Abstract

Fuzzy inference is a method of reasoning with imprecise information. The mathematical operations of fuzzy inference can be stated in terms of generalized vector algebra, in which multiplication and summation are generalized to min and max operations. An optoelectronic H-tree architecture is ideally suited to perform these generalized vector operations in parallel and requires only a simple imaging optical interconnection. Appropriate data encodings and electronic circuitry permit large scale, pipe-lined systems.

© 1996 Optical Society of America

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References

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  1. L. A. Zadeh, “Fuzzy sets,” Inf. Control 6, 338–353 (1965).
    [CrossRef]
  2. G. J. Klir, T. A. Folger, Fuzzy Sets, Uncertainty, and Information (Prentice-Hall, Englewood Cliffs, N.J., 1988).
  3. M. Sugeno, “An introductory survey of fuzzy control,” Inf. Sci. 36, 59–83 (1985).
    [CrossRef]
  4. A. Kandel, Fuzzy Techniques in Pattern Recognition (Wiley, New York, 1982).
  5. L. A. Zadeh, “The role of fuzzy logic in the management of uncertainty in expert systems,” Fuzzy Sets Syst. 11, 199–227 (1983).
    [CrossRef]
  6. M. Togai, H. Watanabe, “Expert system on a chip: an engine for real-time approximate reasoning,” IEEE Expert 1, 56–62 (1986).
    [CrossRef]
  7. M. A. Eshera, S. C. Barash, “Parallel rule-based fuzzy inference on mesh-connected systolic arrays,” IEEE Expert 4, 27–35 (1989).
    [CrossRef]
  8. H. Watanabe, J. R. Symon, W. D. Detloff, K. E. Young, “VLSI fuzzy chip and inference accelerator board systems,” in Fuzzy Logic for the Management of Uncertainty, L. Zadeh, J. Kacprzyk, eds. (Wiley, New York, 1992).
  9. L. Liu, “Optical implementation of parallel fuzzy logic,” Opt. Commun. 73, 183–187 (1989).
    [CrossRef]
  10. G. C. Marsden, B. Olson, S. C. Esener, S. H. Lee, “Optoelectronic fuzzy logic system,” in Optical Computing, Vol. 6 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991) p. 212.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  15. T. Konishi, J. Tanida, Y. Ichioka, “Visual-area coding technique (VACT): optical parallel implementation of fuzzy logic and its visualization with the digital-halftoning process,” Appl. Opt. 34, 3097–3102 (1995).
    [CrossRef] [PubMed]
  16. R. Lopez de Mantaras, Approximate Reasoning Models (Halsted, New York, 1990).
  17. G. C. Marsden, F. Kiamilev, S. Esener, S. H. Lee, “Highly parallel consistent labeling algorithm suitable for optoelectronic implementation,” Appl. Opt. 30, 185–194 (1991).
    [CrossRef] [PubMed]
  18. L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning—I,” Inf. Sci. 8, 199–249 (1975).
    [CrossRef]
  19. L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning—II,” Inf. Sci. 8, 301–357 (1975).
    [CrossRef]
  20. L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning—III,” Inf. Sci. 9, 43–80 (1975).
    [CrossRef]
  21. A. V. Krishnamoorthy, G. Yayla, S. C. Esener, “A scalable neural system using free-space optical interconnects,” IEEE Trans. Neural Net. 3, 404–413 (1992).
    [CrossRef]
  22. S. Esener, F. McCormick, “Current status of optical storage systems,” in Conference on Lasers and Electro-Optics and Quantum Electronics and Laser Science, Vols. 15 and 16 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995).
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    [CrossRef]
  24. G. C. Marsden, “Optoelectronic array processors with applications in machine intelligence and database management,” Ph. D. dissertation (University of California, San Diego, La Jolla, Calif., 1993).
  25. C. Fan, B. Mansoorian, D. A. Van Blerkom, M. W. Hansen, V. H. Ozguz, S. C. Esener, G. C. Marsden, “Digital free-space optical interconnections: a comparison of transmitter technologies,” Appl. Opt. 34, 3103–3115 (1995).
    [CrossRef] [PubMed]
  26. M. Hansen, D. Shih, C. Fan, S. Esener, W. Cheng, E. Yablonovitch, U. Efron, “16×16 SLM with silicon CMOS drivers and III-V modulators,” in Spatial Light Modulators and Applications, Vol. 9 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 5.
  27. S. Esener, “Optical memory systems for 3-D computing,” presented at the Lasers and Electro-Optics Society (LEOS) Annual Meeting, San Francisco, Calif., 16 November 1995.
  28. P. J. Marchand, A. V. Krishnamoorthy, K. S. Urquhart, P. Ambs, S. C. Esener, S. H. Lee, “Motionless-head parallel readout optical disk,” Appl. Opt. 32, 190–203 (1993).
    [CrossRef] [PubMed]

