Optical processing based on conditional higher-order trinary modified signed-digit symbolic substitution

M. S. Alam; M. A. Karim; A. A. S. Awwal; J. J. Westerkamp

doi:10.1364/AO.31.005614

Optical processing based on conditional higher-order trinary modified signed-digit symbolic substitution

M. S. Alam, M. A. Karim, A. A. S. Awwal, and J. J. Westerkamp

Applied Optics
Vol. 31,
Issue 26,
pp. 5614-5621
(1992)
•https://doi.org/10.1364/AO.31.005614

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M. S. Alam, M. A. Karim, A. A. S. Awwal, and J. J. Westerkamp, "Optical processing based on conditional higher-order trinary modified signed-digit symbolic substitution," Appl. Opt. 31, 5614-5621 (1992)

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History
- Original Manuscript: February 28, 1991
- Published: September 10, 1992

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Abstract

Techniques for higher-order modified signed-digit trinary arithmetic by using optical symbolic substitution are presented. This method provides fast multibit computation by adopting a two-step symbolic-substitution scheme. Since more information is represented in fewer digits, this technique leads to a compact design. A content-addressable memory-based and a joint transform correlator-based optical implemention for the proposed technique are also presented.

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Condition Number	Decimal Representation	Trinary MSD Representation
1	−16	$\bar{2} 0 2, \bar{2} 1 \bar{1}$
2	−15	$\bar{2} 1 0, \bar{1} \bar{2} 0$
3	−14	$\bar{2} 2 \bar{2}, \bar{2} 1 1, \bar{1} \bar{2} 1, \bar{1} \bar{1} \bar{2}$
4	−13	$\bar{2} 2 1, \bar{1} \bar{2} 2, \bar{1} \bar{1} 1$
5	−12	$\bar{2} 2 0, \bar{1} \bar{1} 0$
6	−11	$\bar{2} 2 1$ , 1 0 2, $1 1 \bar{1}$
7	−10	$\bar{2} 2 2, \bar{1} 0 \bar{1}, \bar{1} \bar{1} 2$
8	−9	$\bar{1} 0 0$
9	−8	$\bar{1} 0 1, 0 \bar{2} \bar{2}, \bar{1} 1 \bar{2}$
10	−7	$\bar{1} 0 2, 0 \bar{2} \bar{1}, \bar{1} 1 \bar{1}$
11	−6	$\bar{1} 1 0, 0 \bar{2} 0$
12	−5	$\bar{1} 2 \bar{2}, 0 \bar{2} 1, \bar{1} 1 1, 0 \bar{1} \bar{2}$
13	−4	$0 \bar{2} 2, 0 \bar{1} \bar{1}, \bar{1} 2 \bar{1}, \bar{1} 1 2$
14	−3	$\bar{1} 2 0, 0 \bar{1} 0$
15	−2	$\bar{1} 2 1, 0 0 \bar{2}, 0 \bar{1} 1$
16	−1	$\bar{1} 2 2, 0 0 \bar{1}, 0 \bar{1} 2$
17	0	0 0 0
