Basic MBF Blocks Properties and Rank 6 Blocks

Considered all the Monotone Boolean Functions (MBFs) blocks of the sixth rank, it proved a series of new properties MBF blocks. The proposed methods can be used to analyze large ranks MBFs. The tables which describe all of the blocks from 4th to 6th rank are provided. On the basis of these tables are counted typical depending to these blocks.


Introduction
In 1897 R. Dedekind published the article [1], in which he found the number of free distributive lattice elements with four generators. The number of free distributive lattice ψ (n) with n generators coincides with the antichain number in a unit n-dimensional cube. In terms of logical algebra, D (n) = ψ (n) + 2 -is the number of Monotone Boolean Functions (MBFs), depending on n variables x 1 ,..., x n . The ψ (n) calculation problem is usually called Dedekind problem. D (0) -D (5) were received by Church in 1940, D (6) was calculated by Ward in 1946, D (7) was calculated by Church in 1965 and D(8) was received by Wiedemann [2] in 1991. As it turned out, this problem is rather difficult and it defies solution within the frames of traditional method of generating functions.
However, the method of MBFs classification and analysis based on MBF blocks generation (MBF sets connected by three operations discussed below) has not been considered in literature, excluding the studies [3] and [4], which in a number of cases allows cancelling the MBF search due to the exclusion of all isomorphic blocks and considering only nonisomorphic ones. The MBF blocks analysis allows considering this problem from a new angle.
In [5] the first MBF blocks of various ranks were shown. The aim of the article is conclusion the basic properties of MBFs and analysis of all the blocks from 4 to 6 ranks.

Results
Let us recall the main concepts in connection with MBF. is also correct. Among elementary Boolean functions, for example, conjunction and disjunction are monotonic. Any function received from MBFs with the aid of superimposition operation is monotonic by itself. This is why the function generated with the operations of disjunction and conjunction shall be also monotonic. Say, conjunction В is absorbed by conjunction А if conjunction В contains all variables that are present in conjunction А. We shall consider MBF as a Disjunctive Normal Form (DNF), i.e. a sum of products, in which no one is absorbed by the other one. In other words, one set of conjunctions absorbs the other set if all conjunctions of one set are absorbed by the conjunctions of the other set.
We shall designate the Boolean function of n variables ( n rank) -( ) i f n , here i -is the MBF ordinal number. We shall use a binary form to present functions. Boolean where 2 n t = , i a can take either the value of 0 or 1. At that, we shall write the value of the function taken at the lowest set on the right side, while the value taken at the highest set -on the left side. The sets shall be arranged in the following order: the lowest variable 1 x , on the right, the highest variablen x on the left. For example, we shall take a monotonic function from 3 variables ( ) 1  Definition. MBF block is a set of MBFs completed as regards three operations: duality, disjunctive complement and conjunctive complement, so that any MBF of the block can be obtained from any MBF of the same block by using a certain sequence of these three operations.
For example, for ranks 0 and 1 there exists only one block which consists in the first case of two functions and in the second case -of three functions Proof. It is known that dual function g can be obtained by replacing conjunction operations with disjunction operations in initial f operation of conjunction, subject to preserving the operations priority; and vice versa, if the same replacement of operations is carried out in a dual function, we shall obtain the initial function written down in conjunctive normal form (CDF). It follows here from that conjunctive complement of f function and disjunctive complement of g function shall be dual to one another. The lemma is proved.
Consequence. A block can be defined as a set of MBFs closed in relation to three operations: duality, disjunctive complement and conjunctive complement and such that any MBF of the block from any function of the block by using a certain sequence of only two operations 1 ϕ − and ϕ .
In this way, a block of functions can be constructed of one function f by using to this function successively the operation of disjunctive complement and then the duality operation. Each function can belong to only one block.
We shall introduce some notions. Block potency is the number of MBFs included. Two blocks are similar if they possess equal potency and provided the MBFs included in them are disregarded, they are identical. Two blocks are isomorphic if any MBF of one block can be obtained from a certain MBF of another block by some variables substitution. By convention, isomorphic blocks are similar.
All MBF blocks can be divided into 4 types. It follows from lemma 1 that all MBFs can be connected with a sequence of two operations -duality and disjunctive complement. The sequence of these two operations can be disconnected or cyclic. In the first case obtain three types of blocks. Blocks of first type is called a block, in which at the  (4)  Example of such 4 rank block: Here Computer Science and Information Technology 5(1): 27-36, 2017 3. Containing 2 self-dual MBFs.