1995 (2)

1994 (1)

1993 (4)

1992 (1)

A. V. Krishnamoorthy, G. Yayla, S. C. Esener, “A scalable neural system using free-space optical interconnects,” IEEE Trans. Neural Net. 3, 404–413 (1992).
[CrossRef]

1991 (1)

1989 (2)

M. A. Eshera, S. C. Barash, “Parallel rule-based fuzzy inference on mesh-connected systolic arrays,” IEEE Expert 4, 27–35 (1989).
[CrossRef]

L. Liu, “Optical implementation of parallel fuzzy logic,” Opt. Commun. 73, 183–187 (1989).
[CrossRef]

1986 (1)

M. Togai, H. Watanabe, “Expert system on a chip: an engine for real-time approximate reasoning,” IEEE Expert 1, 56–62 (1986).
[CrossRef]

1985 (1)

M. Sugeno, “An introductory survey of fuzzy control,” Inf. Sci. 36, 59–83 (1985).
[CrossRef]

1983 (1)

L. A. Zadeh, “The role of fuzzy logic in the management of uncertainty in expert systems,” Fuzzy Sets Syst. 11, 199–227 (1983).
[CrossRef]

1979 (1)

C. Mead, M. Rem, “Cost and performance of VLSI computing structures,” IEEE J. Solid-State Circuits SC-14, 455–462 (1979).
[CrossRef]

1975 (3)

L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning—I,” Inf. Sci. 8, 199–249 (1975).
[CrossRef]

L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning—II,” Inf. Sci. 8, 301–357 (1975).
[CrossRef]

L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning—III,” Inf. Sci. 9, 43–80 (1975).
[CrossRef]

1965 (1)

L. A. Zadeh, “Fuzzy sets,” Inf. Control 6, 338–353 (1965).
[CrossRef]

Ambs, P.

Barash, S. C.

M. A. Eshera, S. C. Barash, “Parallel rule-based fuzzy inference on mesh-connected systolic arrays,” IEEE Expert 4, 27–35 (1989).
[CrossRef]

Campbell, S.

Chen, C.

Cheng, W.

M. Hansen, D. Shih, C. Fan, S. Esener, W. Cheng, E. Yablonovitch, U. Efron, “16×16 SLM with silicon CMOS drivers and III-V modulators,” in Spatial Light Modulators and Applications, Vol. 9 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 5.

Detloff, W. D.

H. Watanabe, J. R. Symon, W. D. Detloff, K. E. Young, “VLSI fuzzy chip and inference accelerator board systems,” in Fuzzy Logic for the Management of Uncertainty, L. Zadeh, J. Kacprzyk, eds. (Wiley, New York, 1992).

Efron, U.

M. Hansen, D. Shih, C. Fan, S. Esener, W. Cheng, E. Yablonovitch, U. Efron, “16×16 SLM with silicon CMOS drivers and III-V modulators,” in Spatial Light Modulators and Applications, Vol. 9 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 5.

Esener, S.

G. C. Marsden, F. Kiamilev, S. Esener, S. H. Lee, “Highly parallel consistent labeling algorithm suitable for optoelectronic implementation,” Appl. Opt. 30, 185–194 (1991).
[CrossRef] [PubMed]

S. Esener, “Optical memory systems for 3-D computing,” presented at the Lasers and Electro-Optics Society (LEOS) Annual Meeting, San Francisco, Calif., 16 November 1995.

M. Hansen, D. Shih, C. Fan, S. Esener, W. Cheng, E. Yablonovitch, U. Efron, “16×16 SLM with silicon CMOS drivers and III-V modulators,” in Spatial Light Modulators and Applications, Vol. 9 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 5.

S. Esener, F. McCormick, “Current status of optical storage systems,” in Conference on Lasers and Electro-Optics and Quantum Electronics and Laser Science, Vols. 15 and 16 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995).

Esener, S. C.

