Abstract

Based on the negabinary number representation, parallel one-step arithmetic operations (that is, addition and subtraction), logical operations, and matrix-vector multiplication on data have been optically implemented, by use of a two-dimensional spatial-encoding technique. For addition and subtraction, one of the operands in decimal form is converted into the unsigned negabinary form, whereas the other decimal number is represented in the signed negabinary form. The result of operation is obtained in the mixed negabinary form and is converted back into decimal. Matrix-vector multiplication for unsigned negabinary numbers is achieved through the convolution technique. Both of the operands for logical operation are converted to their signed negabinary forms. All operations are implemented by use of a unique optical architecture. The use of a single liquid-crystal-display panel to spatially encode the input data, operational kernels, and decoding masks have simplified the architecture as well as reduced the cost and complexity.

© 2002 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  35. G. Li, L. Liu, H. Cheng, X. Yan, “Parallel optical quaternary signed-digit multiplication and its use for matrix-vector operation,” Optik (Stuttgart) 107, 165–172 (1998).
  36. H. Huang, M. Itoh, T. Yatagai, “Optical scalable parallel modified signed-digit algorithms for large-scale array addition and multiplication using digit-decomposition-plane representation,” Opt. Eng. 38, 432–440 (1999).
    [CrossRef]
  37. S. Bandyopadhyay, A. K. Datta, S. S. Bawa, A. M. Biradar, S. Chandra, “Realization of digital optical matrix-vector multiplication,” J. Phys. D 28, 7–11 (1995).
    [CrossRef]
  38. G. Li, L. Liu, “Complex-valued matrix-vector multiplication using twos complement representation,” Opt. Comm. 105, 161–166 (1994).
    [CrossRef]
  39. G. Li, L. Liu, “Negabinary encoding for optical complex matrix operation,” Opt. Comm. 113, 15–19 (1994).
    [CrossRef]
  40. L. Liu, G. Li, Y. Yin, “Optical complex matrix-vector multiplication with negative binary inner products,” Opt. Lett. 19, 1759–1761 (1994).
    [CrossRef] [PubMed]

1999

S. Zhang, M. A. Karim, “Programmable modified-signed-digit addition module based on binary logic gates,” Opt. Eng. 38, 456–461 (1999).
[CrossRef]

S. Zhang, M. A. Karim, “Optical arithmetic processing using improved redundant binary algorithms,” Opt. Eng. 38, 415–421 (1999).
[CrossRef]

A. K. Cherri, “Signed-digit arithmetic for optical computing: digit grouping and pixel assignment for spatial encoding,” Opt. Eng. 38, 422–431 (1999).
[CrossRef]

M. S. Alam, “Parallel optoelectronic trinary signed-digit division,” Opt. Eng. 38, 441–448 (1999).
[CrossRef]

G. Li, F. Qian, H. Ruan, L. Liu, “Parallel optical negabinary signed-digit computing: algorithm and optical implementation,” Opt. Eng. 38, 403–414 (1999).
[CrossRef]

H. Huang, M. Itoh, T. Yatagai, “Optical scalable parallel modified signed-digit algorithms for large-scale array addition and multiplication using digit-decomposition-plane representation,” Opt. Eng. 38, 432–440 (1999).
[CrossRef]

1998

G. Li, L. Liu, H. Cheng, X. Yan, “Parallel optical quaternary signed-digit multiplication and its use for matrix-vector operation,” Optik (Stuttgart) 107, 165–172 (1998).

