Abstract
The object of solving the problem of minimizing the Boolean function in this work is a block diagram with repetition, what is the truth table of the given function. This allows to leave the minimization principle within the function calculation protocol and, thus, dispense with auxiliary objects like algebraic expressions, Karnaugh map, Veitch diagram, acyclic graph, etc. The algebraic transformations of conjunctors are limited to the verbal form of information, they require active decoding, processing and the addition of algebraic data, therefore, as the number of variable variables increases and the resource of such minimization method is quickly exhausted. In turn, the mathematical apparatus of the combinatorial block diagram with repetition gives more information about the orthogonality, contiguity, uniqueness of truth table blocks, so the application of such minimization system of the Boolean function is more efficient. Equivalent transformations by graphic images, in their properties have a large information capacity, capable of effectively replacing verbal procedures of algebraic transformations. The increased information capacity of the combinatorial method makes it possible to carry out manual minimization of 4, 5-bit Boolean functions quite easily.Using a block diagram with repetition in tasks of minimizing Boolean function is more advantageous in comparison with analogues for the following factors:– lower cost of development and implementation, since the principle of minimization of the method remains within the truth table of this function and does not require other auxiliary objects;– increasing the performance of the manual minimization procedure for 4-, 5-bit functions and increasing the performance of automated minimization with a greater number of variable functions, in particular due to the fact that several search options give the same minimum function.The combinatorial method for minimizing Boolean functions can find practical application in the development of electronic computer systems, because:– DNF minimization is one of the multiextremal logical-combinatorial problems, the solution of which is, in particular, the combinatorial device of block-schemes with repetition;– expands the possibilities of Boolean functions minimization technology for their application in information technology;– improves the algebraic method of minimizing the Boolean function due to the tabular organization of the method and the introduction of the device of figurative numeration;– the minimum function can be obtained by several search options that reduces the complexity of the search algorithm, and is the rationale for developing a corresponding function minimization protocol.
Highlights
The problems and shortcomings of the known methods for minimizing Boolean functions are associated with a rapid growth in the amount of computation, which results in an increase in the number of computational operations, and, in the increase in the number of variables of the logical function
The Boolean function f (x1,..., xn) that describes the operation of a logical device can be realized with the help of a disjunctive normal form (DNF), which in this case describe the scheme of the corresponding logical device
The object of research is the problem of minimizing the Boolean function by a combinatorial method – a blockdiagram with repetition
Summary
The problems and shortcomings of the known methods for minimizing Boolean functions are associated with a rapid growth in the amount of computation, which results in an increase in the number of computational operations, and, in the increase in the number of variables of the logical function. Functions with a large number of variables (more than 16 variables) can be minimized only in a certain sense, not guaranteeing the achievement of the optimal solution with the help of the heuristic Espresso algorithm, which today is documented by the world standard [6]. Since Espresso minimization algorithm does not guarantee optimal minimization of Boolean function with increasing number of variables, the search for new minimization methods remains relevant. The disadvantages of the combinatorial method of manual minimization are associated with the growth of the number of variables (more than seven or eight) of the logical function. Minimizing a function with a large number of variables requires updating the library of submatrices on which the figurative calculus of the combinatorial method is based
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