Blocks of Monotone Boolean Functions

This paper first proposed the method of constructing blocks of monotone Boolean functions (MBFs) is developed for classification and the analysis of these functions. Use of only nonisomorphic blocks considerably simplifies enumeration MBFs. Application of the method of construction blocks on the example of classification and analysis of MBFs from 0 to 4 variables is considered. This method can be used at counting of all MBFs of a given rank n .


Introduction
In 1897 R. Dedekind has published article [1] in which the number of elements of a free distributive lattice with four generators has been found. The number ψ(n) of elements of a free distributive lattice with n generators coincides with number of anti-chains in the unit n-dimensional cube. In the language of logic algebra K(n) = ψ(n) + 2 -the number of the monotone Boolean functions (MBFs), which depend on n variables x 1 , ..., x n . The problem of calculating ψ(n) is called Dedekind's problem. D(0) -D(4) computed by R. Dedekind (1897). D (5) are given by Church (1940). D (6) was calculated by Ward (1946), D (7) was calculated by Church (1965) and D (8) by Wiedemann [2] (1991). As it turned out, this problem is quite difficult and cannot be examine through the traditional method of generating functions. At present known ways of calculation D(n). One way of partitioning the number of distinct monotone functions of n variables is to classify them according to the number of distinct input states, at which the function is equal to 1. Another way of splitting up D(n) is according with the number of conjunctions in the disjunctive normal form. [3] In [4,5] developed a classification of the types of MBFs and enumeration of maximum types of MBFs. In [6] proved the expression for enumeration of types MBFs as a product of matrices.
However, the literature is not considered a method of analysis and classification of MBFs -based structure MBFs blocks (sets of MBFs connected by three operations considered further), which in some cases can reduce sorting MBFs due to that we can to exclude all isomorphic blocks and consider only nonisomorphic blocks. The analysis of MBFs blocks allows examining this problem in a new way.
This paper first proposed the classification method and research MBFs with the help of the blocks. This method is not limited to the number of variables, because for any MBF with any number of variables n using the disjunctive complement s and duality can construct a block containing this MBF. For descriptions of all MBFs of n variables it is sufficient to construct only nonisomorphic blocks. This very significantly reduces the description of all MBFs of n variables. For example, all MBFs of 5 variables (these MBFs blocks will be described in the following paper) can be divided into 522 blocks, but you can only choose 23 pairwise nonisomorphic. Nonisomorphic blocks are directly related to Dedekind numbers.
The aim of paper is development of a method of the analysis and MBFs classification on the basis of construction of MBFs blocks.