1997

1995

S. Zhou, S. Campbell, P. Yeh, H. K. Liu, “Two-stage modified signed-digit optical computing by spatial data encoding and polarization multiplexing,” Appl. Opt. 34, 793–802 (1995).
[CrossRef] [PubMed]

A. K. Datta, M. Seth, “Parallel arithmetic operations in an optical architecture using a modified iterative technique,” Opt. Comm. 115, 245–250 (1995).
[CrossRef]

S. Bandyopadhyay, A. K. Datta, S. S. Bawa, A. M. Biradar, S. Chandra, “Realization of digital optical matrix-vector multiplication,” J. Phys. D 28, 7–11 (1995).
[CrossRef]

1994

G. Li, L. Liu, “Complex-valued matrix-vector multiplication using twos complement representation,” Opt. Comm. 105, 161–166 (1994).
[CrossRef]

G. Li, L. Liu, “Negabinary encoding for optical complex matrix operation,” Opt. Comm. 113, 15–19 (1994).
[CrossRef]

L. Liu, G. Li, Y. Yin, “Optical complex matrix-vector multiplication with negative binary inner products,” Opt. Lett. 19, 1759–1761 (1994).
[CrossRef] [PubMed]

M. S. Alam, “Parallel optical computing using recoded trinary signed-digit numbers,” Appl. Opt. 33, 4392–4397 (1994).
[CrossRef] [PubMed]

M. S. Alam, K. Jemili, M. A. Karim, “Optical higher-order quaternary signed-digit arithmetic,” Opt. Eng. 33, 3419–3426 (1994).
[CrossRef]

K. W. Wong, L. M. Cheng, “Optical modified signed-digit addition based on binary logical operations,” Opt. Laser Technol. 26, 213–217 (1994).
[CrossRef]

H. Huang, M. Itoh, T. Yatagai, “Modified signed-digit arithmetic based on redundant bit representation,” Appl. Opt. 33, 6146–6156 (1994).
[CrossRef] [PubMed]

G. Li, L. Liu, L. Shao, Z. Wang, “Negative binary arithmetic algorithms for digital parallel optical computation,” Opt. Lett. 19, 1337–1339 (1994).
[CrossRef] [PubMed]

J. Tanida, M. Iwata, Y. Ichioka, “Extended coding for optical array logic,” Appl. Opt. 33, 3363–3369 (1994).
[CrossRef]

A. K. Datta, M. Seth, “Multi-input optical parallel logic processing with the shadow-casting technique,” App. Opt. 33, 8146–8152 (1994).
[CrossRef]

1993

Y. Wu, Z. Zhang, L. Liu, Z. Wang, “Arithmetic operation using binary encoding modified-signed-digit system,” Opt. Comm. 100, 53–58 (1993).
[CrossRef]

1992

1990

1989

1988

A. S. P. Kozaitis, “Higher-ordered rules for symbolic substitution,” Opt. Comm. 65, 339–342 (1988).
[CrossRef]

1987

1986

1985

S. Cartwright, S. C. Gustafson, “Convolver based optical systolic processing architecture,” Opt. Eng. 24, 59–62 (1985).
[CrossRef]

1984

S. Cartwright, “New optical matrix-vector multiplier,” Appl. Opt. 23, 1683–1684 (1984).
[CrossRef] [PubMed]

P. S. Guilfoyle, “Systolic acousto-optic binary convolver,” Opt. Eng. 23, 20–25 (1984).
[CrossRef]

1983

1979

Alam, M. S.

Athale, R. A.

Awwal, A. A. S.

Bandyopadhyay, S.

S. Bandyopadhyay, A. K. Datta, S. S. Bawa, A. M. Biradar, S. Chandra, “Realization of digital optical matrix-vector multiplication,” J. Phys. D 28, 7–11 (1995).
[CrossRef]

Basuray, A.

S. Mukhopadhyay, A. Basuray, A. K. Datta, “New technique of arithmetic operation using positional residue system,” App. Opt. 29, 2981–2893 (1990).
[CrossRef]

A. K. Datta, A. Basuray, S. Mukhopadhyay, “Arithmetic operations in optical computations using a modified trinary number system,” Opt. Lett. 14, 426–428 (1989).
[CrossRef] [PubMed]

Bawa, S. S.

S. Bandyopadhyay, A. K. Datta, S. S. Bawa, A. M. Biradar, S. Chandra, “Realization of digital optical matrix-vector multiplication,” J. Phys. D 28, 7–11 (1995).
[CrossRef]

Biradar, A. M.

