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Complexity of systems of functions of Boolean algebra and systems of functions of three-valued logic in classes of polarized polynomial forms

  • Svetlana N. Selezneva EMAIL logo
Published/Copyright: April 26, 2016
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Discrete Mathematics and Applications
From the journal Discrete Mathematics and Applications Volume 26 Issue 2

Abstract

A polarized polynomial form (PPF) (modulo k) is a modulo k sum of products of variables x1, . . . , xn or their Post negations, where the number of negations of each variable is determined by the polarization vector of the PPF. The length of a PPF is the number of its pairwise distinct summands. The length of a function f(x1, . . . , xn)of k-valued logic in the class of PPFs is the minimum length among all PPFs realizing the function. The paper presents a sequence of symmetric functions fn(x1, . . . , xn)of three-valued logic such that the length of each function fn in the class of PPFs is not less than ⌊3n+1/4⌋, where ⌊a⌋ denotes the greatest integer less or equal to the number a. The complexity of a system of PPFs sharing the same polarization vector is the number of pairwise distinct summands entering into all of these PPFs. The complexity LkPPF(F) of a system F ={f1,..., fm} of functions of k-valued logic depending on variables x1,..., xn in the class of PPFs is the minimum complexity among all systems of PPFs {p1,...,pm}such that all PPFs p1,...,pm share the same polarization vector and the PPF pj realizes the function fj, j = 1,...,m. Let LkPPF(m,n) = maxFL2PPF(F), where F runs through all systems consisting of m functions of k-valued logic depending on variables x1,..., xn. For prime values of k it is easy to derive the estimate LkPPF(m,n) ≤ kn. In this paper it is shown that LkPPF(m,n) = 2n and LkPPF(m,n) = 3n for all m ≥ 2, n= 1, 2, . . . Moreover, it is demonstrated that the estimates remain valid when consideration is restricted to systems of symmetric functions only.

Keywords: Boolean function; function of three-valued logic; function of k-valued logic; polarized polynomial form (PPF); complexity; upper estimate; lower estimate

Originally published in Diskretnaya Matematika (2015) 27,No1, 118–130 (in Russian).

Funding source: Russian Foundation for Basic Research

Award Identifier / Grant number: 13-01-00684-a

Funding statement: This work was supported by the Russian Foundation for Basic Research, projects no. 13-01-00684-a and 13-01-00958-a.

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Received: 2014-10-9
Published Online: 2016-4-26
Published in Print: 2016-4-1

2016 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/dma-2016-0009
Keywords for this article
Boolean function; function of three-valued logic; function of k-valued logic; polarized polynomial form (PPF); complexity; upper estimate; lower estimate

Articles in the same Issue

  1. Frontmatter
  2. Article
  3. Functional limit theorems for the decomposable branching process with two types of particles
  4. Article
  5. Completeness problem for the class of linear automata functions
  6. Article
  7. Estimates of accuracy of the Poisson approximation for the distribution of number of runs of long string repetitions in a Markov chain
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  9. Complexity of systems of functions of Boolean algebra and systems of functions of three-valued logic in classes of polarized polynomial forms
  10. Article
  11. A Markov chain with number-theoretic limit distribution
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