2004 Fiscal Year Final Research Report Summary
Optimal Expansion and Characterization of Logic Functions and Classes
| Project/Area Number |
13640136
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | College of Liberal Arts, International Christian University |
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Principal Investigator |
POGOSYAN Grant R. International Christian University, College of Liberal Arts, Professor, 教養学部, 教授 (90234640)
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| Co-Investigator(Kenkyū-buntansha) |
TAKAHASHI Masako International Christian University, College of Liberal Arts, Professor, 教養学部, 教授 (00015588)
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| Project Period (FY) |
2001 – 2004
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| Keywords | boolean functions / optimal expansions / clones / multiple-valued logic / functional terms / logic functions / universal algebra |
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| 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).
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