S. Bandyopadhyay, A. K. Datta, S. S. Bawa, A. M. Biradar, S. Chandra, “Realization of digital optical matrix-vector multiplication,” J. Phys. D 28, 7–11 (1995).
[CrossRef]

Bocker, R. P.

Campbell, S.

Carlotto, M.

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate numerical computation by optical convolution,” in Optical Computing II, W. T. Rhodes, ed., Proc. SPIE232, 151–156 (1980).

Cartwright, S.

S. Cartwright, S. C. Gustafson, “Convolver based optical systolic processing architecture,” Opt. Eng. 24, 59–62 (1985).
[CrossRef]

S. Cartwright, “New optical matrix-vector multiplier,” Appl. Opt. 23, 1683–1684 (1984).
[CrossRef] [PubMed]

Casasent, D.

C. Perlee, D. Casasent, “Negative base encoding in optical linear algebra processors,” Appl. Opt. 25, 168–169 (1986).
[CrossRef] [PubMed]

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate numerical computation by optical convolution,” in Optical Computing II, W. T. Rhodes, ed., Proc. SPIE232, 151–156 (1980).

Chandra, S.

S. Bandyopadhyay, A. K. Datta, S. S. Bawa, A. M. Biradar, S. Chandra, “Realization of digital optical matrix-vector multiplication,” J. Phys. D 28, 7–11 (1995).
[CrossRef]

Cheng, H.

G. Li, L. Liu, H. Cheng, X. Yan, “Parallel optical quaternary signed-digit multiplication and its use for matrix-vector operation,” Optik (Stuttgart) 107, 165–172 (1998).

Cheng, L. M.

K. W. Wong, L. M. Cheng, “Optical modified signed-digit addition based on binary logical operations,” Opt. Laser Technol. 26, 213–217 (1994).
[CrossRef]

Cherri, A. K.

A. K. Cherri, “Signed-digit arithmetic for optical computing: digit grouping and pixel assignment for spatial encoding,” Opt. Eng. 38, 422–431 (1999).
[CrossRef]

Datta, A. K.

A. K. Datta, M. Seth, “Parallel arithmetic operations in an optical architecture using a modified iterative technique,” Opt. Comm. 115, 245–250 (1995).
[CrossRef]

S. Bandyopadhyay, A. K. Datta, S. S. Bawa, A. M. Biradar, S. Chandra, “Realization of digital optical matrix-vector multiplication,” J. Phys. D 28, 7–11 (1995).
[CrossRef]

A. K. Datta, M. Seth, “Multi-input optical parallel logic processing with the shadow-casting technique,” App. Opt. 33, 8146–8152 (1994).
[CrossRef]

S. Mukhopadhyay, A. Basuray, A. K. Datta, “New technique of arithmetic operation using positional residue system,” App. Opt. 29, 2981–2893 (1990).
[CrossRef]

A. K. Datta, A. Basuray, S. Mukhopadhyay, “Arithmetic operations in optical computations using a modified trinary number system,” Opt. Lett. 14, 426–428 (1989).
[CrossRef] [PubMed]

Drake, B. L.

Eichmann, G.

Goodman, J. W.

Guilfoyle, P. S.

P. S. Guilfoyle, “Systolic acousto-optic binary convolver,” Opt. Eng. 23, 20–25 (1984).
[CrossRef]

Gustafson, S. C.

S. Cartwright, S. C. Gustafson, “Convolver based optical systolic processing architecture,” Opt. Eng. 24, 59–62 (1985).
[CrossRef]

Heinrich, M. L.

Henderson, T. B.

Hua, J.

Huang, A.

Huang, H.