Cyclic blocks.
Example of such 4 rank block: Here All functions of the block are connected by three operations.
Blocks consisting of two functions can be of two types, consisting of three functions -of one type, consisting of four functions -of three types, consisting of six functions -of three types: Here Other functions numbers are taken from studies [3] and [4].
The block containing functions ( ) 0 f n and f 1 (n) shall be called base block.
For functions of zero and first rank, there is one block, which is the base (see. Figure 1).
All MBFs of rank of 2 can be presented as 2 blocks, one of them consisting of four MBFs (base) and the other -of two MBFs: Here Base blocks can be of two types: either the first one, or the second one. To be more precise -either with one self-dual and one disjunctive self-complementary for odd ranks, or with two disjunctive self-dual and one function for even ranks. For example, base blocks of 3 and 4 ranks: comes of 1 f by substituting S and, consequently, it is isomorphic in relation to 1 f . Thus, we have obtained two isomorphic functions 1 f and 1 g of two isomorphic functions f and g with the aid of dual operation.
We can see that MBF can be obtained from 2 f by substituting S and, consequently it is isomorphic in relation to 2 f Lemma 4. Suppose function f is given, using some substitution α we shall obtain isomorphic g . If we use any succession of three operations: duality, conjunctive and disjunctive complements to these two isomorphic functions, we shall obtain two isomorphic functions 1 f and 1 g . At that, one of them shall result from the other by substituting α .
Proof. Let us remark that the operation of conjunctive complement is equivalent to the sequence of operations: duality, disjunctive complement and again duality (lemma 1). Successively applying to isomorphic MBFs f and g lemmas 2 and 3 we obtain isomorphic 1 f and 1 g . This shall be more than one function, because each of the three operations is unary and mutually reversible, i.e. an involution, and we cannot obtain different functions out of one function using the same sequence of operations. From lemmas 2, 3 and 4 it follows immediately that: Lemma 5. By applying a sequence of three above mentioned operations to any number of isomorphic functions we shall obtain the same number of isomorphic functions. Theorem 1. Any two MBFs of one block have equal number of isomorphic.
Proof. As a consequence of lemmas we derive that out of two isomorphic MBFs f and g with the aid of any sequence of the three operations described above we also obtain isomorphic MBFs 1 f and 1 g .
Let us choose arbitrary MBFs f and 1 f from block 2.
Suppose that f is more isomorphic than 1 f . By the definition of MBF block, 1 f can be obtained from MBF f by some sequence of operations P . Let us apply the same sequence P to all functions isomorphic to f . According to the proved above, as the result we shall obtain functions isomorphic to 1 f . Consequently, we have obtained the number of isomorphic the number of isomorphic MBFs to 1 f equal to the number of isomorphic MBFs to f , which contradicts to the presumption. The theorem is proved.
Theorem 2. For each MBF in the block the number of isomorphic MBFs in the block is the same. Proof. Suppose an MBF of block f has maximal set of isomorphic functions in the block. We shall designate this set through А. We shall take any other nonisomorphic to f function g . Select the sequence of operations, under which function f turns into function g . According to the block definition, such sequence can be always realized. From function g and all its isomorphic we shall obtain a set В functions with the same sequence of operations. All MBFs from the set B by definition of block are in the same block. It follows from lemma 5 that MBFs in set В shall be isomorphic to one another and their number shall be equal to the number in set А. The theorem is proved.
The number of isomorphic MBFs for one function in the block shall be called a block index. It follows from the definition of isomorphic blocks and from lemmas 2, 3, 4 and 5 that the statement is correct: Consequence 1. Isomorphic blocks have identical index. Consequence 2. Blocks of the first and third type have an index no more than 2. Since the function of self-dual is isomorphic must also be self-dual self-complementary and isomorphic disjunctive must also be self-complementary, and these blocks are only with two of this type of MBF. While blocks of type 2 have index 1, because they have one self-dual and one disjunctive self-complementary and isomorphic to it in the block does not exist. Blocks of the fourth type may be of any index.
Because for any nonisomorphic MBFs of one block, f and g according to theorem 2 possess equal number of isomorphic for each function f and g , consequently, the number of isomorphic for any function of the block is the divider of the number of all MBFs in the block. This is why the block index is the divider of the number of all MBFs in the block. The result of the division is the number of nonisomorphic MBFs in the block. It also follows that if we take all isomorphic MBFs to this function, we shall obtain the number of isomorphic blocks.

Conclusions
With this classification we can define various properties of MBFs.
For example, we shall consider partitioning all MBF blocks of 4 th rank according to such classification. The number of nonisomorphic blocks (or isomorphic blocks groups) of the 4th rank, i.e. the sum of values of the last columns in all 4 tables ( )    The number of inequivalent MBFs of the 4th rank [7] ( ) The same for the 5th rank.   1  4  6  2  1  2  6  5  6  1  3  14  10  1  1  4  14  15  1  1  5  14  15  2  1  6  14  20  1  1  7  14  30  1  2  8  14  30  2  3  9  14  60  1  2  10  14  60  2  1  11  32  30  2  1  12  54  10  6  1 32 Basic MBF Blocks Properties and Rank 6 Blocks Let's calculate for the 5th rank the same expressions as for the 4th rank:    This method is not limited by the number of variables because for each MBF with any number of variables n we can build a block containing this MBF using operations of disjunctive complement and duality. In this case, it is sufficient to build only nonisomorphic blocks to describe all MBFs of n variables. This significantly facilitates the description of all MBFs of n variables. For example, all MBFs of 6 variables can be split into 189182 blocks but only 775 of them can be selected as mutually nonisomorphic. Classes of equivalence and nonisomorphic blocks are directly connected with the Dedekind numbers.