18	1	$1 \bar{2} \bar{2}$ , 0 0 1, $0 1 \bar{2}$
19	2	$1 \bar{2} \bar{1}$ , 0 0 2, $0 1 \bar{1}$
20	3	$1 \bar{2} 0$ , 0 1 0
21	4	$0 2 \bar{2}$ , 0 1 1, $1 \bar{2} 1, 1 \bar{1} \bar{2}$
22	5	$1 \bar{2} 2, 0 2 \bar{1}, 1 \bar{1} \bar{1}$ , 012
23	6	$1 \bar{1} 0$ , 0 2 0
24	7	$1 0 \bar{2}$ , 0 2 1, $1 \bar{1} 1$
25	8	$1 0 \bar{1}$ , 0 2 2, $1 \bar{1} 2$
26	9	1 0 0
27	10	$2 \bar{2} \bar{2}$ , 1 0 1, $1 1 \bar{2}$
28	11	$2 \bar{2} \bar{1}$ , 1 0 2, $1 1 \bar{1}$
29	12	$2 \bar{2} 0$ , 1 1 0
30	13	$2 \bar{2} 1$ , 1 1 1, $1 2 \bar{2}$
31	14	$2 \bar{2} 2, 2 \bar{1} \bar{1}, 1 2 \bar{1}$ , 1 1 2
32	15	$2 \bar{1} 0$ , 1 2 0
33	16	$2 0 \bar{2}, 2 \bar{1} 1$

Condition Number	Addend A_iA_i₋₁	Augend B_iB_i₋₁	A_i₋₂B_i₋₂	Carry C_i	Sum S_iS_i₋₁
1	$\bar{2} \bar{2}$	$\bar{2} \bar{2}$	All negative	$\bar{2}$	0 2
			Otherwise	$\bar{2}$	1 $\bar{1}$
2	$\bar{2} \bar{2}$	$\bar{2} \bar{1}$	Don’t care	$\bar{2}$	1 0
	$\bar{2} \bar{1}$	$\bar{2} \bar{2}$
3	$\bar{2} \bar{2}$	$\bar{2} 0$	All negative	$\bar{2}$	1 1
	$\bar{2} 0$	$\bar{2} \bar{2}$	Otherwise	$\bar{2}$	$2 \bar{2}$
	$\bar{2} \bar{1}$	$\bar{2} \bar{1}$
4	$\bar{2} \bar{2}$	$\bar{1} \bar{2} (\bar{2} 1)$	All negative	$\bar{1}$	$\bar{2} 2$
	$\bar{1} \bar{2} (\bar{2} 1)$	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	$\bar{2} 0$	Otherwise	$\bar{1}$	$\bar{1} \bar{1}$
	$\bar{2} 0$	$\bar{2} \bar{1}$
5	$\bar{2} \bar{2}$	$\bar{1} \bar{1} (\bar{2} 2)$	Don’t care	$\bar{1}$	$\bar{1} 0$
	$\bar{1} \bar{1} (\bar{2} 2)$	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	$\bar{2} 1 (\bar{1} \bar{2})$
	$\bar{2} 1 (\bar{1} \bar{2})$	$\bar{2} \bar{1}$
	$\bar{2} 0$	$\bar{2} 0$
6	$\bar{2} \bar{2}$	$\bar{1} 0$	All negative	$\bar{1}$	$0 \bar{2}$
	$\bar{1} 0$	$\bar{2} \bar{2}$
	$\bar{2} 0$	$\bar{1} \bar{2} (\bar{2} 1)$
	$\bar{1} \bar{2} (\bar{2} 1)$	$\bar{2} 0$	Otherwise	$\bar{1}$	$\bar{1} 1$
	$\bar{2} \bar{1}$	$\bar{1} \bar{1} (\bar{2} 2)$
	$\bar{1} \bar{1} (\bar{2} 2)$	$\bar{2} \bar{1}$
7	$\bar{2} \bar{2}$	$0 \bar{2} (\bar{1} 1)$	All negative	$\bar{2}$	2 2
	$0 \bar{2} (\bar{1} 1)$	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	$\bar{1} 0$
	$\bar{1} 0$	$\bar{2} \bar{1}$
	$\bar{2} 0$	$\bar{1} \bar{1} (\bar{2} 2)$	Otherwise	$\bar{1}$	$0 \bar{1}$
	$\bar{1} \bar{1} (\bar{2} 2)$	$\bar{2} 0$
	$\bar{1} \bar{2} (\bar{2} 1)$	$\bar{1} \bar{2} (\bar{2} 1)$
