How do we simplify logic functions?

2022

A new algorithm for simplifying Boolean logic expressions is presented. It is a top-down guided search algorithm.

Technology transfer: fundamental principles and innovative technical solutions, 2018

The object of research is the method of figurative transformations for Boolean functions minimization. One of the most problematic places to minimize Boolean functions is the complexity of the minimization algorithm and the guarantee of obtaining a minimal function. During the study, the method of equivalent figurative transformations was used, which is based on the laws and axioms of the algebra of logic; minimization protocols for Boolean functions that are used when the truth table of a given function has a complete binary combinatorial system with repetition or an incomplete binary combinatorial system with repetition. A reduction in the complexity of the minimization process for Boolean functions is obtained, new criteria for finding minimal functions are established. This is due to the fact that the proposed method of Boolean functions minimization has a number of peculiarities of solving the problem of finding minimal logical functions, in particular: – mathematical apparatus...

Eastern-European Journal of Enterprise Technologies, 2021

This paper reports a study that has established the possibility of improving the effectiveness of the method of figurative transformations in order to minimize symmetrical Boolean functions in the main and polynomial bases. Prospective reserves in the analytical method were identified, such as simplification of polynomial function conjuncterms using the created equivalent transformations based on the method of inserting the same conjuncterms followed by the operation of super-gluing the variables. The method of figurative transformations was extended to the process of minimizing the symmetrical Boolean functions with the help of algebra in terms of rules for simplifying the functions of the main and polynomial bases and developed equivalent transformations of conjuncterms. It was established that the simplification of symmetric Boolean functions by the method of figurative transformations is based on a flowchart with repetition, which is the actual truth table of the assigned functi...

IEE Proceedings - Computers and Digital Techniques, 1996

An algorithm called XOF.GA is presented which minimises Boolean mult i-output logic functions as multilevel ANDEXOR networks of two-input logic gates. It carries out symbolic simplification, and works from tlhe bottom of a binary variable decision tree to tlhe top, with variable choice determined using a genetic algorithm. Since the algorithm is multilevel in nature, it delivers more compact circuits than two-level ESOP minimisation algorithrrts, such as EXMIN2. it also finds more economical representations than the fixed polarity Ree& Muller method.

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.

IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 1994

A6stract-We describe PBS, a formally proven implementation of multi-level logic synthesis based on the weak division algorithm. We have proved that for all legal input circuits, PBS generates an output circuit that is functionally correct and has minimal size. PBS runs on large examples in reasonable time for a prototype system. PBS, was verified using the Nuprl proof development system. The proof of PBS, which required several months, was well worth the effort since the benefits are realized every time the program is run. Wheo engineers use a verified synthesis tool, they get the idcreased confidence of applying Pormal methods to their designs without the cost of verying each design produced.

CBEMS-18,DMCCIA-18,EHSSM-18,BLEIS-18,ICLTET-18 March 21-23, 2018 Istanbul (Turkey), 2018

In this paper, a new heuristic method to minimize multiple-valued logic (MVL) functions and its implementation results will be introduced. The motivation for this work has been to develop a light weight MVL minimization algorithm which is simpler than existing algorithms so that it can serve as a basis for the future work. MVL-MIN is designed and implemented from scratch and it is compared with MVSIS. MVSIS is a MVL minimization program developed by the Electronic Systems Design group at Berkeley. The advantages of the new algorithm include a new cube calculus based technique for detecting and eliminating the totally redundant prime implicants. A comparison with MVSIS approved that the proposed algorithm is able to solve all test files within a fixed allocation of computer resources. Since conventional testbenches are fairly small, we generated 3 sets of testbench in blifmv format. Each set includes 8 multiple-valued logic functions. MVL-MIN is able to solve all test-benches about 5 times faster than MVSIS on average.

The Karnaugh map (K-map) stands as a foundational graphical method for systematically simplifying Boolean algebra expressions. Introduced in 1953, this technique provides an organized approach crucial for transforming complex truth tables into analogous, simplified logic circuits. This report delves into the fundamental concepts underpinning K-maps, including the definitions and roles of minterms and maxterms, and the essential principles of grouping terms. It meticulously differentiates between the Sum of Product (SOP) and Product of Sum (POS) forms, detailing their characteristics, logical implementations, and comparative aspects. Furthermore, the report illustrates the various types of K-maps based on the number of variables, highlighting their structural nuances. By offering an intuitive visual approach, K-maps significantly reduce the complexity often associated with algebraic manipulation, thereby enhancing the efficiency and reliability of digital circuit design. This method remains a cornerstone in digital electronics, providing a practical and accessible tool for engineers and students alike.

2003

Optimization can make at least two contributions to boolean logic. Its solution methods can address inference and satisfiability problems, and its style of analysis can reveal tractable classes of boolean problems that might otherwise have gone unnoticed. They key to linking optimization with logic is to provide logical formulas a numerical interpretation or semantics. While syntax concerns the structure of logical expressions, semantics gives them meaning. Boolean semantics, for instance, focuses on truth functions that capture the meaning of logical propositions. To take an example, the function

International Journal of Computer Mathematics, 2007

A pedagogical treatment of two-level multiple-output logic minimization is presented through a compact exposition of a novel manual fast procedure. This procedure is a purely map technique which generalizes the map procedure for single-output minimization. It requires neither the generation of the set of all paramount prime implicants, nor the construction of a cover matrix. Instead, it utilizes certain visual interactions between various groups of maps placed at distinct levels of a Hasse diagram, which is conveniently drawn in a Karnaugh map layout so that any parent map is easily visualized as adjacent to all its children maps. The present exposition is believed to enhance what is currently available in undergraduate texts, and is intended as a supplement to, rather than a replacement of, automated computational experience. An illustrative example demonstrates the proposed procedure for the dual cases of AND-OR and OR-AND minimizations.