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Fuzzy machines in a category

Published online by Cambridge University Press:  17 April 2009

Michael A. Arbib  and
Ernest G. Manes
Michael A. Arbib
Affiliation:
Department of Computer and Information Science, University of Massachusetts, Amherst, Massachusetts, USA
Ernest G. Manes
Affiliation:
Department of Mathematics, University of Massachusetts, Amherst, Massachusetts, USA.
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Abstract

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“Fuzzy theories” and “distributive laws” are used to define “fuzzy systems” in an arbitrary category. The resulting minimal realization theory provides new insights even in classical cases (so that, for non-deterministic sequential machines, the minimal realization problem is reformulated in terms of the structure of join-irreducibles in finite lattices). The definition of “fuzzy theory” is of independent interest and meshes well with philosophical aspects of fuzzy set theory.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 13 , Issue 2 , October 1975 , pp. 169 - 210
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Arbib, Michael A. and Manes, Ernest G., “Machines in a category: an expository introduction”, SIAM Rev. 16 (1974), 163–192.CrossRefGoogle Scholar
[2]Arbib, Michael A., Manes, Ernest G., Arrows, structures, and functors. The categorical imperative (Academic Press, New York, San Francisco, London, 1975).Google Scholar
[3]Arbib, M.A., Manes, E.G., “A categorist's view of automata and systems”, Category theory applied to computation and control, 51–64 (Lecture Notes in Computer Science, 25. Springer-Verlag, Berlin, Heidelberg, New York, 1975).CrossRefGoogle Scholar
[4]Arbib, Michael A., Manes, Ernest G., “Adjoint machines, state-behavior machines, and duality”, J. Pure Appl. Algebra (to appear).Google Scholar
[5]Bainbridge, Edwin Stewart, “A unified minimal realization theory with duality (PhD thesis, University of Michigan, Michigan, 1972).Google Scholar
[6]Beck, Jon, “Distributive laws”, Seminar on triples and categorical homology theory, 119–140 (Lecture Notes in Mathematics, 80. Springer-Verlag, Berlin, Heidelberg, New York, 1969).CrossRefGoogle Scholar
[7]Burroni, Élisabeth, “Algèbres relatives à une loi distributive”, C.R. Acad. Sci. Paris Sér. A 276 (1973), 443–446.Google Scholar
[8]Ehrig, H., Kiermeier, K.-D., Kreowski, H.-J. and Kühnel, W., Universal theory of automata. A categorical approach (Teubner, Stuttgart, 1974).CrossRefGoogle Scholar
[9]Goguen, J.A., “L-fuzzy sets”, J. Math. Anal. Appl. 18 (1967), 145–174.CrossRefGoogle Scholar
[10]Goguen, J.A., “Minimal realization of machines in closed categories”, Bull. Amer. Math. Soc. 78 (1972), 777–783.CrossRefGoogle Scholar
[11]Goguen, J.A., “Concept representation in natural and artificial languages: axioms, extensions and applications for fuzzy sets”, Internat. J. Man-Machine Studies 6 (1974), 513–561.CrossRefGoogle Scholar
[12]Kleisli, H., “Every standard construction is induced by a pair of adjoint functors”, Proc. Amer. Math. Soc. 16 (1965), 544–546.CrossRefGoogle Scholar
[13]Lane, S. Mac, Categories for the working mathematician (Graduate Texts in Mathematics, 5. Springer-Verlag, New York, Heidelberg, Berlin, 1971).CrossRefGoogle Scholar
[14]Manes, Ernest G., Algebraic theories (Graduate Texts in Mathematics. Springer-Verlag, New York, Heidelberg, Berlin, to appear).Google Scholar
[15]Rabin, M.O., Scott, D., “Finite automata and their decision problems”, IBM J. Res. Develop. 3 (1959), 114–125.CrossRefGoogle Scholar
[16]Raney, George N., “Sequential functions”, J. Assoc. Comput. Mach. 5 (1958), 177–180.CrossRefGoogle Scholar
[17]Schützenberger, M.P., “On a theorem of R. Jungen”, Proc. Amer. Math. Soc. 13 (1962), 885–890.CrossRefGoogle Scholar
[18]Zadeh, L.A., “Fuzzy sets”, Inform. Control 8 (1965), 338–353.CrossRefGoogle Scholar