H. Huang, M. Itoh, T. Yatagai, “Optical scalable parallel modified signed-digit algorithms for large-scale array addition and multiplication using digit-decomposition-plane representation,” Opt. Eng. 38, 432–440 (1999).
[CrossRef]

H. Huang, M. Itoh, T. Yatagai, “Modified signed-digit arithmetic based on redundant bit representation,” Appl. Opt. 33, 6146–6156 (1994).
[CrossRef] [PubMed]

Huang, M. W.

Hwang, K.

K. Hwang, Computer Arithmetic: Principles, Architecture and Design (Wiley, New York, 1979).

Ichioka, Y.

J. Tanida, M. Iwata, Y. Ichioka, “Extended coding for optical array logic,” Appl. Opt. 33, 3363–3369 (1994).
[CrossRef]

J. Tanida, Y. Ichioka, “Optical logic array processor using shadowgrams,” J. Opt. Soc. Am. 73, 800–809 (1983).
[CrossRef]

Ishihara, S.

Itoh, M.

H. Huang, M. Itoh, T. Yatagai, “Optical scalable parallel modified signed-digit algorithms for large-scale array addition and multiplication using digit-decomposition-plane representation,” Opt. Eng. 38, 432–440 (1999).
[CrossRef]

H. Huang, M. Itoh, T. Yatagai, “Modified signed-digit arithmetic based on redundant bit representation,” Appl. Opt. 33, 6146–6156 (1994).
[CrossRef] [PubMed]

Iwata, M.

J. Tanida, M. Iwata, Y. Ichioka, “Extended coding for optical array logic,” Appl. Opt. 33, 3363–3369 (1994).
[CrossRef]

Jemili, K.

M. S. Alam, K. Jemili, M. A. Karim, “Optical higher-order quaternary signed-digit arithmetic,” Opt. Eng. 33, 3419–3426 (1994).
[CrossRef]

Karim, M. A.

S. Zhang, M. A. Karim, “Programmable modified-signed-digit addition module based on binary logic gates,” Opt. Eng. 38, 456–461 (1999).
[CrossRef]

S. Zhang, M. A. Karim, “Optical arithmetic processing using improved redundant binary algorithms,” Opt. Eng. 38, 415–421 (1999).
[CrossRef]

M. S. Alam, K. Jemili, M. A. Karim, “Optical higher-order quaternary signed-digit arithmetic,” Opt. Eng. 33, 3419–3426 (1994).
[CrossRef]

M. S. Alam, M. A. Karim, A. A. S. Awwal, J. J. Westerkamp, “Optical processing based on conditional higher-order trinary modified signed-digit symbolic substitution,” Appl. Opt. 31, 5614–5621 (1992).
[CrossRef] [PubMed]

Kim, D. H.

Kostrzewski, A.

Kozaitis, A. S. P.

A. S. P. Kozaitis, “Higher-ordered rules for symbolic substitution,” Opt. Comm. 65, 339–342 (1988).
[CrossRef]

Lasher, M. E.

Li, G.

G. Li, F. Qian, H. Ruan, L. Liu, “Parallel optical negabinary signed-digit computing: algorithm and optical implementation,” Opt. Eng. 38, 403–414 (1999).
[CrossRef]

G. Li, L. Liu, H. Cheng, X. Yan, “Parallel optical quaternary signed-digit multiplication and its use for matrix-vector operation,” Optik (Stuttgart) 107, 165–172 (1998).

G. Li, L. Liu, Y. Yin, J. Hua, “Parallel optical negabinary arithmetic based on logic operations,” Appl. Opt. 36, 1011–1016 (1997).
[CrossRef] [PubMed]

G. Li, L. Liu, L. Shao, Z. Wang, “Negative binary arithmetic algorithms for digital parallel optical computation,” Opt. Lett. 19, 1337–1339 (1994).
[CrossRef] [PubMed]

G. Li, L. Liu, “Negabinary encoding for optical complex matrix operation,” Opt. Comm. 113, 15–19 (1994).
[CrossRef]

L. Liu, G. Li, Y. Yin, “Optical complex matrix-vector multiplication with negative binary inner products,” Opt. Lett. 19, 1759–1761 (1994).
[CrossRef] [PubMed]

G. Li, L. Liu, “Complex-valued matrix-vector multiplication using twos complement representation,” Opt. Comm. 105, 161–166 (1994).
[CrossRef]

Li, Y.