8	$\bar{2} \bar{2}$	$0 \bar{1} (\bar{1} 2)$	Don’t care	$\bar{1}$	0 0
	$0 \bar{1} (\bar{1} 2)$	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	$0 \bar{2} (\bar{1} 1)$
	$0 \bar{2} (\bar{1} 1)$	$\bar{2} \bar{1}$
	$\bar{2} 0$	$\bar{1} 0$
	$\bar{1} 0$	$\bar{2} 0$
	$\bar{1} 2 (\bar{2} 1)$	$\bar{1} \bar{1} (\bar{2} 2)$
	$\bar{1} \bar{1} (\bar{2} 2)$	$\bar{1} \bar{2} (\bar{2} 1)$
9	$\bar{2} \bar{2}$	0 0	All negative	$\bar{1}$	0 1
	0 0	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	$0 \bar{1} (\bar{1} 2)$
	$0 \bar{1} (\bar{1} 2)$	$\bar{2} \bar{1}$
	$\bar{2} 0$	$0 \bar{2} (\bar{1} 1)$
	$0 \bar{2} (\bar{1} 1)$	$\bar{2} 0$	Otherwise	0	$\bar{2} \bar{2}$
	$\bar{1} \bar{2} (\bar{2} 1)$	$\bar{1} 0$
	$\bar{1} 0$	$\bar{1} \bar{2} (\bar{2} 1)$
	$\bar{1} \bar{1} (\bar{2} 2)$	$\bar{1} \bar{1} (\bar{2} 2)$
10	$\bar{2} \bar{2}$	$0 1 (1 \bar{2})$	All negative	$\bar{1}$	0 2
	$0 1 (1 \bar{2})$	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	0 0
	0 0	$\bar{2} \bar{1}$
	$\bar{2} 0$	$0 \bar{1} (\bar{1} 2)$
	$0 \bar{1} (\bar{1} 2)$	$\bar{2} 0$	Otherwise	0	$\bar{2} \bar{1}$
	$\bar{1} \bar{2} (\bar{2} 1)$	$0 \bar{2} (\bar{1} 1)$
	$0 \bar{2} (\bar{1} 1)$	$\bar{1} \bar{2} (\bar{2} 1)$
	$\bar{1} 0$	$\bar{1} \bar{1} (\bar{2} 2)$
	$\bar{1} \bar{1} (\bar{2} 2)$	$\bar{1} 0$
11	$\bar{2} \bar{2}$	$0 2 (1 \bar{1})$	Don’t care	0	$\bar{2} 0$
	$0 2 (1 \bar{1})$	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	$0 1 (1 \bar{2})$
	$0 1 (1 \bar{2})$	$\bar{2} \bar{1}$
	$\bar{2} 0$	0 0
	0 0	$\bar{2} 0$
	$0 \bar{2} (\bar{1} 1)$	$\bar{1} \bar{1} (\bar{2} 2)$
	$\bar{1} \bar{1} (\bar{2} 2)$	$0 \bar{2} (\bar{1} 1)$
	$0 \bar{1} (\bar{1} 2)$	$\bar{2} 1 (\bar{1} \bar{2})$
	$\bar{2} 1 (\bar{1} \bar{2})$	$0 \bar{1} (\bar{1} 2)$
	$\bar{1} 0$	$\bar{1} 0$
12	$\bar{2} \bar{2}$	1 0	All negative	0	$\bar{2} 1$
	1 0	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	$0 2 (1 \bar{1})$
	$0 2 (1 \bar{1})$	$\bar{2} \bar{1}$
	$\bar{2} 0$	$0 1 (1 \bar{2})$
	$0 1 (1 \bar{2})$	$\bar{2} 0$
	$\bar{1} \bar{1} (\bar{2} 2)$	$0 \bar{1} (\bar{1} 2)$	Otherwise	0	$\bar{1} \bar{2}$
	$0 \bar{1} (\bar{1} 2)$	$\bar{1} \bar{1} (\bar{2} 2)$
	0 0	$\bar{2} 1 (\bar{1} \bar{2})$
	$\bar{2} 1 (\bar{1} \bar{2})$	0 0
	$\bar{1} 0$	$0 \bar{2} (\bar{1} 1)$
	$0 \bar{2} (\bar{1} 1)$	$\bar{1} 0$
13	$\bar{2} \bar{2}$	$1 1 (2 \bar{2})$	All negative	0	$\bar{2} 2$