Results
Let's remind the main concepts connected with MBFs.
For a disjunctive normal form of MBFs it means that in it there is no negation operation, and there are only conjunction and disjunction operations. The vector P = (a n ,…, a i ,…, a 1 , a 0 ), the components of which take values from the set {0,1} is called [5] input set of Boolean function of n variables. The set of all input sets forms Boolean cube of rank n. themselves input sets P are tops of the Boolean cube. Any Boolean function is defined by a set of vertices of the Boolean cube, in which the function is equal to unity. Any set of incomparable tops of a Boolean cube is called as an antichain to set the MBFs [5] is sufficient to indicate some antichain in a Boolean cube. Each top of an antichain (except tops corresponding to entrance sets (0..., 0) and (1..., 1)) defines conjunction in a disjunctive normal form of MBFs corresponding to this antichain.
Consider one of the ways to describe the MBFs as a minimum input sets or the corresponding family of subsets Sperner. (Any family of subsets of a set is called a family of subsets Sperner, if none of the subsets of the family is not contained in any other subset of the same family.) In this case, if MBFs of n variables, then an arbitrary subset of a family of subsets Sperner may contain from 0 to n elements.
Let's consider all MBFs of ranks from 0 to 4. There are only two MBFs of rank 0. It is f 0 (0) identically equal 0 and f 1 (0) identically equal 1. There are three MBFs of rank 1. It is f 0 (1) identically equal 0, f 1 (1) identically equal 1 and f 2 (1) = x 1 . There are six MBFs of rank 2. It is f 0 (2) identically equal 0, In the following table all 168 MBFs of rank 4 are given in the minimal disjunctive form. MBFs with number 0, that is f 0 (4) is zero MBFs (it is equal 0 at all values of entrance variables), and f 1 (4) -unit MBFs.
All MBFs of one rank form a distributive lattice with respect to the operations of conjunction and disjunction. Such lattices of R 0 , R 1 , R 2 and R 3 for MBFs ranks from 0 to 3 are represented on fig. 1. Lattices of R 1 , R 2 and R 3 differ from free distributive lattices of the same rank complement of the highest and lowest tops. Block MBFs rank n is a subset of all MBFs rank n, closed with respect to the three operations: duality, conjunctive complement and disjunctive complement. We introduce some definitions. Block power is the number of MBFs which enter into it. Two blocks are similar, if the same power and abstraction from their member of MBFs, these blocks are indistinguishable. Two blocks are isomorphic if any MBFs one block can be obtained from some other MBFs block certain substitution variables. In complement, if both of the MBFs to perform one of three operations defined for the block, then the resulting MBFs first block is obtained from the resultant MBFs another unit of the same substitution variables. By definition isomorphic blocks are similar.
MBFs of ranks 0 and 1 are grouped in one block consisting and two and of three MBFs respectively with respect to considered three operations. MBFs rank 2 may be represented as two blocks, one of which consists of four MBFs, and another -of the two MBFs. All these blocks are shown in Fig. 3. Here, the opera tion of duality is represented by the solid line, the operation of disjunctive complement -a dashed line and the operation of conjunctive complementdash-dotted line. For example, in a unit consisting of MBFs f 0 (2), f 1 (2), f 2 (2) and f 3 (2) we have: f 1 (2) = f 0 -1 (2), f 3 (2) = 0 f (2) and f 2 (2) = 1 f (2). x 1 x 3 ∨ x 2 x 3 respectively. These blocks, as shown in Fig. 4, i.e. with the same number and the same form MBFs said to be similar.
On fig. 6 Two MBFs blocks of rank 4 powers of 6 (a) the block 4 and b) the block 5) are shown. The block 4 has no isomorphic and consists of MBFs f 5 (4), f 4 (4), f 3 (4), f 2 (4), f 0 (4) and f 1 (4). The first 4 MBFs and unit MBFs f 1 (4) are the maximum MBFs of rank 4 weights 1. MBFs f 4 (4) and zero MBFs f 0 (4) are own disjunctive complement, and MBFs f 5 (4) and MBFs f 1 (4) are own conjunctive complement. The block 5 consists of MBFs f 6 (4)  1 x 4 and f 156 (4) = x 1 x 3 ∨ x 2 x 3. These blocks contain 72 MBFs. In total there are 9 blocks, isomorphic to blocks on fig. 3. These blocks contain 108 MBFs. Thus, all 168 MBFs of rank 4 form 24 blocks. Among these blocks only 7 not isomorphic among themselves and 6 are not similar each other. Power block of rank 4 can be one of 5 numbers: 2, 3, 4, 6 and 12. All MBFs of any block can be obtained from the generator MBFs using two operations, for example, disjunctive complement and a duality. Blocks, isomorphic to some block, it is possible to receive substitution of variables in MBFs entering into this block. Operation of substitution of variables is realised much more simply, than duality operation. Therefore, for complete search of MBFs of the set rank by a method of creation of MBFs blocks it is enough to construct not isomorphic with each other blocks, and the others turn out from them substitution of variables. Already for rank 4 instead of 24 blocks it is enough to construct 7. With rank growth the relation of all blocks does not isomorphic increases.
As can be seen from Fig. 3 -7 all MBFs of similar blocks turn out equally by means of the considered three operations. In some cases, as for the blocks generated MBFs f 5 (3) = x 1 , f 10 (3) = x 2 and f 15 (3) = x 3 , all of these MBFs blocks obtained from each other by a change of variables. Search of all MBFs of rank n can be arranged as follows. Take MBFs arbitrary rank n. by considering the three operations for her construction block. Then taken MBFs arbitrary rank n, is not included in the block, and it is also construct block. These actions are repeated until there are MBFs are not included inconstructed blocks. Thus, all MBFs rank n will be divided into blocks. The advantage of this method is that all MBFs inside the block 3 connected with each other operations and the construction of blocks of the similar blocks can be not build enumeration reduced.

Conclusions
In summary we will note the following. The method of the analysis and MBFs classification on the basis of creation of MBFs blocks is developed. Application of this method to MBFs analysis with number of variables from 0 to 4 is shown. Application of a method of creation of MBFs blocks becomes simpler, if the number of blocks and number of similar blocks is in advance known. Further it is necessary to study splitting into MBFs blocks of separate ranks, and also to look for regularities of splitting into blocks the general for MBFs of all ranks.
Regularities for the blocks found enough, but they cannot be presented in a single paper. They will be discussed in following papers. In particular, they discovered that using blocks is convenient to assume self-dual and disjunctive self-complementary MBFs. At the same time one cannot say that we will consider all the regularities. Some of them, which relate to the number and size of nonisomorphic to MBFs blocks given rank may still require the efforts of many mathematicians.