Novel method to simplify Boolean functions

Arabian Journal for …, 2007

In this study an Off-set based direct-cover minimization method for single-output logic functions is proposed represented in a sum-of-products form. To find the sufficient set of prime implicants including the given On-cube with the existing direct-cover minimization methods, this cube is expanded for one coordinate at a time. The correctness of each expansion is controlled by the way in which the cube being expanded intersects with all of K<2 n Off-cubes. If we take into consideration that the expanding of one cube has a polynomial complexity, then the total complexity of this approach can be expressed as O(n p )O(2 n ), that is, the product of polynomial and exponential complexities. To obtain the complete set of prime implicants including the given On-cube, the proposed method uses Off-cubes expanded by this On-cube. The complexity of this operation is approximately equivalent to the complexity of an intersection of one On-cube expanded by existing methods for one coordinate. Therefore, the complexity of the process of calculating of the complete set of prime implicants including given On-cube is reduced approximately to O(n p ) times. The method is tested on several different kinds of problems and on standard MCNC benchmarks, results of which are compared with ESPRESSO.

In previous work ), a fast systematic method for minimization of the Boolean functions was presented. Such method is a simple because there is no need for any visual representation such as Karnough map or arrangement technique such as Tabulation method. Furthermore, it is suitable for boolean function with large number of variables (more than 4 variable). Moreover, it is very simple to understand and use. In this paper, the simplified functions are implemented with minimum amount of components. A powerful solution for realization of more complex functions is given. This is done by using modular neural nets (MNNs) that divide the input space into several homogenous regions. Such approach is applied to implement XOR functions, 16 logic function on one bit level, and 2-bit digital multiplier. Compared to previous non-modular designs, a clear reduction in the order of computations and hardware requirements is achieved.

Proceedings Ninth IEEE International Conference on Tools with Artificial Intelligence, 1997

The computation of prime implicants has several and significant applications in different areas, including Automated Reasoning, Non-Monotonic Reasoning, Electronic Design Automation, among others. In this paper we describe a new model and algorithm for computing minimum-size prime implicants of propositional formulas. The proposed approach is based on creating an integer linear program (ILP) formulation for computing the minimumsize prime implicant, which simplifies existing formulations. In addition, we introduce two new algorithms for solving ILPs, both of which are built on top of an algorithm for propositional satisfiability (SAT). Given the organization of the proposed SAT algorithm, the resulting ILP procedures implement powerful search pruning techniques, including a non-chronological backtracking search strategy, clause recording procedures and identification of necessary assignments. Experimental results, obtained on several benchmark examples, indicate that the proposed model and algorithms are significantly more efficient than other existing solutions.

Proceedings of the IEEE 1995 National Aerospace and Electronics Conference. NAECON 1995, 1995

A composite mapping technique using a newly proposed lGgic minimization scheme (KH-map) has been investigated here. This paper presents an extended feature of KH-map that can combine multiple maps for better representation of switching functions within limited space, for a relatively large number of variables. The combined KH-rcap can be efficiently used to simplify Boolean expressions to be realized in two-level logic. The technique is simpler, more generalized and more efficient than conventional minimization methods and is easily applicable f o r any number of variables. been proposed [11,12], that presents a very intutive approach to the logic minimizatioin problem. In this paper concatenatior, of multiple KH-map will be introduced with its application in finding minimal s u m of product. For each combination of n variables or bits there exists exactly n different binary codes with Hamming distance one, i.e., each binary combination has n adjacent combinations. Representation of such adjacency relation by Karnaugh map is verv difficult for large values of n. The proposed KH-map method, howeve r , w i l l b e a b l e t o o v e r c o m e s u c h difficulties.

