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RDNF Oriented Analytics to Random Boolean Functions

Levon Aslanyan, Irina Arsenyan, Vilik Karakhanyan, Hasmik Sahakyan

TL;DR

The paper addresses estimating the average complexity of the reduced disjunctive normal form for Boolean functions defined on the hypercube, focusing on maximal $k$-dimensional intervals. It develops exact expressions for $r_k(n,p)$ under the $F_p$ distribution and conducts asymptotic analysis of $i_k(n,p)$ to reveal how interval counts scale with $n$ and $k$. The results clarify when larger-dimensional intervals become likely or rare, informing Boolean-function minimization, logic-based pattern recognition, and related computational tasks. By linking hypercube geometry, probabilistic combinatorics, and r.d.n.f. representations, the work provides analytical tools for constraint logic programming and decision-making domains.

Abstract

Dominant areas of computer science and computation systems are intensively linked to the hypercube-related studies and interpretations. This article presents some transformations and analytics for some example algorithms and Boolean domain problems. Our focus is on the methodology of complexity evaluation and integration of several types of postulations concerning special hypercube structures. Our primary goal is to demonstrate the usual formulas and analytics in this area, giving the necessary set of common formulas often used for complexity estimations and approximations. The basic example under considered is the Boolean minimization problem, in terms of the average complexity of the so-called reduced disjunctive normal form (also referred to as complete, prime irredundant, or Blake canonical form). In fact, combinatorial counterparts of the disjunctive normal form complexities are investigated in terms of sets of their maximal intervals. The results obtained compose the basis of logical separation classification algorithmic technology of pattern recognition. In fact, these considerations are not only general tools of minimization investigations of Boolean functions, but they also prove useful structures, models, and analytics for constraint logic programming, machine learning, decision policy optimization and other domains of computer science.

RDNF Oriented Analytics to Random Boolean Functions

TL;DR

The paper addresses estimating the average complexity of the reduced disjunctive normal form for Boolean functions defined on the hypercube, focusing on maximal -dimensional intervals. It develops exact expressions for under the distribution and conducts asymptotic analysis of to reveal how interval counts scale with and . The results clarify when larger-dimensional intervals become likely or rare, informing Boolean-function minimization, logic-based pattern recognition, and related computational tasks. By linking hypercube geometry, probabilistic combinatorics, and r.d.n.f. representations, the work provides analytical tools for constraint logic programming and decision-making domains.

Abstract

Dominant areas of computer science and computation systems are intensively linked to the hypercube-related studies and interpretations. This article presents some transformations and analytics for some example algorithms and Boolean domain problems. Our focus is on the methodology of complexity evaluation and integration of several types of postulations concerning special hypercube structures. Our primary goal is to demonstrate the usual formulas and analytics in this area, giving the necessary set of common formulas often used for complexity estimations and approximations. The basic example under considered is the Boolean minimization problem, in terms of the average complexity of the so-called reduced disjunctive normal form (also referred to as complete, prime irredundant, or Blake canonical form). In fact, combinatorial counterparts of the disjunctive normal form complexities are investigated in terms of sets of their maximal intervals. The results obtained compose the basis of logical separation classification algorithmic technology of pattern recognition. In fact, these considerations are not only general tools of minimization investigations of Boolean functions, but they also prove useful structures, models, and analytics for constraint logic programming, machine learning, decision policy optimization and other domains of computer science.
Paper Structure (5 sections, 2 theorems, 11 equations, 3 figures)

This paper contains 5 sections, 2 theorems, 11 equations, 3 figures.

Table of Contents

  1. Hypercube and Related Structures
  2. Concepts and definitions in the binary domain
  3. On the Maximum Number of $k$-Dimensional Maximal Intervals of RBF
  4. On the Dependency of Number of $k$-Dimensional Maximal Intervals on $k$
  5. Conclusion

Key Result

Theorem 1

$r_k(n,p)$ is a concave function of the parameter $k$ in the interval $[0,n]$.

Figures (3)

  • Figure 1: Geometry of hypercube
  • Figure 2: This figure presents the bipartite graph of functions and $k$-dimensional maximal intervals. Upper line functions are placed in order of the number of their "true" values, from $0$ to $2^k.$ Different functions include different numbers of $k$-dimensional maximal intervals and have different probabilities under the distribution $\ F_p. \ $Instead, each interval presented in the bottom line is connected to the same number of functions. This is because the sizes of intervals is the same. The order of intervals is by groups of intervals, that belong to the same direction. Numeration inside the functions with the same number of "ones" and inside the groups of intervals of the same direction is arbitrary.
  • Figure 4: Differential of growing $r_k(n,p).$

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2