C. Fan, B. Mansoorian, D. A. Van Blerkom, M. W. Hansen, V. H. Ozguz, S. C. Esener, G. C. Marsden, “Digital free-space optical interconnections: a comparison of transmitter technologies,” Appl. Opt. 34, 3103–3115 (1995).
[CrossRef] [PubMed]

P. J. Marchand, A. V. Krishnamoorthy, K. S. Urquhart, P. Ambs, S. C. Esener, S. H. Lee, “Motionless-head parallel readout optical disk,” Appl. Opt. 32, 190–203 (1993).
[CrossRef] [PubMed]

A. V. Krishnamoorthy, G. Yayla, S. C. Esener, “A scalable neural system using free-space optical interconnects,” IEEE Trans. Neural Net. 3, 404–413 (1992).
[CrossRef]

G. C. Marsden, B. Olson, S. C. Esener, S. H. Lee, “Optoelectronic fuzzy logic system,” in Optical Computing, Vol. 6 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991) p. 212.

Eshera, M. A.

M. A. Eshera, S. C. Barash, “Parallel rule-based fuzzy inference on mesh-connected systolic arrays,” IEEE Expert 4, 27–35 (1989).
[CrossRef]

Fan, C.

C. Fan, B. Mansoorian, D. A. Van Blerkom, M. W. Hansen, V. H. Ozguz, S. C. Esener, G. C. Marsden, “Digital free-space optical interconnections: a comparison of transmitter technologies,” Appl. Opt. 34, 3103–3115 (1995).
[CrossRef] [PubMed]

M. Hansen, D. Shih, C. Fan, S. Esener, W. Cheng, E. Yablonovitch, U. Efron, “16×16 SLM with silicon CMOS drivers and III-V modulators,” in Spatial Light Modulators and Applications, Vol. 9 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 5.

Folger, T. A.

G. J. Klir, T. A. Folger, Fuzzy Sets, Uncertainty, and Information (Prentice-Hall, Englewood Cliffs, N.J., 1988).

Hansen, M.

M. Hansen, D. Shih, C. Fan, S. Esener, W. Cheng, E. Yablonovitch, U. Efron, “16×16 SLM with silicon CMOS drivers and III-V modulators,” in Spatial Light Modulators and Applications, Vol. 9 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 5.

Hansen, M. W.

Ichioka, Y.

Itoh, H.

Kandel, A.

A. Kandel, Fuzzy Techniques in Pattern Recognition (Wiley, New York, 1982).

Kiamilev, F.

Klir, G. J.

G. J. Klir, T. A. Folger, Fuzzy Sets, Uncertainty, and Information (Prentice-Hall, Englewood Cliffs, N.J., 1988).

Konishi, T.

Krishnamoorthy, A. V.

P. J. Marchand, A. V. Krishnamoorthy, K. S. Urquhart, P. Ambs, S. C. Esener, S. H. Lee, “Motionless-head parallel readout optical disk,” Appl. Opt. 32, 190–203 (1993).
[CrossRef] [PubMed]

A. V. Krishnamoorthy, G. Yayla, S. C. Esener, “A scalable neural system using free-space optical interconnects,” IEEE Trans. Neural Net. 3, 404–413 (1992).
[CrossRef]

Lee, S. H.

Lin, S.

Liu, L.

Lopez de Mantaras, R.

R. Lopez de Mantaras, Approximate Reasoning Models (Halsted, New York, 1990).

Mansoorian, B.

Marchand, P. J.

Marsden, G. C.

C. Fan, B. Mansoorian, D. A. Van Blerkom, M. W. Hansen, V. H. Ozguz, S. C. Esener, G. C. Marsden, “Digital free-space optical interconnections: a comparison of transmitter technologies,” Appl. Opt. 34, 3103–3115 (1995).
[CrossRef] [PubMed]

G. C. Marsden, F. Kiamilev, S. Esener, S. H. Lee, “Highly parallel consistent labeling algorithm suitable for optoelectronic implementation,” Appl. Opt. 30, 185–194 (1991).
[CrossRef] [PubMed]

G. C. Marsden, B. Olson, S. C. Esener, S. H. Lee, “Optoelectronic fuzzy logic system,” in Optical Computing, Vol. 6 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991) p. 212.

G. C. Marsden, “Optoelectronic array processors with applications in machine intelligence and database management,” Ph. D. dissertation (University of California, San Diego, La Jolla, Calif., 1993).

McCormick, F.

S. Esener, F. McCormick, “Current status of optical storage systems,” in Conference on Lasers and Electro-Optics and Quantum Electronics and Laser Science, Vols. 15 and 16 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995).

Mead, C.

