Abstract

A functional correspondence is shown among neural networks, symbolic substitution, digital computers, the permutation group SN, and optical correlator devices. Group and element networks are postulated. The dimensionality of the group number N is interpreted as a measure of the parallel capacity of a network. The principle of symbolic substitution is used to design S2 rules for a full binary adder, and neural model techniques are applied to produce a neural net implementing the adder. Optical neuromorphs are described for the processing nodes of the network.

© 1988 Optical Society of America

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References

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  1. A. Huang, “Parallel Algorithms for Optical Digital Computers,” Proc. IEEE 13 (1983).
  2. A. Huang, “Architectural Considerations Involved in the Design of an Optical Digital Computer,” Proc., IEEE 72, 780 (1984).
    [CrossRef]
  3. K.-H. Brenner, A. Huang, “An Optical Processor Based on Symbolic Substitution,” in Technical Digest, Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1985), paper WA4.
  4. D. Casasent, E. Botha, “Knowledge in Optical Symbolic Pattern Recognition Processors,” Opt. Eng. 26, 034 (1987).
    [CrossRef]
  5. S. Grossberg, Studies of Mind and Brain, Boston Studies in the Philosophy of Science, Vol. 70 (Reidel, Norwell, MA, 1982).
    [CrossRef]
  6. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  7. F. Byron, R. Fuller, Mathematics of Classical and Quantum Physics, Vol. 2 (Addison-Wesley, Reading, MA, 1970), Chap. 10.
  8. G. Carpenter, S. Grossberg, “A Massively Parallel Architecture for a Self-Organizing Neural Pattern Recognition Machine,” Comput Vision Graphics Image Proc. 37, 54 (1987).
    [CrossRef]
  9. Ref. 5, Chap. 1, Appendix D.
  10. H. Szu, “Globally Connected Network Models for Computing Using Fine-Grained Processing Elements,” in Proceedings, International Conference on Lasers ’85 (1985), pp. 92–97.
  11. A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb Devices: Transphasors with High Gain, Bistable Switches, and Sequential Logic Gates,” Philos. Trans. A. Soc. London Ser. A 313, 249 (1984).
    [CrossRef]

1987 (2)

D. Casasent, E. Botha, “Knowledge in Optical Symbolic Pattern Recognition Processors,” Opt. Eng. 26, 034 (1987).
[CrossRef]

G. Carpenter, S. Grossberg, “A Massively Parallel Architecture for a Self-Organizing Neural Pattern Recognition Machine,” Comput Vision Graphics Image Proc. 37, 54 (1987).
[CrossRef]

1984 (2)

A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb Devices: Transphasors with High Gain, Bistable Switches, and Sequential Logic Gates,” Philos. Trans. A. Soc. London Ser. A 313, 249 (1984).
[CrossRef]

A. Huang, “Architectural Considerations Involved in the Design of an Optical Digital Computer,” Proc., IEEE 72, 780 (1984).
[CrossRef]

1983 (1)

A. Huang, “Parallel Algorithms for Optical Digital Computers,” Proc. IEEE 13 (1983).

Botha, E.

D. Casasent, E. Botha, “Knowledge in Optical Symbolic Pattern Recognition Processors,” Opt. Eng. 26, 034 (1987).
[CrossRef]

Brenner, K.-H.

K.-H. Brenner, A. Huang, “An Optical Processor Based on Symbolic Substitution,” in Technical Digest, Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1985), paper WA4.

Byron, F.

F. Byron, R. Fuller, Mathematics of Classical and Quantum Physics, Vol. 2 (Addison-Wesley, Reading, MA, 1970), Chap. 10.

Carpenter, G.

G. Carpenter, S. Grossberg, “A Massively Parallel Architecture for a Self-Organizing Neural Pattern Recognition Machine,” Comput Vision Graphics Image Proc. 37, 54 (1987).
[CrossRef]

Casasent, D.

D. Casasent, E. Botha, “Knowledge in Optical Symbolic Pattern Recognition Processors,” Opt. Eng. 26, 034 (1987).
[CrossRef]

Fuller, R.

F. Byron, R. Fuller, Mathematics of Classical and Quantum Physics, Vol. 2 (Addison-Wesley, Reading, MA, 1970), Chap. 10.

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Grossberg, S.

G. Carpenter, S. Grossberg, “A Massively Parallel Architecture for a Self-Organizing Neural Pattern Recognition Machine,” Comput Vision Graphics Image Proc. 37, 54 (1987).
[CrossRef]

S. Grossberg, Studies of Mind and Brain, Boston Studies in the Philosophy of Science, Vol. 70 (Reidel, Norwell, MA, 1982).
[CrossRef]

Huang, A.

