CC BY
6
Journal of Siberian Federal University. Mathematics & Physics 2016, 9(1), 119—122
УДК 519.716
On Decomposition of Sub-definite Partial Boolean Functions
Ivan K. Sharankhaev*
Institute of Mathematics and Computer Science Buryat State University Smolin, 24a, Ulan-Ude, 670000
Russia
Received 02.11.2015, received in revised form 06.12.2015, accepted 15.01.2016 In this article we study Boolean functions with two kinds of indeterminacy. We prove criterion of decomposition of this functions including separating decomposition. As a result we have method that allows to obtain representation of an arbitrary function using superposition of functions that have smaller dimentions.
Keywords: incompletely defined Boolean function, sub-definite partial Boolean function, decomposition, superposition.
DOI: 10.17516/1997-1397-2016-9-1-119-122.
Introduction
In the theory of discrete functions rapidly developing area, is engaged study of functions defined on a finite set A and receiving as their values subsets of A, including 0. Such maps are found in the mathematical modeling of information processing, in the case where the set A = {0,1} are incompletely defined Boolean functions. As can be seen, there are two kinds of indeterminacy. For the first type of indeterminacy on the sets on which the function value is not defined, the indeterminacy is understood as the ability to adopt and value 0 and value 1, i.e. image of these sets is the set {0,1}. Boolean functions with this kind of indeterminacy are considered, for example, in [1]. The second type of indeterminacy are associated with the empty set, typically means taboo data and studied, for example, in [2].
In this paper we consider incompletely defined Boolean functions with two kinds of indeterminacy, following [3], we call them sub-definite partial Boolean functions.
The problem of representation of an arbitrary sub-definite partial Boolean function by the functions of lower dimension is very important. We proved criterion of decomposition sub-definite partial Boolean functions, including separating decomposition, which generalizes the criterion of the functional separability of Boolean functions by G. N. Povarov [4] and provides a method of obtaining representations sub-definite partial Boolean functions by the functions of lower dimension.
The work [5] is dedicated to finding of repetition-free representations of sub-definite partial Boolean functions in a special basic set. We note that the obtaining of the results of [5] is greatly simplified by the criterion of separating decomposition. Moreover, our method can be used to construct algorithms of repetition-free representations of sub-definite partial Boolean functions in other basic sets.
* goran5@mail.ru © Siberian Federal University. All rights reserved
1. Basic concepts and definitions
We preface the description of the main results with the needed definitions and notation. We note that the terminology which used to sub-definite partial Boolean functions, completely preserved from the theory of Boolean functions, which can be seen in [6]. We use the following notation: variables are denoted by the symbols x, y, u, v, w, maybe with subscripts; constants are denoted by the symbols a, a, j, maybe with subscripts; the symbol X denotes the tuple (xi,... ,x„); \X\ is the length of a tuple X.
Let is the power of set A, 2a is the set of all subsets of A, E2 = {0,1}. We define the following sets of functions:
Pin = {f \f : En ^ 2e}, P.2 = U Pln,
n
P.n = {f \f g Pin and \f(a) \ = 1 for every a G En}, P. = U P.,n-
n
Functions from P2 are called Boolean functions, and functions from Pi are called sub-definite partial Boolean functions. Below sub-definite partial Boolean functions are simply called functions.
By definition we believe that the superposition
f (fi(xi,... , xm ), . . .,fn(xi, . . . , xm)), where f, f1, ..., fn g Pi, represents some function g(x1,..., xm), if for every (a1,..., am) g E™
0, if fi(a1,..., am) = 0 for some i G {1,..., m}; g(ai,...,am)= <j u f(3i,...,3n), otherwise.
The function obtained from f (x1, ...,xn) by the substitution of a constant a g {0,1} for a variable xi is called the remainder function and is denoted . This definition is extended to a subset of variables by induction.
For simplicity we will use the following code:
0 ^ *, {0} v 0, {1} v 1, {0,1} v 2. The function which on all tuples is equal to * will be denoted by *.