C. Mead, M. Rem, “Cost and performance of VLSI computing structures,” IEEE J. Solid-State Circuits SC-14, 455–462 (1979).
[CrossRef]

Mukai, S.

Olson, B.

G. C. Marsden, B. Olson, S. C. Esener, S. H. Lee, “Optoelectronic fuzzy logic system,” in Optical Computing, Vol. 6 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991) p. 212.

Ozguz, V. H.

Rem, M.

C. Mead, M. Rem, “Cost and performance of VLSI computing structures,” IEEE J. Solid-State Circuits SC-14, 455–462 (1979).
[CrossRef]

Shih, D.

M. Hansen, D. Shih, C. Fan, S. Esener, W. Cheng, E. Yablonovitch, U. Efron, “16×16 SLM with silicon CMOS drivers and III-V modulators,” in Spatial Light Modulators and Applications, Vol. 9 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 5.

Sugeno, M.

M. Sugeno, “An introductory survey of fuzzy control,” Inf. Sci. 36, 59–83 (1985).
[CrossRef]

Symon, J. R.

H. Watanabe, J. R. Symon, W. D. Detloff, K. E. Young, “VLSI fuzzy chip and inference accelerator board systems,” in Fuzzy Logic for the Management of Uncertainty, L. Zadeh, J. Kacprzyk, eds. (Wiley, New York, 1992).

Tanida, J.

Togai, M.

M. Togai, H. Watanabe, “Expert system on a chip: an engine for real-time approximate reasoning,” IEEE Expert 1, 56–62 (1986).
[CrossRef]

Urquhart, K. S.

Van Blerkom, D. A.

Watanabe, H.

M. Togai, H. Watanabe, “Expert system on a chip: an engine for real-time approximate reasoning,” IEEE Expert 1, 56–62 (1986).
[CrossRef]

H. Watanabe, J. R. Symon, W. D. Detloff, K. E. Young, “VLSI fuzzy chip and inference accelerator board systems,” in Fuzzy Logic for the Management of Uncertainty, L. Zadeh, J. Kacprzyk, eds. (Wiley, New York, 1992).

Wu, W.

Yablonovitch, E.

M. Hansen, D. Shih, C. Fan, S. Esener, W. Cheng, E. Yablonovitch, U. Efron, “16×16 SLM with silicon CMOS drivers and III-V modulators,” in Spatial Light Modulators and Applications, Vol. 9 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 5.

Yajima, H.

Yayla, G.

A. V. Krishnamoorthy, G. Yayla, S. C. Esener, “A scalable neural system using free-space optical interconnects,” IEEE Trans. Neural Net. 3, 404–413 (1992).
[CrossRef]

Yeh, P.

Young, K. E.

H. Watanabe, J. R. Symon, W. D. Detloff, K. E. Young, “VLSI fuzzy chip and inference accelerator board systems,” in Fuzzy Logic for the Management of Uncertainty, L. Zadeh, J. Kacprzyk, eds. (Wiley, New York, 1992).

Zadeh, L. A.

L. A. Zadeh, “The role of fuzzy logic in the management of uncertainty in expert systems,” Fuzzy Sets Syst. 11, 199–227 (1983).
[CrossRef]

L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning—III,” Inf. Sci. 9, 43–80 (1975).
[CrossRef]

L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning—I,” Inf. Sci. 8, 199–249 (1975).
[CrossRef]

L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning—II,” Inf. Sci. 8, 301–357 (1975).
[CrossRef]

L. A. Zadeh, “Fuzzy sets,” Inf. Control 6, 338–353 (1965).
[CrossRef]

Zhang, S.

Zhou, S.

Zhu, Z.

Appl. Opt. (6)

Fuzzy Sets Syst. (1)

L. A. Zadeh, “The role of fuzzy logic in the management of uncertainty in expert systems,” Fuzzy Sets Syst. 11, 199–227 (1983).
[CrossRef]

IEEE Expert (2)

M. Togai, H. Watanabe, “Expert system on a chip: an engine for real-time approximate reasoning,” IEEE Expert 1, 56–62 (1986).
[CrossRef]

M. A. Eshera, S. C. Barash, “Parallel rule-based fuzzy inference on mesh-connected systolic arrays,” IEEE Expert 4, 27–35 (1989).
[CrossRef]

IEEE J. Solid-State Circuits (1)

C. Mead, M. Rem, “Cost and performance of VLSI computing structures,” IEEE J. Solid-State Circuits SC-14, 455–462 (1979).
[CrossRef]