Liu, H. K.

Liu, L.

G. Li, F. Qian, H. Ruan, L. Liu, “Parallel optical negabinary signed-digit computing: algorithm and optical implementation,” Opt. Eng. 38, 403–414 (1999).
[CrossRef]

G. Li, L. Liu, H. Cheng, X. Yan, “Parallel optical quaternary signed-digit multiplication and its use for matrix-vector operation,” Optik (Stuttgart) 107, 165–172 (1998).

G. Li, L. Liu, Y. Yin, J. Hua, “Parallel optical negabinary arithmetic based on logic operations,” Appl. Opt. 36, 1011–1016 (1997).
[CrossRef] [PubMed]

G. Li, L. Liu, “Negabinary encoding for optical complex matrix operation,” Opt. Comm. 113, 15–19 (1994).
[CrossRef]

G. Li, L. Liu, L. Shao, Z. Wang, “Negative binary arithmetic algorithms for digital parallel optical computation,” Opt. Lett. 19, 1337–1339 (1994).
[CrossRef] [PubMed]

L. Liu, G. Li, Y. Yin, “Optical complex matrix-vector multiplication with negative binary inner products,” Opt. Lett. 19, 1759–1761 (1994).
[CrossRef] [PubMed]

G. Li, L. Liu, “Complex-valued matrix-vector multiplication using twos complement representation,” Opt. Comm. 105, 161–166 (1994).
[CrossRef]

Y. Wu, Z. Zhang, L. Liu, Z. Wang, “Arithmetic operation using binary encoding modified-signed-digit system,” Opt. Comm. 100, 53–58 (1993).
[CrossRef]

Mukhopadhyay, S.

S. Mukhopadhyay, A. Basuray, A. K. Datta, “New technique of arithmetic operation using positional residue system,” App. Opt. 29, 2981–2893 (1990).
[CrossRef]

A. K. Datta, A. Basuray, S. Mukhopadhyay, “Arithmetic operations in optical computations using a modified trinary number system,” Opt. Lett. 14, 426–428 (1989).
[CrossRef] [PubMed]

Neft, D.

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate numerical computation by optical convolution,” in Optical Computing II, W. T. Rhodes, ed., Proc. SPIE232, 151–156 (1980).

Perlee, C.

Psaltis, D.

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate numerical computation by optical convolution,” in Optical Computing II, W. T. Rhodes, ed., Proc. SPIE232, 151–156 (1980).

Qian, F.

G. Li, F. Qian, H. Ruan, L. Liu, “Parallel optical negabinary signed-digit computing: algorithm and optical implementation,” Opt. Eng. 38, 403–414 (1999).
[CrossRef]

Ruan, H.

G. Li, F. Qian, H. Ruan, L. Liu, “Parallel optical negabinary signed-digit computing: algorithm and optical implementation,” Opt. Eng. 38, 403–414 (1999).
[CrossRef]

Seth, M.

A. K. Datta, M. Seth, “Parallel arithmetic operations in an optical architecture using a modified iterative technique,” Opt. Comm. 115, 245–250 (1995).
[CrossRef]

A. K. Datta, M. Seth, “Multi-input optical parallel logic processing with the shadow-casting technique,” App. Opt. 33, 8146–8152 (1994).
[CrossRef]

Shao, L.

Swartzlander, E.

Tanida, J.

J. Tanida, M. Iwata, Y. Ichioka, “Extended coding for optical array logic,” Appl. Opt. 33, 3363–3369 (1994).
[CrossRef]

J. Tanida, Y. Ichioka, “Optical logic array processor using shadowgrams,” J. Opt. Soc. Am. 73, 800–809 (1983).
[CrossRef]

Tsunoda, Y.