	$1 1 (2 \bar{2})$	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	1 0
	1 0	$\bar{2} \bar{1}$
	$\bar{2} 0$	$0 2 (1 \bar{1})$
	$0 2 (1 \bar{1})$	$\bar{2} 0$
	$\bar{2} 1 (\bar{1} \bar{2})$	$0 1 (1 \bar{2})$
	$0 1 (1 \bar{2})$	$\bar{2} 1 (\bar{1} \bar{2})$	Otherwise	0	$\bar{1} \bar{1}$
	$\bar{1} \bar{1} (\bar{2} 2)$	0 0
	0 0	$\bar{1} \bar{1} (\bar{2} 2)$
	$\bar{1} 0$	$0 \bar{1} (\bar{1} 2)$
	$0 \bar{1} (\bar{1} 2)$	$\bar{1} 0$
	$0 \bar{2} (\bar{1} 1)$	$\bar{1} \bar{2} (\bar{2} 1)$
	$\bar{1} \bar{2} (\bar{2} 1)$	$0 \bar{2} (\bar{1} 1)$
14	$\bar{2} \bar{2}$	$1 2 (2 \bar{1})$	Don’t care	0	$\bar{1} 0$
	$1 2 (2 \bar{1})$	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	$1 1 (2 \bar{2})$
	$1 1 (2 \bar{2})$	$\bar{2} \bar{1}$
	$\bar{2} 0$	1 0
	1 0	$\bar{2} 0$
	$\bar{2} 1 (\bar{1} \bar{2})$	$0 2 (1 \bar{1})$
	$0 2 (1 \bar{1})$	$\bar{2} 1 (\bar{1} \bar{2})$
	$\bar{1} \bar{1} (\bar{2} 2)$	0 1(1 2)
	$0 1 (1 \bar{2})$	$\bar{1} \bar{1} (\bar{2} 2)$
	$0 \bar{2} (\bar{1} 1)$	$0 \bar{1} (\bar{1} 2)$
	$0 \bar{1} (\bar{1} \bar{2})$	$0 \bar{2} (\bar{1} 1)$
	$\bar{1} 0$	0 0
	0 0	$\bar{1} 0$
15	$\bar{2} \bar{2}$	2 0	All negative	0	$\bar{1} 1$
	2 0	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	$1 2 (2 \bar{1})$
	$1 2 (2 \bar{1})$	$\bar{2} \bar{1}$
	$\bar{2} 0$	$1 1 (2 \bar{2})$
	$1 1 (2 \bar{2})$	$\bar{2} 0$
	$\bar{2} 1 (\bar{1} 2)$	1 0
	1 0	$\bar{2} 1 (\bar{1} \bar{2})$
	$\bar{1} \bar{1} (\bar{2} 2)$	$0 2 (1 \bar{1})$	Otherwise	0	$0 \bar{2}$
	$0 2 (1 \bar{1})$	$\bar{1} \bar{1} (\bar{2} 2)$
	$\bar{1} 0$	$0 1 (1 \bar{2})$
	$0 1 (1 \bar{2})$	$\bar{1} 0$
	$0 \bar{2} (\bar{1} 1)$	0 0
	0 0	$0 \bar{2} (\bar{1} 1)$
	$0 \bar{1}$	$0 \bar{1}$
16	$\bar{2} \bar{2}$	2 1	All negative	0	$\bar{1} 2$
	2 1	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	2 0
	2 0	$\bar{2} \bar{1}$
	$\bar{2} 0$	$1 2 (2 \bar{1})$
	$1 2 (2 \bar{1})$	$\bar{2} 0$
	$\bar{2} 1 (\bar{1} \bar{2})$	$1 1 (2 \bar{2})$
	$1 1 (2 \bar{2})$	$\bar{2} 1 (\bar{1} \bar{2})$
	$\bar{1} \bar{1} (\bar{2} 2)$	1 0	Otherwise	0	$0 \bar{1}$
	1 0	$\bar{1} \bar{1} (\bar{2} \bar{2})$
	$\bar{1} 0$	$0 2 (1 \bar{1})$
	$0 2 (1 \bar{1})$	$\bar{1} 0$
	$0 \bar{2} (\bar{1} 1)$	$0 1 (1 \bar{2})$
	$0 1 (1 \bar{2})$	$0 \bar{2} (\bar{1} 1)$
	$0 \bar{1} (\bar{1} 2)$	0 0
	0 0	$0 \bar{1} (\bar{1} 2)$
17	0 0	0 0	Don’t care	0	0 0