IEEE Trans. Neural Net. (1)

A. V. Krishnamoorthy, G. Yayla, S. C. Esener, “A scalable neural system using free-space optical interconnects,” IEEE Trans. Neural Net. 3, 404–413 (1992).
[CrossRef]

Inf. Control (1)

L. A. Zadeh, “Fuzzy sets,” Inf. Control 6, 338–353 (1965).
[CrossRef]

Inf. Sci. (4)

M. Sugeno, “An introductory survey of fuzzy control,” Inf. Sci. 36, 59–83 (1985).
[CrossRef]

L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning—I,” Inf. Sci. 8, 199–249 (1975).
[CrossRef]

L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning—II,” Inf. Sci. 8, 301–357 (1975).
[CrossRef]

L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning—III,” Inf. Sci. 9, 43–80 (1975).
[CrossRef]

Opt. Commun. (1)

L. Liu, “Optical implementation of parallel fuzzy logic,” Opt. Commun. 73, 183–187 (1989).
[CrossRef]

Opt. Lett. (2)

Other (9)

S. Esener, F. McCormick, “Current status of optical storage systems,” in Conference on Lasers and Electro-Optics and Quantum Electronics and Laser Science, Vols. 15 and 16 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995).

G. C. Marsden, B. Olson, S. C. Esener, S. H. Lee, “Optoelectronic fuzzy logic system,” in Optical Computing, Vol. 6 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991) p. 212.

G. C. Marsden, “Optoelectronic array processors with applications in machine intelligence and database management,” Ph. D. dissertation (University of California, San Diego, La Jolla, Calif., 1993).

R. Lopez de Mantaras, Approximate Reasoning Models (Halsted, New York, 1990).

M. Hansen, D. Shih, C. Fan, S. Esener, W. Cheng, E. Yablonovitch, U. Efron, “16×16 SLM with silicon CMOS drivers and III-V modulators,” in Spatial Light Modulators and Applications, Vol. 9 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 5.

S. Esener, “Optical memory systems for 3-D computing,” presented at the Lasers and Electro-Optics Society (LEOS) Annual Meeting, San Francisco, Calif., 16 November 1995.

A. Kandel, Fuzzy Techniques in Pattern Recognition (Wiley, New York, 1982).

G. J. Klir, T. A. Folger, Fuzzy Sets, Uncertainty, and Information (Prentice-Hall, Englewood Cliffs, N.J., 1988).

H. Watanabe, J. R. Symon, W. D. Detloff, K. E. Young, “VLSI fuzzy chip and inference accelerator board systems,” in Fuzzy Logic for the Management of Uncertainty, L. Zadeh, J. Kacprzyk, eds. (Wiley, New York, 1992).

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Figures (18)

Fig. 1
Fig. 1

(a) Crisp subsets have membership functions restricted to 0 or 1, such as the subset of ages over 50. (b) Fuzzy subsets allow membership function values between 0 and 1. Fuzzy subsets can be used to represent imprecise information, such as old age.

Fig. 2
Fig. 2

(a) α match is used to determine the degree to which the consequent is asserted. The fuzzy scalar α is determined at the largest intersection of the antecedent and premise, shown within the circle. (b) β match is used to determine the degree to which the conclusion is restricted. The fuzzy scalar β is determined at the largest intersection of the complement of the antecedent and premise, shown within the circle.

Fig. 3
Fig. 3

Rule conclusions are combined disjunctively to obtain the accumulative conclusion for necessity measures.

Fig. 4
Fig. 4

Rule conclusions are combined conjunctively to obtain the accumulative conclusion for possibility measures.

Fig. 5
Fig. 5

Standard MSB first, unsigned binary encoding permits a dynamic range of M-1 with log2 M bits.

Fig. 6
Fig. 6

PWM encoding uses M-1 bits for the same dynamic range with the values encoded as an initial sequence of high bits.

Fig. 7
Fig. 7

Binary min/max unit detects and passes the smaller of the two binary encoded fuzzy values on the min port ∧ and the larger of the two fuzzy values on the max port (∨).

Fig. 8
Fig. 8

Timing diagrams for the binary min/max gate show the temporal relationships between the reset, input, and output signals.

Fig. 9
Fig. 9

Circuit block diagram of a binary min/max unit.

Fig. 10
Fig. 10

max operation for PWM encoded data can be achieved with a simple or gate. min operation is achieved with an and gate.