Wang, Z.

G. Li, L. Liu, L. Shao, Z. Wang, “Negative binary arithmetic algorithms for digital parallel optical computation,” Opt. Lett. 19, 1337–1339 (1994).
[CrossRef] [PubMed]

Y. Wu, Z. Zhang, L. Liu, Z. Wang, “Arithmetic operation using binary encoding modified-signed-digit system,” Opt. Comm. 100, 53–58 (1993).
[CrossRef]

Westerkamp, J. J.

Wong, K. W.

K. W. Wong, L. M. Cheng, “Optical modified signed-digit addition based on binary logical operations,” Opt. Laser Technol. 26, 213–217 (1994).
[CrossRef]

Wu, Y.

Y. Wu, Z. Zhang, L. Liu, Z. Wang, “Arithmetic operation using binary encoding modified-signed-digit system,” Opt. Comm. 100, 53–58 (1993).
[CrossRef]

Yan, X.

G. Li, L. Liu, H. Cheng, X. Yan, “Parallel optical quaternary signed-digit multiplication and its use for matrix-vector operation,” Optik (Stuttgart) 107, 165–172 (1998).

Yatagai, T.

H. Huang, M. Itoh, T. Yatagai, “Optical scalable parallel modified signed-digit algorithms for large-scale array addition and multiplication using digit-decomposition-plane representation,” Opt. Eng. 38, 432–440 (1999).
[CrossRef]

H. Huang, M. Itoh, T. Yatagai, “Modified signed-digit arithmetic based on redundant bit representation,” Appl. Opt. 33, 6146–6156 (1994).
[CrossRef] [PubMed]

Yeh, P.

Yin, Y.

Zhang, S.

S. Zhang, M. A. Karim, “Programmable modified-signed-digit addition module based on binary logic gates,” Opt. Eng. 38, 456–461 (1999).
[CrossRef]

S. Zhang, M. A. Karim, “Optical arithmetic processing using improved redundant binary algorithms,” Opt. Eng. 38, 415–421 (1999).
[CrossRef]

Zhang, Z.

Y. Wu, Z. Zhang, L. Liu, Z. Wang, “Arithmetic operation using binary encoding modified-signed-digit system,” Opt. Comm. 100, 53–58 (1993).
[CrossRef]

Zhou, S.

App. Opt.

S. Mukhopadhyay, A. Basuray, A. K. Datta, “New technique of arithmetic operation using positional residue system,” App. Opt. 29, 2981–2893 (1990).
[CrossRef]

A. K. Datta, M. Seth, “Multi-input optical parallel logic processing with the shadow-casting technique,” App. Opt. 33, 8146–8152 (1994).
[CrossRef]

Appl. Opt.

A. Huang, Y. Tsunoda, J. W. Goodman, S. Ishihara, “Optical computation using residue arithmetic,” Appl. Opt. 18, 149–162 (1979).
[CrossRef] [PubMed]

C. Perlee, D. Casasent, “Negative base encoding in optical linear algebra processors,” Appl. Opt. 25, 168–169 (1986).
[CrossRef] [PubMed]

R. P. Bocker, B. L. Drake, M. E. Lasher, T. B. Henderson, “Modified signed-digit addition and subtraction using optical symbolic substitution,” Appl. Opt. 25, 2456–2457 (1986).
[CrossRef] [PubMed]

E. Swartzlander, “Digital optical arithmetic,” Appl. Opt. 25, 3021–3032 (1986).
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Figures (9)

Fig. 1
Fig. 1

Unified experimental optical architecture for arithmetic, logical, and matrix-vector operations.

Fig. 2
Fig. 2

Spatially encoded (3 × 3) patterns of the operands a and A.

Fig. 3
Fig. 3

Superimposed (3 × 3) input patterns of nth bit of operands a and A required for arithmetic operations when (a) operands a and A are represented as signed negabinary and unsigned negabinary numbers, respectively, and (b) operands a and A are represented as unsigned negabinary and signed negabinary numbers, respectively.

