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On the non-monotone complexity of Boolean and k-value logic functions

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Název: On the non-monotone complexity of Boolean and k-value logic functions
Autoři: Vadim Vasilievich Kochergin, Anna Vitalievna Mihajlovich
Zdroj: Mathematical Problems of Cybernetics. :51-151
Informace o vydavateli: Keldysh Institute of Applied Mathematics, 2024.
Rok vydání: 2024
Popis: The paper discusses major advances in various extensions of inversion complexity problem accomplished in the past ten years. In the 1950s, A.A. Markov solved the problem of inversion complexity for Boolean functions. He determined the exact number of inverters in logic circuits of a special type for an arbitrary Boolean function. Precisely, these circuits contained inverters with unit weight and monotone functions with zero weights. In the paper we present, among other findings: the exact value of non-monotone complexity for any Boolean function; the exact value of inversion complexity for k-valued logic functions over a basis containing only monotone functions along with either Post or Lucasiewicz negations; upper and lower bounds of the non-monotone complexity for every k-valued logical function. The difference between these upper and lower bounds is constant and independent of the basis used. We also touch upon the problem of Boolean function circuit complexity over infinite bases. First, we present a sufficiently thorough overview of this problem. Second, we determine the exact value of complexity for an arbitrary Boolean function over a basis that consists of negation and all monotone functions.
Druh dokumentu: Article
DOI: 10.20948/mvk-2024-51
Rights: CC BY
Přístupové číslo: edsair.doi...........dcc2c38d0a27dd65606c79c50fe7bccc
Databáze: OpenAIRE
Popis
Abstrakt:The paper discusses major advances in various extensions of inversion complexity problem accomplished in the past ten years. In the 1950s, A.A. Markov solved the problem of inversion complexity for Boolean functions. He determined the exact number of inverters in logic circuits of a special type for an arbitrary Boolean function. Precisely, these circuits contained inverters with unit weight and monotone functions with zero weights. In the paper we present, among other findings: the exact value of non-monotone complexity for any Boolean function; the exact value of inversion complexity for k-valued logic functions over a basis containing only monotone functions along with either Post or Lucasiewicz negations; upper and lower bounds of the non-monotone complexity for every k-valued logical function. The difference between these upper and lower bounds is constant and independent of the basis used. We also touch upon the problem of Boolean function circuit complexity over infinite bases. First, we present a sufficiently thorough overview of this problem. Second, we determine the exact value of complexity for an arbitrary Boolean function over a basis that consists of negation and all monotone functions.
DOI:10.20948/mvk-2024-51