Discrete Applied Mathematics, 2008

Read-once functions have gained recent, renewed interest in the fields of theory and algorithms of Boolean functions, computational learning theory and logic design and verification. In an earlier paper [M.C. Golumbic, A. Mintz, U. Rotics, Factoring and recognition of read-once functions using cographs and normality, and the readability of functions associated with partial k-trees, Discrete Appl. Math. 154 (2006) 1465-1677], we presented the first polynomial-time algorithm for recognizing and factoring read-once functions, based on a classical characterization theorem of Gurvich which states that a positive Boolean function is read-once if and only if it is normal and its co-occurrence graph is P 4-free. In this note, we improve the complexity bound by showing that the method can be modified slightly, with two crucial observations, to obtain an O(n|f |) implementation, where |f | denotes the length of the DNF expression of a positive Boolean function f, and n is the number of variables in f. The previously stated bound was O(n 2 k), where k is the number of prime implicants of the function. In both cases, f is assumed to be given as a DNF formula consisting entirely of the prime implicants of the function.

2009

This paper introduces a new fast systematic method for minimization of the Boolean functions. This method is a very simple because there is no need for any visual representation such as Karnough map or arrangement technique such as Tabulation method and is very easy for programming. Furthermore, it is very suitable for boolean function with large number of variables (more than 4 variable). In adition, it is very simple for student's understanding. Moreover, the work presented in [23] is developed. In order to accelerate the operation of the proposed method, a new approach for fast term detection is presented. Such approach uses fast neural networks (FNNs) implemented in the frequency domain. The operation of these networks relies on performing cross correlation in the frequency domain rather than spatial one. It is proved mathematically and practically that the number of computation steps required for the presented FNNs is less than that needed by conventional neural networks (CNNs). Simulation results using MATLAB confirm the theoretical computations.

1996

The author has granted a nonexclusive licence allowing the National Librrily of Canada to reproduce, loan, distribute or sell copies of this thesis in microfonn, paper or electronic formats. The author retains ownership of the copyright in this thesis. Neither the thesis nor substantial extracts fiom it may be printed or othenvise reproduced without the author7 s permission. L'auteur a accordé une licence non exclusive permettant à la Bibliothèque nationale du Canada de reproduire, prêter, distribuer ou vendre des copies de cette thèse sous la forme de microfiche/film, de reproduction sur papier ou sur format électronique. L'auteur conserve la propriété du droit d'auteur qui protège cette thèse. Ni la thèse ni des extraits substantiels de celle-ci ne doivent être imprimés ou autrement reproduits sans son autorisation.

IEEE Transactions on Computers, 1978

ArXiv, 2019

Many approaches such as Quine-McCluskey algorithm, Karnaugh map solving, Petrick&#39;s method and McBoole&#39;s method have been devised to simplify Boolean expressions in order to optimize hardware implementation of digital circuits. However, the algorithmic implementations of these methods are hard-coded and also their computation time is proportional to the number of minterms involved in the expression. In this paper, we propose KarNet, where the ability of Convolutional Neural Networks to model relationships between various cell locations and values by capturing spatial dependencies is exploited to solve Karnaugh maps. In order to do so, a Karnaugh map is represented as an image signal, where each cell is considered as a pixel. Experimental results show that the computation time of KarNet is independent of the number of minterms and is of the order of one-hundredth to one-tenth that of the rule-based methods. KarNet being a learned system is found to achieve nearly a hundred per...

1995

One of the crucial problems multi-level logic synthesis techniques for multi-output boolean functions f = (f1; . . . ; fm) : f0; 1g n ! f0;1g m have to deal with is nding sublogic which can be shared by di erent outputs, i.e., nding boolean functions = ( 1; . . . ; h) : f0;1g p ! f0; 1g h which can be used as common sublogic of good realizations of f1; . . . ; fm. In this paper we present an e cient robdd based implementation of this Common Decomposition Functions Problem (cdf). The key concept of our method is the exploitation of equivalences of the functions f1; . . . ; fm which considerably reduces the running time of the tool. Formally, cdf is de ned as follows: Given m boolean functions f1; . . . ; fm : f0;1g n ! f0;1g, and two natural numbers p and h, nd h boolean functions 1; . . . ; h : f0;1g p ! f0;1g such that 81 k m there is a decomposition of fk of the form fk(x1; . . . ; xn) = g (k) ( 1(x1; . . . ; xp); . . . ; h(x1; . . . ; xp); (k) h+1 (x1; . . . ; xp); . . . ; (k) r k (x1; . . . ; xp); xp+1; . . . ; xn) using a minimal number rk of single-output boolean decomposition functions.