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Diagnostic tests under shifts with fixed filling tuple

  • Grigorii V. Antiufeev EMAIL logo
Veröffentlicht/Copyright: 13. Oktober 2021
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Discrete Mathematics and Applications
Aus der Zeitschrift Discrete Mathematics and Applications Band 31 Heft 5

Abstract

We consider a fault source under which the fault functions are obtained from the original function f(x̃n) ∈ P2n by a left shift of values of the Boolean variables by at most n. For the vacant positions of the variables, the values are selected from a given filling tuple γ̃ = (γ1, γ2, …, γn) ∈ E2n , which also moves to the left by the number of positions corresponding to a specific fault function. The problem of diagnostic of faults of this kind is considered. We show that the Shannon function Lγ~shifts,diagn(n), which is equal to the smallest necessary test length for diagnostic of any n-place Boolean function with respect to a described fault source, satisfies the inequality n2≤Lγ~shifts,diagn(n)≤n.

Keywords: shifts; tests; the Shannon function

Originally published in Diskretnaya Matematika (2020) 32,№4, 3–9 (in Russian).

References

[1] Antiufeev G., “On a property of Boolean functions, which guarantees the existence of logarithmical diagnostic tests in case of variable shifting”, Problems of radio electronics. Ser. “Electronic computing technology (EVT)”, 2014, N 4,71-77 (in Russian).Suche in Google Scholar

[2] Antiufeev G., “Shift-fault detection and diagnosis in Boolean functions”, Problems of radio electronics. Ser. “Electronic computing technology (EVT)”, 2015, N 2,14-17 (in Russian).Suche in Google Scholar

[3] Lozhkin C. A., Lectures on fundamentals of cybernetics: tutorial, Publ. dep. CMC faculty of Lomonosov Moscow State University, Moscow, 2004 (in Russian), 256 pp.Suche in Google Scholar

[4] Romanov D. S., Antyufeev G. V., “About tests for primitive shifts of variables in Boolean functions”, Problems of radio electronics. Ser. “Electronic computing technology (EVT), 2013, N 2, 64-68.Suche in Google Scholar

[5] Shur A. M., Combinatorics on words, Izd-vo Ural'skogo un-ta, Ekaterinburg, 2003 (in Russian), 96 pp.Suche in Google Scholar

[6] Fine N. J., Wilf H. S., “Uniqueness theorem for periodic functions”, Proc. Amer. Math. Soc., 16:1 (1965), 109-114.10.1090/S0002-9939-1965-0174934-9Suche in Google Scholar
Received: 2019-10-11
Published Online: 2021-10-13
Published in Print: 2021-10-26

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Diagnostic tests under shifts with fixed filling tuple
  3. On the average-case complexity of Boolean functions under binomial distribution on their domains
  4. Boolean analogues of the Pascal triangle with maximal possible number of ones
  5. The limited deficit method and the problem of constructing orthomorphisms and almost orthomorphisms of Abelian groups
  6. On the complexity of monotone circuits for threshold symmetric Boolean functions
  7. Properties of multitype subcritical branching processes in random environment
https://doi.org/10.1515/dma-2021-0027
Schlagwörter für diesen Artikel
shifts; tests; the Shannon function

Artikel in diesem Heft

  1. Frontmatter
  2. Diagnostic tests under shifts with fixed filling tuple
  3. On the average-case complexity of Boolean functions under binomial distribution on their domains
  4. Boolean analogues of the Pascal triangle with maximal possible number of ones
  5. The limited deficit method and the problem of constructing orthomorphisms and almost orthomorphisms of Abelian groups
  6. On the complexity of monotone circuits for threshold symmetric Boolean functions
  7. Properties of multitype subcritical branching processes in random environment
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