18	2 2	$\bar{2} \bar{1}$	All negative	0	0 1
	$\bar{2} \bar{1}$	2 2
	2 1	$\bar{2} 0$
	$\bar{2} 0$	2 1
	2 0	$\bar{1} \bar{2} (\bar{2} 1)$
	$\bar{1} \bar{2} (\bar{2} 1)$	2 0
	$2 \bar{1} (1 2)$	$\bar{1} \bar{1} (\bar{2} 2)$
	$\bar{1} \bar{1} (\bar{2} 2)$	$2 \bar{1} (1 2)$
	$1 1 (2 \bar{2})$	$\bar{1} 0$	Otherwise	0	$1 \bar{2}$
	$\bar{1} 0$	$1 1 (2 \bar{2})$
	1 0	$0 \bar{2} (\bar{1} 1)$
	$0 \bar{2} (\bar{1} 1)$	1 0
	$0 2 (1 \bar{1})$	$0 \bar{1} (\bar{1} 2)$
	$0 \bar{1} (\bar{1} 2)$	$0 2 (1 \bar{1})$
	$0 1 (1 \bar{2})$	0 0
	0 0	$0 1 (1 \bar{2})$
·	·		·	·	·
·	·		·	·	·
·	·		·	·	·
32	2 2	2 1	Don’t care	1	2 0
	2 1	2 2
33	2 2	2 2	All negative	2	$\bar{1} 1$
			Otherwise	2	$0 \bar{2}$

Condition Number	Decimal Representation	Trinary MSD Representation
1	−16	$\bar{2} 0 2, \bar{2} 1 \bar{1}$
2	−15	$\bar{2} 1 0, \bar{1} \bar{2} 0$
3	−14	$\bar{2} 2 \bar{2}, \bar{2} 1 1, \bar{1} \bar{2} 1, \bar{1} \bar{1} \bar{2}$
4	−13	$\bar{2} 2 1, \bar{1} \bar{2} 2, \bar{1} \bar{1} 1$
5	−12	$\bar{2} 2 0, \bar{1} \bar{1} 0$
6	−11	$\bar{2} 2 1$ , 1 0 2, $1 1 \bar{1}$
7	−10	$\bar{2} 2 2, \bar{1} 0 \bar{1}, \bar{1} \bar{1} 2$
8	−9	$\bar{1} 0 0$
9	−8	$\bar{1} 0 1, 0 \bar{2} \bar{2}, \bar{1} 1 \bar{2}$
10	−7	$\bar{1} 0 2, 0 \bar{2} \bar{1}, \bar{1} 1 \bar{1}$
11	−6	$\bar{1} 1 0, 0 \bar{2} 0$
12	−5	$\bar{1} 2 \bar{2}, 0 \bar{2} 1, \bar{1} 1 1, 0 \bar{1} \bar{2}$
13	−4	$0 \bar{2} 2, 0 \bar{1} \bar{1}, \bar{1} 2 \bar{1}, \bar{1} 1 2$
14	−3	$\bar{1} 2 0, 0 \bar{1} 0$
15	−2	$\bar{1} 2 1, 0 0 \bar{2}, 0 \bar{1} 1$
16	−1	$\bar{1} 2 2, 0 0 \bar{1}, 0 \bar{1} 2$
17	0	0 0 0
18	1	$1 \bar{2} \bar{2}$ , 0 0 1, $0 1 \bar{2}$
19	2	$1 \bar{2} \bar{1}$ , 0 0 2, $0 1 \bar{1}$
20	3	$1 \bar{2} 0$ , 0 1 0
21	4	$0 2 \bar{2}$ , 0 1 1, $1 \bar{2} 1, 1 \bar{1} \bar{2}$
22	5	$1 \bar{2} 2, 0 2 \bar{1}, 1 \bar{1} \bar{1}$ , 012
23	6	$1 \bar{1} 0$ , 0 2 0
24	7	$1 0 \bar{2}$ , 0 2 1, $1 \bar{1} 1$
25	8	$1 0 \bar{1}$ , 0 2 2, $1 \bar{1} 2$
26	9	1 0 0
27	10	$2 \bar{2} \bar{2}$ , 1 0 1, $1 1 \bar{2}$
28	11	$2 \bar{2} \bar{1}$ , 1 0 2, $1 1 \bar{1}$
29	12	$2 \bar{2} 0$ , 1 1 0
30	13	$2 \bar{2} 1$ , 1 1 1, $1 2 \bar{2}$
31	14	$2 \bar{2} 2, 2 \bar{1} \bar{1}, 1 2 \bar{1}$ , 1 1 2
32	15	$2 \bar{1} 0$ , 1 2 0
33	16	$2 0 \bar{2}, 2 \bar{1} 1$

Condition Number	Addend A_iA_i₋₁	Augend B_iB_i₋₁	A_i₋₂B_i₋₂	Carry C_i	Sum S_iS_i₋₁
