Characterization of Multiple-Valued Logic for Dealing with Ambiguity

Noboru Takagi

doi:10.20965/jaciii.2013.p0377

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JACIII Vol.17 No.3 pp. 377-385

doi: 10.20965/jaciii.2013.p0377

(2013)

Paper:

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Characterization of Multiple-Valued Logic for Dealing with Ambiguity

Noboru Takagi

Department of Intelligent Systems Design Engineering, Toyama Prefectural University, 5180 Kurokawa Imizu-shi, Toyama 939-0398, Japan

Received:

July 30, 2012

Accepted:

February 14, 2013

Published:

May 20, 2013

Keywords:

multiple-valued logic, fuzzy logic, ambiguity, monotone function, hazard detection

Abstract

This paper focuses on r-valued functions monotonic in a partial order relation, which can be interpreted as a relation expressing something about ambiguity. First, necessary and sufficient conditions for an r-valued function to be monotonic in the relation are clarified. After that, minimal and maximal information loss functions are defined as a special part of r-valued functions monotonic in the relation. We describe a practical application of minimal information loss functions that correct input failures in multiple-valued logic circuits.

Cite this article as:

N. Takagi, “Characterization of Multiple-Valued Logic for Dealing with Ambiguity,” J. Adv. Comput. Intell. Intell. Inform., Vol.17 No.3, pp. 377-385, 2013.

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References

[1] S. C. Kleene, “Introduction to Metamathematics,” North-Holland Pub., pp. 332-340, 1952.
[2] M. Mukaidono, “Regular Ternary Logic Functions – Ternary Logic Functions Suitable for Treating Ambiguity,” IEEE Trans. on Computers, Vol.C-35, No.2, pp. 179-183, 1986.
[3] M. Mukaidono, “On the Mathematical Structure of the C-type Fail Safe Logic,” IECE Trans., Vol.52-C, No.12, pp. 812-819, 1969.
[4] M. Mukaidono, “The B-ternary logic and its application to the detection of hazards in combinational switching circuits,” Proc. of the 8th Int. Symposium on Multiple-Valued Logic, IEEE, pp. 269-275, 1978.
[5] Y. Yamamoto and M. Mukaidono, “P-functions – ternary logic functions capable of correcting input failures and suitable for treating ambiguities,” IEEE Trans. on Computers, Vol.41, No.1, pp. 28-35, 1992.
[6] M. Mukaidono and I. G. Rosenberg, “k-valued Function for Treating Ambiguities – Their Clone and a Normal Form,” Proc. of the 16th Int. Symposium onMultiple-Valued Logic, IEEE, pp. 204-211, 1986.
[7] I. G. Rosenberg, “Completeness properties of multiple-valued logic algebras,” D. C. Rine (Ed.), Computer Science and Multiple-Valued Logic, North-Holland, pp. 150-192, 1984.
[8] E. L. Post, “Introduction to a general theory of elementary propositions,” American J. of Mathematics, Vol.43, pp. 163-185, 1921.
[9] K. Ibuki, K. Naemura, and A. Nozaki, “The general theory of complete sets of logical functions,” IECE Trans., Vol.46, No.7, pp. 42-48, 1963.
[10] J. C. Muzio and T. C. Wesselkamper, “Multiple-Valued Switching Theory,” Adam Hilger Ltd., pp. 63-100, 1986.
[11] G. Epstein, “Multiple-Valued Logic Design: An Introduction,” Institute of Physics Publishing, pp. 124-153, 1993.
[12] J. A. Brzozowski and C. H. Seger, “Asynchronous Circuits,” Springer-Verlag, 1995.
[13] J. A. Brzozowski and Z. Esik, “Hazard algebras,” Formal Methods in System Design, Vol.23, Issue 3, pp. 233-256, 2003.
[14] N. Takagi, “A delay model of multiple-valued logic circuits consisting of min, max, and literal operations,” IEICE Trans. on Information and Systems, Vol.E93-D, No.8, pp. 2040-2047, 2010.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.

[1] [1] S. C. Kleene, “Introduction to Metamathematics,” North-Holland Pub., pp. 332-340, 1952.

[2] [2] M. Mukaidono, “Regular Ternary Logic Functions – Ternary Logic Functions Suitable for Treating Ambiguity,” IEEE Trans. on Computers, Vol.C-35, No.2, pp. 179-183, 1986.

[3] [3] M. Mukaidono, “On the Mathematical Structure of the C-type Fail Safe Logic,” IECE Trans., Vol.52-C, No.12, pp. 812-819, 1969.

[4] [4] M. Mukaidono, “The B-ternary logic and its application to the detection of hazards in combinational switching circuits,” Proc. of the 8th Int. Symposium on Multiple-Valued Logic, IEEE, pp. 269-275, 1978.

[5] [5] Y. Yamamoto and M. Mukaidono, “P-functions – ternary logic functions capable of correcting input failures and suitable for treating ambiguities,” IEEE Trans. on Computers, Vol.41, No.1, pp. 28-35, 1992.

[6] [6] M. Mukaidono and I. G. Rosenberg, “k-valued Function for Treating Ambiguities – Their Clone and a Normal Form,” Proc. of the 16th Int. Symposium onMultiple-Valued Logic, IEEE, pp. 204-211, 1986.

[7] [7] I. G. Rosenberg, “Completeness properties of multiple-valued logic algebras,” D. C. Rine (Ed.), Computer Science and Multiple-Valued Logic, North-Holland, pp. 150-192, 1984.

[8] [8] E. L. Post, “Introduction to a general theory of elementary propositions,” American J. of Mathematics, Vol.43, pp. 163-185, 1921.

[9] [9] K. Ibuki, K. Naemura, and A. Nozaki, “The general theory of complete sets of logical functions,” IECE Trans., Vol.46, No.7, pp. 42-48, 1963.

[10] [10] J. C. Muzio and T. C. Wesselkamper, “Multiple-Valued Switching Theory,” Adam Hilger Ltd., pp. 63-100, 1986.

[11] [11] G. Epstein, “Multiple-Valued Logic Design: An Introduction,” Institute of Physics Publishing, pp. 124-153, 1993.

[12] [12] J. A. Brzozowski and C. H. Seger, “Asynchronous Circuits,” Springer-Verlag, 1995.

[13] [13] J. A. Brzozowski and Z. Esik, “Hazard algebras,” Formal Methods in System Design, Vol.23, Issue 3, pp. 233-256, 2003.

[14] [14] N. Takagi, “A delay model of multiple-valued logic circuits consisting of min, max, and literal operations,” IEICE Trans. on Information and Systems, Vol.E93-D, No.8, pp. 2040-2047, 2010.