Fig. 11
Fig. 11

Parallel architecture consists of an optical memory, a memory interface, and an inference tree. One-to-one optical interconnections exist between the optical memory and interface and between the interface and inference tree. A binary tree structure is used to propagate instructions to the elements of the interface. The inference tree is also a binary tree.

Fig. 12
Fig. 12

During the calculation of an α or β match, the fuzzy vectors representing the antecedent and premise are transmitted bit serially from the interface to the leaf units of the inference tree. The resulting scalar is stored temporarily in the root unit. The white and black arrows represent optical and electrical interconnections, respectively.

Fig. 13
Fig. 13

During the calculation of the rule conclusion, the consequent is transmitted from the interface to the leaf units of the inference tree, where the vector elements are combined disjunctively or conjunctively with the α or β match transmitted from the root unit. The result is transmitted from the inference tree to the interface.

Fig. 14
Fig. 14

During the update of the accumulative conclusion, the rule conclusion and previous accumulative conclusion are transmitted from the interface to the leaf units of the inference tree. The updated accumulative conclusion is transmitted from the leaf units to the interface.

Fig. 15
Fig. 15

Leaf units of the inference tree each consist of two optical detectors for receiving data from the interface, an optical transmitter for transmitting data to the interface, electrical connections to the fanning units, and computational circuitry.

Fig. 16
Fig. 16

Each fanning unit consists of electrical connections, a min/max unit, and a flip–flop for propagating the reset signal directly ahead of the data stream.

Fig. 17
Fig. 17

H-tree layout used for the inference tree, thereby reducing capacitive delay to O(N 1/2) and (b) for the interface.

Fig. 18
Fig. 18

Optical interconnection between the interface and the inference tree requires simple imaging. The optical paths of the signals yield optical delays of O(N 1/2) time complexity, which are consistent with the capacitive delays of the H tree.

Tables (2)

Tables Icon

Table 1 Combinations of α and β Matches of Multiple Antecedents a

Tables Icon

Table 2 Simple Two-Bit Instruction a

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

S ( x ) = 1 if x S = 0 if x S ,
S : { x | x X } [ 0 , 1 ] .
S T ( x ) = S ( x ) AND T ( x ) .
S T ( x ) = S ( x ) OR T ( x ) .
S T ( x ) = S ( x ) T ( x ) = MIN [ S ( x ) , T ( x ) ] ,
S T ( x ) = S ( x ) T ( x ) = MAX [ S ( x ) , T ( x ) ] .
¬ S ( x ) = 1 S ( x ) .
p q p q .
A C A C .
C ( z ) = [ x = 1 N A ( x ) A ( x ) ] C ( z ) ,
x = 1 N
C ( z ) = [ x = 1 N ¬ A ( x ) A ( x ) ] C ( z ) ,
C ( z ) = [ x = 1 N A ( x ) A ( x ) ] C ( z )
C ( z ) = [ x = 1 N ¬ A ( x ) A ( x ) ] C ( z )
α ( A , A ) = x = 1 N A ( x ) A ( x ) ,
β ( A , A ) = x = 1 N ¬ A ( x ) A ( x ) ,
C ( z ) = α C ( z ) ,
C ( z ) = β C ( z ) .
A , B C A B C
A , B C A B C .
C ( z ) = [ x = 1 N y = 1 N A ( x ) A ( x ) B ( y ) B ( y ) ] C ( z )
C ( z ) = α ( A , A ) α ( B , B ) C ( z ) .
C ( z ) = [ x = 1 N y = 1 N [ A ( x ) A ( x ) ] [ B ( y ) B ( y ) ] ] C ( z )
C ( z ) = [ α ( A , A ) α ( B , B ) ] C ( z )
C ( z ) = [ x = 1 N y = 1 N [ ¬ A ( x ) A ( x ) ] [ ¬ B ( y ) B ( y ) ] ] C ( z )
C ( z ) = [ β ( A , A ) β ( B , B ) ] C ( z )
C ( z ) = [ x = 1 N y = 1 N [ ¬ A ( x ) A ( x ) ] [ ¬ B ( y ) B ( y ) ] ] C ( z )
C ( z ) = [ β ( A , A ) β ( B , B ) ] C ( z ) .
C ( z ) = r C r ( z ) ,
C ( z ) = r C r ( z ) .
C r ( z ) = x M r ( z , x ) A ( x ) ,
M r ( z , x ) = A r ( x ) C r ( z ) ,
M r ( z , x ) = ¬ A r ( x ) C r ( z ) .

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