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Optimal Expansion and Characterization of Logic Functions and Classes

Research Project

Project/Area Number 13640136
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionCollege of Liberal Arts, International Christian University

Principal Investigator

POGOSYAN Grant R.  International Christian University, College of Liberal Arts, Professor, 教養学部, 教授 (90234640)

Co-Investigator(Kenkyū-buntansha) TAKAHASHI Masako  International Christian University, College of Liberal Arts, Professor, 教養学部, 教授 (00015588)
Project Period (FY) 2001 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2004: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2003: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2002: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2001: ¥1,400,000 (Direct Cost: ¥1,400,000)
Keywordsboolean functions / optimal expansions / clones / multiple-valued logic / functional terms / logic functions / universal algebra
Research Abstract

The following optimization and characterization problems for functions and classes were in focus :
(1) Optimal Expansion of Boolean and Multiple-Valued Logic Functions : The notion of efficiently irreducible bases (introduced earlier by the head researcher) was used to choose a set of generators for an optimal expansion of logic functions. For Boolean functions and one such generator base, namely AND, XOR NOT, an algorithm for building a shortest, in terms of the number of literals, ESOP (Exclusive Sum of Products) was designed and implemented.
(2) Functional Terms for Characterizing Classes of Logic Functions : A formal expression called ‘functional term' was introduced and used to neatly characterize functional classes. Various properties and criteria for such characterization were obtained. The method was applied to the closed classes of Boolean functions achieving complete description of the Post's lattice in such terms.
(3) Clones and Universal Algebra : Clones (special closed classes of finite-valued logic functions) form an algebraic lattice which is still widely unknown for non-boolean cases. The clones which are irreducible in terms of lattice operations (join and/or meet) were studied. It was shown that the lattice can be fully generated by both join- and meet-irreducible clones. Some special chains of such irreducible clones were described (a joint work with I.G.Rosenberg).

Report

(5 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (32 results)

All 2004 2003 2002 Other

All Journal Article (16 results) Publications (16 results)

  • [Journal Article] Post Classes Characterized by Functional Terms2004

    • Author(s)
      Stephan Foldes, Grant R. Pogosyan
    • Journal Title

      Discrete Applied Mathematics. Elsevier Publishers, North Holland Vol.142

      Pages: 35-51

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Algebraic Properties of Totally Irreducible Elements of Clone Lattices2004

    • Author(s)
      Grant R.Pogosyan, Ivo G.Rosenberg
    • Journal Title

      Proceedings of the 34^th International Symposium on Multiple-Valued Logic. IEEE Press ISMVL-2004

      Pages: 109-114

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Building Minimum ESOPs Through Redundancy Elimination2004

    • Author(s)
      G.Pogosyan, I.Rosenberg, S.Takada
    • Journal Title

      Proceedings of 6th International Workshop on Boolean Problems. WSBP-2004

      Pages: 201-207

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Post Classes Characterized by Functional Terms2004

    • Author(s)
      Stephan Foldes, Grant R.Pogosyan
    • Journal Title

      Discrete Applied Mathematics, Elsevier Publishers (North Holland) Vol.142

      Pages: 35-51

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Algebraic Properties of Totally Irreducible Elements of Clone Lattices2004

    • Author(s)
      Grant R.Pogosyan, Ivo G.Rosenberg
    • Journal Title

      Proceedings of 34^<th> International Symposium on Multiple-Valued Logic (IEEE Press)

      Pages: 109-114

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Building Minimum ESOPs through redundancy Elimination2004

    • Author(s)
      Grant R.Pogosyan, Ivo G.Rosenberg, Seiya Takada
    • Journal Title

      Proceedings of 6^<th> International Workshop in Boolean Problems

      Pages: 201-207

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Post Classes Characterized by Functional Terms2004

    • Author(s)
      Stephan Foldes, Grant R.Pogpsyan
    • Journal Title

      Discrete Applied Mathematics. (Elsevier Publishers, North Holland) Vol.142

      Pages: 35-51

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Algebraic Properties of Totally Irreducible Elements of Clone Lattices2004

    • Author(s)
      Grant R.Pogosyan, Ivo G.Rosenberg
    • Journal Title

      Proceedings of 34rd International Symposium on Multiple-Valued Logic. (IEEE Press) ISMVL-2004

      Pages: 109-114

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Building Minimum ESOPs Through Redundancy Elimination2004

    • Author(s)
      Grant R.Pogosyan, Ivo G.Rosenberg, Seiya Takada
    • Journal Title

      Proceedings of 6th International Workshop on Boolean Problems. (IEEE Press) WSBP-2004

      Pages: 201-207

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On General Recursive Functions2004

    • Author(s)
      Masako Takahashi
    • Journal Title

      Proceedings of the 38th MLG Meeting MLG-2004

      Pages: 14-16

    • Related Report
      2004 Annual Research Report
  • [Journal Article] An Algorithm for Optimal Representation of a Partial Boolean Function as a mod2 Sum of Products2003

