Transformations of Boolean Functions

Boolean functions are characterized by the unique structure of their solution space. Some properties of the solution space, such as the possible existence of a solution, are well sought after but difficult to obtain. To better reason about such properties, we define transformations as functions that change one Boolean function to another while maintaining some properties of the solution space. We explore transformations of Boolean functions, compactly described as Boolean formulas, where the property is to maintain is the number of solutions in the solution spaces. We first discuss general characteristics of such transformations. Next, we reason about the computational complexity of transforming one Boolean formula to another. Finally, we demonstrate the versatility of transformations by extensively discussing transformations of Boolean formulas to "blocks," which are solution spaces in which the set of solutions makes a prefix of the solution space under a lexicographic order of the variables

Boolean functions minimization by the method of figurative transformations

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 of the block diagram with repetition makes it possible to obtain more information about the orthogonality, adjacency, uniqueness of the truth table blocks; – equivalent figurative transformations due to the greater information capacity are capable of replacing verbal procedures of algebraic transformations; – result of minimization is estimated based on the sign of the minimum function; – minimum DNF or CNF functions are obtained regardless of the given normal form of the logical function, which means that it is necessary to minimize the given function for two normal forms — DNF and CNF using the full truth table; This ensures that it is possible to obtain an optimal reduction in the number of variables of a given function without losing its functionality. The effectiveness of the use of equivalent figurative transformations for Boolean functions minimization is demonstrated by examples of minimization of functions borrowed from other methods for the purpose of comparison. Compared with similar well-known methods of Boolean functions minimization, this provides: – less complexity of the minimization procedure for Boolean functions; – guaranteed Boolean functions minimization; – self-sufficiency of the specified method of Boolean functions minimization due to the introduction of features of the minimal function and minimization of two normal forms – DNF and CNF on the complete truth table of a given Boolean functio

One-way permutations, computational asymmetry and distortion

Computational asymmetry, i.e., the discrepancy between the complexity of transformations and the complexity of their inverses, is at the core of one-way transformations. We introduce a computational asymmetry function that measures the amount of one-wayness of permutations. We also introduce the word-length asymmetry function for groups, which is an algebraic analogue of computational asymmetry. We relate boolean circuits to words in a Thompson monoid, over a fixed generating set, in such a way that circuit size is equal to word-length. Moreover, boolean circuits have a representation in terms of elements of a Thompson group, in such a way that circuit size is polynomially equivalent to word-length. We show that circuits built with gates that are not constrained to have fixed-length inputs and outputs, are at most quadratically more compact than circuits built from traditional gates (with fixed-length inputs and outputs). Finally, we show that the computational asymmetry function is closely related to certain distortion functions: The computational asymmetry function is polynomially equivalent to the distortion of the path length in Schreier graphs of certain Thompson groups, compared to the path length in Cayley graphs of certain Thompson monoids. We also show that the results of Razborov and others on monotone circuit complexity lead to exponential lower bounds on certain distortions.Comment: 33 page

Monoids with tests and the algebra of possibly non-halting programs

We study the algebraic theory of computable functions, which can be viewed as arising from possibly non-halting computer programs or algorithms, acting on some state space, equipped with operations of composition, if-then-else and while-do defined in terms of a Boolean algebra of conditions. It has previously been shown that there is no finite axiomatisation of algebras of partial functions under these operations alone, and this holds even if one restricts attention to transformations (representing halting programs) rather than partial functions, and omits while-do from the signature. In the halting case, there is a natural “fix”, which is to allow composition of halting programs with conditions, and then the resulting algebras admit a finite axiomatisation. In the current setting such compositions are not possible, but by extending the notion of if-then-else, we are able to give finite axiomatisations of the resulting algebras of (partial) functions, with while-do in the signature if the state space is assumed finite. The axiomatisations are extended to consider the partial predicate of equality. All algebras considered turn out to be enrichments of the notion of a (one-sided) restriction semigrou