Fig. 4
Fig. 4

Decoding mask of nth bit for an arithmetic operation.

Fig. 5
Fig. 5

Source structure and cell structure at location R of the LCD panel.

Fig. 6
Fig. 6

(a) Patterns for the operands a = 44 (in 8-bit signed negabinary form) and A = -141 (in 8-bit negabinary form) designed for arithmetic operations. (b) Superimposed pattern of a = 44 and A = -141. (c) Decoding mask pattern.

Fig. 7
Fig. 7

(a) Superimposed (3 × 3) input patterns of nth bit of a and A for logical operations when both operands are represented as signed negabinary numbers. (b) Decoding mask of nth bit for a logical operation. (c) Patterns at location R of the LCD panel for introduction of operational kernels for some logical operations.

Fig. 8
Fig. 8

(a) Superimposed 2-bit patterns of the operands a and A required for logical operations, where (i) a = 0 and A = 0, (ii) a = 0 and A = 1, (iii) a = 1 and A = 0, and (iv) a = 1 and A = 1. (b) Decoding mask pattern for a logical operation.

Fig. 9
Fig. 9

Schematic optical architecture for matrix-vector multiplication.

Tables (1)

Tables Icon

Table 1 Truth Tables for or, and, and Ex-or in the Signed Negabinary Number System

Equations (14)

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x=i=-MN-1 xi-2i, xi0, 1,
y=j=-PQ-1 yj-2j, yj0, 1¯,
z=k=-RS-1 zk-2k, zk0, 1¯, 1,
augend a:1¯ 1¯ 0 1¯ 0 1¯ 0 0decimal 44addend A:1 0 1 1 0 1 1 1¯decimal -141final sum:0 1¯ 1 0 0 0 1 1decimal -97..
0+0=0, 0+1¯=1¯, 1¯+1¯=1¯ OR operation; 0  0=0, 0  1¯=0, 1¯  1¯=1¯ AND operation; 00=0, 01¯=1¯, 1¯1¯=0 ex-OR operation.
00signnega+1¯ 1¯signnega=1¯ 1¯signnega=1dec, 00signnega  1¯ 1¯signnega=00signnega=0dec.
h=aA=m=0L-1n=0M-1 amAn-2m+n,
a=m=0L-1 am-2m, A=n=0M-1 An-2n.
h=h14-214+h13-213++h1-21+h0-20,
h0=a0A0, h1=a1A0+a0A1, h2=a2A0+a1A1+a0A2,  h7=a7A0+a6A1+a5A2+a4A3+a3A4+a2A5+a1A6+a0A7,  h14=a7A7.
abcdefghijklABC= aA+bB+cCdA+eB+fCgA+hB+iCjA+kB+lC,
aA+bB+cC=h+h+h=h14+h14+h14-214++h6+h6+h6-26+h5+h5+h5-25++h0+h0+h0-20,
h0+h0+h0=a0A0+b0B0+c0C0, h1+h1+h1=a0A1+a1A0+b0B1+b1B0+c0C1+c1C0 =a0A1+b0B1+c0C1+a1A0+b1B0+c1C0, h2+h2+h2=a0A2+a1A1+a2A0+b0B2+b1B1+b2B0+c0C2+c1C1+c2C0 =a0A2+b0B2+c0C2+a1A1+b1B1+c1C1+a2A0+b2B0+C2C0, h3+h3+h3=a0A3+a1A2+a2A1+a3A0+b0B3+b1B2+b2B1+b3B0+c0C3+c1C2+c2C1+c3C0 =a0A3+b0B3+c0C3+a1A2+b1B2+c1C2+a2A1+b2B1+c2C1+a3A0+b3B0+c3C0,  h14+h14+h14=a7A7+b7B7+c7C7.
d=ld1/l-d1, z+z1=lz1/l-d1, d1<1,

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