    • Author(s)
      Grant R.Pogosyan, lvo G.Rosenberg
    • Journal Title

      Proceedings of 6th International Symposium on Representations and Methodology of Future Computing Technology RM2003

      Pages: 27-34

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] 「研究成果報告書概要(和文)」より2003

    • Author(s)
      Grant R.Pogosyan, lvo G.Rosenberg
    • Journal Title

      Proceedings of the 33^rd International Symposium on Multiple-Valued Logic. IEEE Press ISMVL2003

      Pages: 309-314

    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] An Algorithm for Optimal Representation of a Partial Boolean Function as a mod2 Sum of Products2003

    • Author(s)
      Grant R.Pogosyan, Ivo G.Rosenberg
    • Journal Title

      Proceedings of 6^<th> International Symposium on Representations and Methodology of future Computing Technology

      Pages: 27-34

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Generation of the Post Lattice by Irreducible Clones2003

    • Author(s)
      Grant R.Pogosyan, Ivo G.Rosenberg
    • Journal Title

      Proceedings of 33^<rd> International Symposium on Multiple-Valued Logic (IEEE Press)

      Pages: 309-314

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Classes of Boolean Functions Defined by Functional Terms2002

    • Author(s)
      Grant R.Pogosyan
    • Journal Title

      Multiple Valued Logic. Gordon and Breach Publishers (currently - Journal of Multiple-Valued Logic and Soft Computing. OCP Science). Vol. 7

      Pages: 417-448

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Classes of Boolean Functions Defined by Functional Terms2002

    • Author(s)
      Grant R.Pogosyan
    • Journal Title

      Multiple Valued Logic - An International Journal (Gordon and Breach Publishers) (currently - Journal of Multiple Valued Logic and Soft Computing, OCP Science) Vol.7

      Pages: 417-448

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] Grant R.Pogosyan, Ivo G.Rosenberg: "An Algorithm for Optimal Representation of a Partial Boolean Function as a mod2 Sum of Products"Proceedings of 6th International Symposium on Representations and Methodology of Future Computing Technology. RM2003. 27-34 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Grant R.Pogosyan, Ivo G.Rosenberg: "Generation of the Post Lattice by Irreducible Clones."Proceedings of 33rd International Symposium on Multiple-Valued Logic. IEEE Press. ISMVL2003. 309-314 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Grant R.Pogosyan, Ivo G.Rosenberg, Seiya Takada: "ESOP最小化アルゴリズムについて"多値論理研究ノート 日本多値論理研究会. 第26巻 第1号. 1-11 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Stephan Foldes, Grant R.Pogosyan: "Post Classes Characterized by Functional Terms"Discrete Applied Mathematics. Elsevier Publishers, North Holland. (to appear). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Grant R.Pogosyan, Ivo G.Rosenberg: "Algebraic Properties of Totally Irreducible Elements of Clone Lattices"Proceedings of 34rd International Symposium on Multiple-Valued Logic. IEEE Press. (to appear in May). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] 高橋 正子: "プログラムの起源を探る"数学教育学会誌. 秋季臨時増刊号. 78-80 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 高橋 正子(分担執筆): "「岩波 数学辞典」第4版"岩波書店(12月刊行予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Grant R.Pogosyan: "Classes of Boolean Functions Defined by Functional Terms"Multiple Valued Logic, Gordon and Breach Publishers. Vol.7. 417-448 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Grant R.Pogosyan, Ivo G.Rosenberg: "Gaussian Eliminations in GF(2) to construct an Optimal ESOP Expression for a Partial Boolean Function"ICU Science Research Reports. MI-0203. 1-20 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Grant R.Pogosyan, Ivo G.Rosenberg: "Irreducible Clones as Generators of the Post Lattice"ICU Science Research Reports. MI-0301. 1-11 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Grant R.Pogosyan, Ivo G.Rosenberg: "An Algorithm for Optimal Representation of a Partial Boolean Function as a mod2 Sum of Products"Proceedings of 6th International Symposium on Representations and Methodology of Future Computing Technology. RM2003(to appear in March 2003). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Stephan Foldes, Grant R.Pogosyan: "Post Classes Characterized by Functional Terms"Discrete Mathematics, Elsevier Publishers, North Holland. (to appear). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Grant R.Pogosyan, Ivo G.Rosenberg: "Generation of the Post Lattice by Irreducible Clones"Proceedings of 33rd International Symposium on Multiple-Valued Logic, IEEE Press. ISMVL2003(to appear in May 2003). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Grant R. Pogosyan: "Classes of Boolean Functions Defined by Functional Terms"Multiple V alued Logic. An International Journal. Gordon and Breach Publishers.. Vol. 8(to appear). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Masako Takahashi: "Lambda- representable functions over term algebras"International Journal of Foundations of Computer Science. 12. 3-29 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Dezani-Ciancaglini, M.Okada, M.Takahashi (co-editors): "Theories of Types and Proofs"Theoretical Computer Science, 272, Elsevier Science. 397 (2002)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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