1	$\bar{2} \bar{2}$	$\bar{2} \bar{2}$	All negative	$\bar{2}$	0 2
			Otherwise	$\bar{2}$	1 $\bar{1}$
2	$\bar{2} \bar{2}$	$\bar{2} \bar{1}$	Don’t care	$\bar{2}$	1 0
	$\bar{2} \bar{1}$	$\bar{2} \bar{2}$
3	$\bar{2} \bar{2}$	$\bar{2} 0$	All negative	$\bar{2}$	1 1
	$\bar{2} 0$	$\bar{2} \bar{2}$	Otherwise	$\bar{2}$	$2 \bar{2}$
	$\bar{2} \bar{1}$	$\bar{2} \bar{1}$
4	$\bar{2} \bar{2}$	$\bar{1} \bar{2} (\bar{2} 1)$	All negative	$\bar{1}$	$\bar{2} 2$
	$\bar{1} \bar{2} (\bar{2} 1)$	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	$\bar{2} 0$	Otherwise	$\bar{1}$	$\bar{1} \bar{1}$
	$\bar{2} 0$	$\bar{2} \bar{1}$
5	$\bar{2} \bar{2}$	$\bar{1} \bar{1} (\bar{2} 2)$	Don’t care	$\bar{1}$	$\bar{1} 0$
	$\bar{1} \bar{1} (\bar{2} 2)$	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	$\bar{2} 1 (\bar{1} \bar{2})$
	$\bar{2} 1 (\bar{1} \bar{2})$	$\bar{2} \bar{1}$
	$\bar{2} 0$	$\bar{2} 0$
6	$\bar{2} \bar{2}$	$\bar{1} 0$	All negative	$\bar{1}$	$0 \bar{2}$
	$\bar{1} 0$	$\bar{2} \bar{2}$
	$\bar{2} 0$	$\bar{1} \bar{2} (\bar{2} 1)$
	$\bar{1} \bar{2} (\bar{2} 1)$	$\bar{2} 0$	Otherwise	$\bar{1}$	$\bar{1} 1$
	$\bar{2} \bar{1}$	$\bar{1} \bar{1} (\bar{2} 2)$
	$\bar{1} \bar{1} (\bar{2} 2)$	$\bar{2} \bar{1}$
7	$\bar{2} \bar{2}$	$0 \bar{2} (\bar{1} 1)$	All negative	$\bar{2}$	2 2
	$0 \bar{2} (\bar{1} 1)$	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	$\bar{1} 0$
	$\bar{1} 0$	$\bar{2} \bar{1}$
	$\bar{2} 0$	$\bar{1} \bar{1} (\bar{2} 2)$	Otherwise	$\bar{1}$	$0 \bar{1}$
	$\bar{1} \bar{1} (\bar{2} 2)$	$\bar{2} 0$
	$\bar{1} \bar{2} (\bar{2} 1)$	$\bar{1} \bar{2} (\bar{2} 1)$
8	$\bar{2} \bar{2}$	$0 \bar{1} (\bar{1} 2)$	Don’t care	$\bar{1}$	0 0
	$0 \bar{1} (\bar{1} 2)$	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	$0 \bar{2} (\bar{1} 1)$
	$0 \bar{2} (\bar{1} 1)$	$\bar{2} \bar{1}$
	$\bar{2} 0$	$\bar{1} 0$
	$\bar{1} 0$	$\bar{2} 0$
	$\bar{1} 2 (\bar{2} 1)$	$\bar{1} \bar{1} (\bar{2} 2)$
	$\bar{1} \bar{1} (\bar{2} 2)$	$\bar{1} \bar{2} (\bar{2} 1)$
9	$\bar{2} \bar{2}$	0 0	All negative	$\bar{1}$	0 1
	0 0	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	$0 \bar{1} (\bar{1} 2)$
	$0 \bar{1} (\bar{1} 2)$	$\bar{2} \bar{1}$
	$\bar{2} 0$	$0 \bar{2} (\bar{1} 1)$
	$0 \bar{2} (\bar{1} 1)$	$\bar{2} 0$	Otherwise	0	$\bar{2} \bar{2}$
	$\bar{1} \bar{2} (\bar{2} 1)$	$\bar{1} 0$
	$\bar{1} 0$	$\bar{1} \bar{2} (\bar{2} 1)$