For arbitrary n-ary functions f h g we define function f U g in the following way:
(f U g)(ai, ...,an) = f (ai, ...,an) U g(ai, ...,an)
for an arbitrary tuple (a1,..., an).
Function f has decomposition by partition of set of variables on u, V, W, if there exist functions h h g such that holds
f (U,V,W) = h(U,W,g(U,V)). (1)
If U = 0, then this decomposition is called separating.
2. The main result
In this section we prove necessary and sufficient condition of existence of decomposition and also separating decomposition for an arbitrary function.
Theorem 1. An arbitrary function f has decomposition by partition of set of variables on U,V,W if and only if for an arbitrary tuple a (\a\ = \U\) there exist no more than four different remainder
functions of fa for variables V, and each of remainder functions is equal to or *, or some function f0, or some function fi, or f0 U fi.
Proof. Necessity. Since function f has decomposition, then
f (u,V,W) = h(u,W, g(u,v)).
For arbitrary tuples a and v we have
f (a, v, W) = h(a, W, g(a, v)).
Because of g(a, v) & {0,1, *, 2}, the remainder function f (a, v, W) is equal to or *, or h(a, W, 0), or h(a, W, 1), or h(a, W, 2) = h(a, W, 0) U h(a, W, 1).
Sufficiency. We define functions g(u,V) and h(u,W,y). For arbitrary tuples a and v
*, if f (a, v, W) = *;
0, if f (a, P, W) = fo(W);
1, if f (a, v, W) = fi(W);
2, if f (a, ~v,w) = fo(W) u fi(W).
g{a, ¡3)
and
~ \ \ fo(W), if y = 0; h(a,W,y) =
{ fi(W), if y = 1.
We show that for such functions g and h the equality (1) holds. We consider h(a,Y,g(a, v)) for arbitrary a, v, v.
If g(a, v) = *, then h(a,j, g(a, v)) = * = f (a, v,v).
If g(a,v) = 0, then h(a, v, g(a, ~v)) = fo(v) = f (a, ~v, y).
If g(a, ~v) = 1, then h(a,v, g(a, ~v)) = h(v) = f (a, ~v,y).
If g(a, ~v) = 2, then h(a, v, g(a, ~v)) = fo(v) u fi(v) = f (a, v,v). □
Corollary 1 (Criterion of separating decomposition). An arbitrary function f has separating decomposition by partition of set of variables on V, W if and only if there exist no more than four different remainder functions of f for variables V, and each of remainder functions is equal to or or some function f0, or some function f\, or f0 U fi.
Proof follows from proof of theorem when u = 0. □
References
[1] V.V.Tarasov, Completeness criterion for partial logic functions, Problemy kibernetiki, 30(1975), 319-325 (in Russian).
[2] R.V.Freivald, Completeness criterion for partial functions of algebra logic and many-valued logics, Dokl. Akad. Nauk SSSR, 167(1966), 1249-1250 (in Russian).
[3] V.I.Panteleev, Completeness criterion for sub-definite partial Boolean functions, Vestnik Novosibirskogo Gos. Univ. Ser. Mathematika, mechanika, informatika, 9(2009), no.3, 95114 (in Russian).
[4] G.N.Povarov, On functional separability of Boolean functions, Dokl. Akad. Nauk SSSR, 94(1954), no. 2, 801-803 (in Russian).
[5] V.L.Semicheva, Methods of finding of repetition-free representations of incompletely defined Boolean functions, Dissertation of candidate in physics and mathematics, Irkutsk, 2008 (in Russian).
[6] S.F.Vinokurov, N.A.Peryazev, Selected questions of theory of Boolean functions, Fizmatlit, Moscow, 2001 (in Russian).
О декомпозиции недоопределенных частичных булевых функций
Иван К. Шаранхаев
В статье 'рассматриваются булевы функции с двумя видами неопределенности. Доказан критерий декомпозиции, в том числе разделительной декомпозиции таких функций, который дает метод, позволяющий получать представление произвольной функции с помощью суперпозиции функций меньших размерностей.
Ключевые слова: не всюду определенная булева функция, недоопределенная частичная булева функция, декомпозиция, суперпозиция.
