21 research outputs found
Representation of Boolean functions in terms of quantum computation
The relationship between quantum physics and discrete mathematics is reviewed
in this article. The Boolean functions unitary representation is considered.
The relationship between Zhegalkin polynomial, which defines the algebraic
normal form of Boolean function, and quantum logic circuits is described. It is
shown that quantum information approach provides simple algorithm to construct
Zhegalkin polynomial using truth table. Developed methods and algorithms have
arbitrary Boolean function generalization with multibit input and multibit
output. Such generalization allows us to use many-valued logic (k-valued logic,
where k is a prime number). Developed methods and algorithms can significantly
improve quantum technology realization. The presented approach is the baseline
for transition from classical machine logic to quantum hardware.Comment: 19 pages, 4 figure
On the effect of variable identification on the essential arity of functions
We show that every function of several variables on a finite set of k
elements with n>k essential variables has a variable identification minor with
at least n-k essential variables. This is a generalization of a theorem of
Salomaa on the essential variables of Boolean functions. We also strengthen
Salomaa's theorem by characterizing all the Boolean functions f having a
variable identification minor that has just one essential variable less than f.Comment: 10 page
Quantum Algorithm for Dynamic Programming Approach for DAGs. Applications for Zhegalkin Polynomial Evaluation and Some Problems on DAGs
In this paper, we present a quantum algorithm for dynamic programming
approach for problems on directed acyclic graphs (DAGs). The running time of
the algorithm is , and the running time of the
best known deterministic algorithm is , where is the number of
vertices, is the number of vertices with at least one outgoing edge;
is the number of edges. We show that we can solve problems that use OR,
AND, NAND, MAX and MIN functions as the main transition steps. The approach is
useful for a couple of problems. One of them is computing a Boolean formula
that is represented by Zhegalkin polynomial, a Boolean circuit with shared
input and non-constant depth evaluating. Another two are the single source
longest paths search for weighted DAGs and the diameter search problem for
unweighted DAGs.Comment: UCNC2019 Conference pape
When nominal analogical proportions do not fail
International audienceAnalogical proportions are statements of the form "a is to b as c is to d", where a, b, c, d are tuples of attribute values describing items. The mechanism of analogical inference, empirically proved to be efficient in classification and reasoning tasks, started to be better understood when the characterization of the class of classification functions with which the analogical inference always agrees was established for Boolean attributes. The purpose of this paper is to study the case of finite attribute domains that are not necessarily two-valued, i.e., when attributes are nominal. In particular, we describe the more stringent class of "hard" analogy preserving (HAP) functions f : X1×· · ·×Xm → X over finite domains X1,. .. , Xm, X for binary classification purposes. This description is obtained in two steps. First we observe that such AP functions are almost affine, that is, their restriction to any S1×· · ·×Sm, where Si ⊆ Xi and |Si| ≤ 2 (1 ≤ i ≤ m), can be turned into an affine function by renaming variable and function values. We then use this result together with some universal algebraic tools to show that they are essentially unary or quasi-linear, which provides a general representation of HAP functions. As a by-product, in the case when X1 = · · · = Xm = X, it follows that this class of HAP functions constitutes a clone on X, thus generalizing several results by some of the authors in the Boolean case
Parallel Factorization of Boolean Polynomials
Polynomial factorization is a classical algorithmic problem in algebra, which has a wide range of applications. Of special interest is factorization over finite fields, among which the field of order two is probably the most important one due to the relationship to Boolean functions. In particular, factorization of Boolean polynomials corresponds to decomposition of Boolean functions given in the Algebraic Normal Form. It has been also shown that factorization provides a solution to decomposition of functions given in the full DNF (i.e., by a truth table), for positive DNFs, and for cartesian decomposition of relational datatables. These applications show the importance of developing fast and practical factorization algorithms. In the paper, we consider some recently proposed polynomial time factorization algorithms for Boolean polynomials and describe a parallel MIMD implementation thereof, which exploits both the task and data level parallelism. We report on an experimental evaluation, which has been conducted on logic circuit synthesis benchmarks and synthetic polynomials, and show that our implementation significantly improves the efficiency of factorization. Finally, we report on the performance benefits obtained from a parallel algorithm when executed on a massively parallel many core architecture (Redefine)