	$\bar{1} \bar{1} (\bar{2} 2)$	$\bar{1} \bar{1} (\bar{2} 2)$
10	$\bar{2} \bar{2}$	$0 1 (1 \bar{2})$	All negative	$\bar{1}$	0 2
	$0 1 (1 \bar{2})$	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	0 0
	0 0	$\bar{2} \bar{1}$
	$\bar{2} 0$	$0 \bar{1} (\bar{1} 2)$
	$0 \bar{1} (\bar{1} 2)$	$\bar{2} 0$	Otherwise	0	$\bar{2} \bar{1}$
	$\bar{1} \bar{2} (\bar{2} 1)$	$0 \bar{2} (\bar{1} 1)$
	$0 \bar{2} (\bar{1} 1)$	$\bar{1} \bar{2} (\bar{2} 1)$
	$\bar{1} 0$	$\bar{1} \bar{1} (\bar{2} 2)$
	$\bar{1} \bar{1} (\bar{2} 2)$	$\bar{1} 0$
11	$\bar{2} \bar{2}$	$0 2 (1 \bar{1})$	Don’t care	0	$\bar{2} 0$
	$0 2 (1 \bar{1})$	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	$0 1 (1 \bar{2})$
	$0 1 (1 \bar{2})$	$\bar{2} \bar{1}$
	$\bar{2} 0$	0 0
	0 0	$\bar{2} 0$
	$0 \bar{2} (\bar{1} 1)$	$\bar{1} \bar{1} (\bar{2} 2)$
	$\bar{1} \bar{1} (\bar{2} 2)$	$0 \bar{2} (\bar{1} 1)$
	$0 \bar{1} (\bar{1} 2)$	$\bar{2} 1 (\bar{1} \bar{2})$
	$\bar{2} 1 (\bar{1} \bar{2})$	$0 \bar{1} (\bar{1} 2)$
	$\bar{1} 0$	$\bar{1} 0$
12	$\bar{2} \bar{2}$	1 0	All negative	0	$\bar{2} 1$
	1 0	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	$0 2 (1 \bar{1})$
	$0 2 (1 \bar{1})$	$\bar{2} \bar{1}$
	$\bar{2} 0$	$0 1 (1 \bar{2})$
	$0 1 (1 \bar{2})$	$\bar{2} 0$
	$\bar{1} \bar{1} (\bar{2} 2)$	$0 \bar{1} (\bar{1} 2)$	Otherwise	0	$\bar{1} \bar{2}$
	$0 \bar{1} (\bar{1} 2)$	$\bar{1} \bar{1} (\bar{2} 2)$
	0 0	$\bar{2} 1 (\bar{1} \bar{2})$
	$\bar{2} 1 (\bar{1} \bar{2})$	0 0
	$\bar{1} 0$	$0 \bar{2} (\bar{1} 1)$
	$0 \bar{2} (\bar{1} 1)$	$\bar{1} 0$
13	$\bar{2} \bar{2}$	$1 1 (2 \bar{2})$	All negative	0	$\bar{2} 2$
	$1 1 (2 \bar{2})$	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	1 0
	1 0	$\bar{2} \bar{1}$
	$\bar{2} 0$	$0 2 (1 \bar{1})$
	$0 2 (1 \bar{1})$	$\bar{2} 0$
	$\bar{2} 1 (\bar{1} \bar{2})$	$0 1 (1 \bar{2})$
	$0 1 (1 \bar{2})$	$\bar{2} 1 (\bar{1} \bar{2})$	Otherwise	0	$\bar{1} \bar{1}$
	$\bar{1} \bar{1} (\bar{2} 2)$	0 0
	0 0	$\bar{1} \bar{1} (\bar{2} 2)$
	$\bar{1} 0$	$0 \bar{1} (\bar{1} 2)$
	$0 \bar{1} (\bar{1} 2)$	$\bar{1} 0$
	$0 \bar{2} (\bar{1} 1)$	$\bar{1} \bar{2} (\bar{2} 1)$
	$\bar{1} \bar{2} (\bar{2} 1)$	$0 \bar{2} (\bar{1} 1)$
14	$\bar{2} \bar{2}$	$1 2 (2 \bar{1})$	Don’t care	0	$\bar{1} 0$
	$1 2 (2 \bar{1})$	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	$1 1 (2 \bar{2})$
