Blocks of Monotone Boolean Functions of Rank 5

Tkachenco V.G,Sinyavsky O.V

doi:10.13189/csit.2016.040402

Abstract

Based on the classification of monotone Boolean functions (MBFs) on the types and the method of building MBFs blocks, an analysis is conducted of the MBFs rank 5. Four matrixes for such MBFs are adduced. It is shown that there are 7581 MBFs rank 5, 276 of them are MBFs of maximal types. These 7581 MBFs are contained in 522 blocks or 23 groups of isomorphic blocks or 6 groups of similar blocks. The offered methods can be used to analyze large MBFs ranks. In previous articles were shown how MBFs used in telecommunications for analyzing networks and building codes for cryptosystems.

Highlights

For each i in minimal disjunctive forms [7] of the monotone Boolean functions (MBFs) contains the same number of conjunctions, which include i variables
The Dedekind numbers count the number of monotone Boolean function as well as the number of antichains or the number of Sperner families
In [7] type of monotone Boolean functions (MBFs) is defined as a vector Т = (a0, a1,...,ai,...,an) of n + 1-th component, which are numbered from left to right from 0 to n, where the i-th component of the vector ai equal to the number of input sets, the MBFs, containing i units

Summary

Introduction

For each i in minimal disjunctive forms [7] of the MBFs contains the same number of conjunctions, which include i variables. In the block 4.181 we have: 1) f3141(5) = x1x2 ∨ x1x3 x1x4x5 ∨ x2x3x4 (0,0,2,2,0,0); 2) f3142(5) = x1x5 ∨ x2x3 ∨ x2x4x5 ∨ x3x4x5 (0,0,2,2,0,0); 3) f3143(5) = x2x5 ∨ x3x5 ∨ x1x2x3 ∨ x1x2x4 ∨ x1x3x4 (0,0,2,3,0,0); 4) f3144(5) = x2x5 ∨ x3x5 ∨ x4x5 ∨ x1x2x4 ∨ x1x3x4 (0,0,3,2,0,0); 5) f3145(5) = x1x5 ∨ x4x5 ∨ x2x3x4 ∨ x2x3x5 (0,0,2,2,0,0); 6) f3146(5) = x1x4 ∨ x1x5 ∨ x1x2x3 ∨ x2x3x4 (0,0,2,2,0,0); 7) f3147(5) = x1x2 ∨ x1x3 ∨ x1x4 ∨ x2x4x5 ∨ x3x4x5 (0,0,3,2,0,0); 8) f3148(5) = x1x2 ∨ x1x3 ∨ x2x3x5 ∨ x2x4x5 ∨ x3x4x5 (0,0,2,3,0,0); 9) f3149(5) = x1x5 ∨ x2x3 ∨ x1x2x4 ∨ x1x3x4 (0,0,2,2,0,0); 10) f3150(5) = x2x5 ∨ x3x5 ∨ x1x4x5 ∨ x2x3x4 (0,0,2,2,0,0); 11) f3151(5) = x2x5 ∨ x3x5 ∨ x4x5 ∨ x1x2x3 ∨ x2x3x4(0,0,3,2,0,0); 12) f3152(5) = x1x5 ∨ x4x5 ∨ x1x2x3 ∨ x1x2x4 ∨ x1x3x4 (0,0,2,3,0,0); 13) f3153(5)=x1x4 ∨ x1x5 ∨ x2x3x5 ∨ x2x4x5 ∨ x3x4x5 (0,0,2,3,0,0); 14) f3154(5) = x1x2 ∨ x1x3 ∨ x1x4 ∨ x2x3x4 ∨ x2x3x5 (0,0,3,2,0,0).The ninth group contains 420 MBFs f3141(5)...f3560(5).

Results

Conclusion