs are the generalizations of existing decision diagrams: BDDs, FDDs, KFDDs and BBDDs. Interestingly, under certain conditions, it is sufficient to consider the above expansions (the classical expansions and their bi-versions). By imposing reduction and ordering rules, BKFDDs are compact and canonical forms of Boolean functions. The experimental results demonstrate that BKFDDs outperform other existing decision diagrams in terms of sizes.

Boolean functions as points on the hypersphere in the Euclidean space

2019 ◽  
Vol 29 (2) ◽  
pp. 89-101 ◽  
Author(s):  
Oleg A. Logachev ◽  
Sergey N. Fedorov ◽  
Valerii V. Yashchenko
Keyword(s):  
Euclidean Space ◽  
Boolean Function ◽  
Boolean Functions ◽  
Nonlinear Functions ◽  
New Approach ◽  
Injective Mapping

Abstract A new approach to the study of algebraic, combinatorial, and cryptographic properties of Boolean functions is proposed. New relations between functions have been revealed by consideration of an injective mapping of the set of Boolean functions onto the sphere in a Euclidean space. Moreover, under this mapping some classes of functions have extremely regular localizations on the sphere. We introduce the concept of curvature of a Boolean function, which characterizes its proximity (in some sense) to maximally nonlinear functions.

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