	$1 1 (2 \bar{2})$	$\bar{2} \bar{1}$
	$\bar{2} 0$	1 0
	1 0	$\bar{2} 0$
	$\bar{2} 1 (\bar{1} \bar{2})$	$0 2 (1 \bar{1})$
	$0 2 (1 \bar{1})$	$\bar{2} 1 (\bar{1} \bar{2})$
	$\bar{1} \bar{1} (\bar{2} 2)$	0 1(1 2)
	$0 1 (1 \bar{2})$	$\bar{1} \bar{1} (\bar{2} 2)$
	$0 \bar{2} (\bar{1} 1)$	$0 \bar{1} (\bar{1} 2)$
	$0 \bar{1} (\bar{1} \bar{2})$	$0 \bar{2} (\bar{1} 1)$
	$\bar{1} 0$	0 0
	0 0	$\bar{1} 0$
15	$\bar{2} \bar{2}$	2 0	All negative	0	$\bar{1} 1$
	2 0	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	$1 2 (2 \bar{1})$
	$1 2 (2 \bar{1})$	$\bar{2} \bar{1}$
	$\bar{2} 0$	$1 1 (2 \bar{2})$
	$1 1 (2 \bar{2})$	$\bar{2} 0$
	$\bar{2} 1 (\bar{1} 2)$	1 0
	1 0	$\bar{2} 1 (\bar{1} \bar{2})$
	$\bar{1} \bar{1} (\bar{2} 2)$	$0 2 (1 \bar{1})$	Otherwise	0	$0 \bar{2}$
	$0 2 (1 \bar{1})$	$\bar{1} \bar{1} (\bar{2} 2)$
	$\bar{1} 0$	$0 1 (1 \bar{2})$
	$0 1 (1 \bar{2})$	$\bar{1} 0$
	$0 \bar{2} (\bar{1} 1)$	0 0
	0 0	$0 \bar{2} (\bar{1} 1)$
	$0 \bar{1}$	$0 \bar{1}$
16	$\bar{2} \bar{2}$	2 1	All negative	0	$\bar{1} 2$
	2 1	$\bar{2} \bar{2}$
	$\bar{2} \bar{1}$	2 0
	2 0	$\bar{2} \bar{1}$
	$\bar{2} 0$	$1 2 (2 \bar{1})$
	$1 2 (2 \bar{1})$	$\bar{2} 0$
	$\bar{2} 1 (\bar{1} \bar{2})$	$1 1 (2 \bar{2})$
	$1 1 (2 \bar{2})$	$\bar{2} 1 (\bar{1} \bar{2})$
	$\bar{1} \bar{1} (\bar{2} 2)$	1 0	Otherwise	0	$0 \bar{1}$
	1 0	$\bar{1} \bar{1} (\bar{2} \bar{2})$
	$\bar{1} 0$	$0 2 (1 \bar{1})$
	$0 2 (1 \bar{1})$	$\bar{1} 0$
	$0 \bar{2} (\bar{1} 1)$	$0 1 (1 \bar{2})$
	$0 1 (1 \bar{2})$	$0 \bar{2} (\bar{1} 1)$
	$0 \bar{1} (\bar{1} 2)$	0 0
	0 0	$0 \bar{1} (\bar{1} 2)$
17	0 0	0 0	Don’t care	0	0 0
18	2 2	$\bar{2} \bar{1}$	All negative	0	0 1
	$\bar{2} \bar{1}$	2 2
	2 1	$\bar{2} 0$
	$\bar{2} 0$	2 1
	2 0	$\bar{1} \bar{2} (\bar{2} 1)$
	$\bar{1} \bar{2} (\bar{2} 1)$	2 0
	$2 \bar{1} (1 2)$	$\bar{1} \bar{1} (\bar{2} 2)$
	$\bar{1} \bar{1} (\bar{2} 2)$	$2 \bar{1} (1 2)$
	$1 1 (2 \bar{2})$	$\bar{1} 0$	Otherwise	0	$1 \bar{2}$
	$\bar{1} 0$	$1 1 (2 \bar{2})$
	1 0	$0 \bar{2} (\bar{1} 1)$
	$0 \bar{2} (\bar{1} 1)$	1 0
	$0 2 (1 \bar{1})$	$0 \bar{1} (\bar{1} 2)$
	$0 \bar{1} (\bar{1} 2)$	$0 2 (1 \bar{1})$
	$0 1 (1 \bar{2})$	0 0
	0 0	$0 1 (1 \bar{2})$
·	·		·	·	·
·	·		·	·	·
·	·		·	·	·
32	2 2	2 1	Don’t care	1	2 0
	2 1	2 2
33	2 2	2 2	All negative	2	$\bar{1} 1$
			Otherwise	2	$0 \bar{2}$

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