A. Huang, “Architectural Considerations Involved in the Design of an Optical Digital Computer,” Proc., IEEE 72, 780 (1984).
[CrossRef]

A. Huang, “Parallel Algorithms for Optical Digital Computers,” Proc. IEEE 13 (1983).

K.-H. Brenner, A. Huang, “An Optical Processor Based on Symbolic Substitution,” in Technical Digest, Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1985), paper WA4.

Kar, A. K.

A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb Devices: Transphasors with High Gain, Bistable Switches, and Sequential Logic Gates,” Philos. Trans. A. Soc. London Ser. A 313, 249 (1984).
[CrossRef]

Mathew, J. G. H.

A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb Devices: Transphasors with High Gain, Bistable Switches, and Sequential Logic Gates,” Philos. Trans. A. Soc. London Ser. A 313, 249 (1984).
[CrossRef]

Prise, M. E.

A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb Devices: Transphasors with High Gain, Bistable Switches, and Sequential Logic Gates,” Philos. Trans. A. Soc. London Ser. A 313, 249 (1984).
[CrossRef]

Smith, S. D.

A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb Devices: Transphasors with High Gain, Bistable Switches, and Sequential Logic Gates,” Philos. Trans. A. Soc. London Ser. A 313, 249 (1984).
[CrossRef]

Szu, H.

H. Szu, “Globally Connected Network Models for Computing Using Fine-Grained Processing Elements,” in Proceedings, International Conference on Lasers ’85 (1985), pp. 92–97.

Taghizadeh, M. R.

A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb Devices: Transphasors with High Gain, Bistable Switches, and Sequential Logic Gates,” Philos. Trans. A. Soc. London Ser. A 313, 249 (1984).
[CrossRef]

Tooley, F. A. P.

A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb Devices: Transphasors with High Gain, Bistable Switches, and Sequential Logic Gates,” Philos. Trans. A. Soc. London Ser. A 313, 249 (1984).
[CrossRef]

Walker, A. C.

A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb Devices: Transphasors with High Gain, Bistable Switches, and Sequential Logic Gates,” Philos. Trans. A. Soc. London Ser. A 313, 249 (1984).
[CrossRef]

Comput Vision Graphics Image Proc. (1)

G. Carpenter, S. Grossberg, “A Massively Parallel Architecture for a Self-Organizing Neural Pattern Recognition Machine,” Comput Vision Graphics Image Proc. 37, 54 (1987).
[CrossRef]

Opt. Eng. (1)

D. Casasent, E. Botha, “Knowledge in Optical Symbolic Pattern Recognition Processors,” Opt. Eng. 26, 034 (1987).
[CrossRef]

Philos. Trans. A. Soc. London Ser. A (1)

A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb Devices: Transphasors with High Gain, Bistable Switches, and Sequential Logic Gates,” Philos. Trans. A. Soc. London Ser. A 313, 249 (1984).
[CrossRef]

Proc. IEEE (1)

A. Huang, “Parallel Algorithms for Optical Digital Computers,” Proc. IEEE 13 (1983).

Proc., IEEE (1)

A. Huang, “Architectural Considerations Involved in the Design of an Optical Digital Computer,” Proc., IEEE 72, 780 (1984).
[CrossRef]

Other (6)

K.-H. Brenner, A. Huang, “An Optical Processor Based on Symbolic Substitution,” in Technical Digest, Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1985), paper WA4.

Ref. 5, Chap. 1, Appendix D.

H. Szu, “Globally Connected Network Models for Computing Using Fine-Grained Processing Elements,” in Proceedings, International Conference on Lasers ’85 (1985), pp. 92–97.

S. Grossberg, Studies of Mind and Brain, Boston Studies in the Philosophy of Science, Vol. 70 (Reidel, Norwell, MA, 1982).
[CrossRef]

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

F. Byron, R. Fuller, Mathematics of Classical and Quantum Physics, Vol. 2 (Addison-Wesley, Reading, MA, 1970), Chap. 10.

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Figures (7)

Fig. 1
Fig. 1

Interrelationships among concepts involving symbolic substitution.

Fig. 2
Fig. 2

Common functional property fg is found in optics, neural nets, group theory, and computers.

Fig. 3
Fig. 3

Grossberg subnetwork constructions of group elements for the first three symmetric groups.

Fig. 4
Fig. 4

Additional connections among group elements, shown for S3, which permit application of the group product rule between any pair.

Fig. 5
Fig. 5

Neural net complexity for symmetric group SN using Grossberg minimal slabs.

Fig. 6
Fig. 6

Binary adder neural subnetworks implementing two symbolic substitution pattern rules.

Fig. 7
Fig. 7

Complete neural network for N-bit full binary adder with carry.

Equations (4)

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

C ( x , y ) = d u d υ A ( u , υ ) B * ( u x , υ y ) .
01 00 01 00 + 01 10 + 01 01 + 00 01 + 00 00 .
( a ) the pattern 1 is replaced by 0 0 1
( b ) the pattern 01 is replaced by 10 1 0 .

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