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Table 3. Relational Composition
o---------o---------o---------o---------o
| # !1! | !1! | !1! |
o=========o=========o=========o=========o
| L # X | Y | |
o---------o---------o---------o---------o
| M # | Y | Z |
o---------o---------o---------o---------o
| L o M # X | | Z |
o---------o---------o---------o---------o
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\(\text{Table 3. Relational Composition}\!\)
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\(\mathit{1}\!\)
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\(\mathit{1}\!\)
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\(\mathit{1}\!\)
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| \(L\!\)
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\(X\!\)
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\(Y\!\)
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| \(M\!\)
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\(Y\!\)
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\(Z\!\)
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| \(L \circ M\)
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\(X\!\)
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\(Z\!\)
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Table 9. Composite of Triadic and Dyadic Relations
o---------o---------o---------o---------o---------o
| # !1! | !1! | !1! | !1! |
o=========o=========o=========o=========o=========o
| G # T | U | | V |
o---------o---------o---------o---------o---------o
| L # | U | W | |
o---------o---------o---------o---------o---------o
| G o L # T | | W | V |
o---------o---------o---------o---------o---------o
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\(\text{Table 9. Composite of Triadic and Dyadic Relations}\!\)
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\(\mathit{1}\!\)
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\(\mathit{1}\!\)
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\(\mathit{1}\!\)
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\(\mathit{1}\!\)
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| \(G\!\)
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\(T\!\)
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\(U\!\)
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\(V\!\)
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| \(L\!\)
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\(U\!\)
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\(W\!\)
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| \(G \circ L\)
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\(T\!\)
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\(W\!\)
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\(V\!\)
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Table 13. Another Brand of Composition
o---------o---------o---------o---------o
| # !1! | !1! | !1! |
o=========o=========o=========o=========o
| G # X | Y | Z |
o---------o---------o---------o---------o
| T # | Y | Z |
o---------o---------o---------o---------o
| G o T # X | | Z |
o---------o---------o---------o---------o
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\(\text{Table 13. Another Brand of Composition}\!\)
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\(\mathit{1}\!\)
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\(\mathit{1}\!\)
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\(\mathit{1}\!\)
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| \(G\!\)
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\(X\!\)
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\(Y\!\)
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\(Z\!\)
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| \(T\!\)
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\(Y\!\)
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\(Z\!\)
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| \(G \circ T\)
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\(X\!\)
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\(Z\!\)
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Table 15. Conjunction Via Composition
o---------o---------o---------o---------o
| # !1! | !1! | !1! |
o=========o=========o=========o=========o
| L, # X | X | Y |
o---------o---------o---------o---------o
| S # | X | Y |
o---------o---------o---------o---------o
| L , S # X | | Y |
o---------o---------o---------o---------o
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\(\text{Table 15. Conjunction Via Composition}\!\)
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\(\mathit{1}\!\)
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\(\mathit{1}\!\)
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\(\mathit{1}\!\)
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| \(L,\!\)
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\(X\!\)
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\(X\!\)
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\(Y\!\)
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| \(S\!\)
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\(X\!\)
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\(Y\!\)
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| \(L,\!S\)
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\(X\!\)
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\(Y\!\)
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Table 18. Relational Composition P o Q
o---------o---------o---------o---------o
| # !1! | !1! | !1! |
o=========o=========o=========o=========o
| P # X | Y | |
o---------o---------o---------o---------o
| Q # | Y | Z |
o---------o---------o---------o---------o
| P o Q # X | | Z |
o---------o---------o---------o---------o
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\(\text{Table 18. Relational Composition}~ P \circ Q\)
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\(\mathit{1}\!\)
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\(\mathit{1}\!\)
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\(\mathit{1}\!\)
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| \(P\!\)
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\(X\!\)
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\(Y\!\)
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| \(Q\!\)
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\(Y\!\)
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\(Z\!\)
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| \(P \circ Q\)
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\(X\!\)
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\(Z\!\)
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Table 20. Arrow: J(L(u, v)) = K(Ju, Jv)
o---------o---------o---------o---------o
| # J | J | J |
o=========o=========o=========o=========o
| K # X | X | X |
o---------o---------o---------o---------o
| L # Y | Y | Y |
o---------o---------o---------o---------o
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\(\text{Table 20. Arrow Equation:}~~ J(L(u, v)) = K(Ju, Jv)\)
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\(J\!\)
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\(J\!\)
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\(J\!\)
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| \(K\!\)
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\(X\!\)
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\(X\!\)
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\(X\!\)
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| \(L\!\)
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\(Y\!\)
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\(Y\!\)
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\(Y\!\)
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Grammar Stuff
Table 13. Algorithmic Translation Rules
| \(\text{Sentence in PARCE}\!\)
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\(\xrightarrow{\operatorname{Parse}}\)
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\(\text{Graph in PARC}\!\)
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| \(\operatorname{Conc}^0\)
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\(\xrightarrow{\operatorname{Parse}}\)
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\(\operatorname{Node}^0\)
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| \(\operatorname{Conc}_{j=1}^k s_j\)
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\(\xrightarrow{\operatorname{Parse}}\)
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\(\operatorname{Node}_{j=1}^k \operatorname{Parse} (s_j)\)
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| \(\operatorname{Surc}^0\)
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\(\xrightarrow{\operatorname{Parse}}\)
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\(\operatorname{Lobe}^0\)
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| \(\operatorname{Surc}_{j=1}^k s_j\)
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\(\xrightarrow{\operatorname{Parse}}\)
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\(\operatorname{Lobe}_{j=1}^k \operatorname{Parse} (s_j)\)
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Table 14.1 Semantic Translation : Functional Form
| \(\operatorname{Sentence}\)
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\(\xrightarrow[\operatorname{~~~~~~~~~~}]{\operatorname{Parse}}\)
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\(\operatorname{Graph}\)
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\(\xrightarrow[\operatorname{~~~~~~~~~~}]{\operatorname{Denotation}}\)
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\(\operatorname{Proposition}\)
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| \(s_j\!\)
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\(\xrightarrow{\operatorname{~~~~~~~~~~}}\)
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\(C_j\!\)
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\(\xrightarrow{\operatorname{~~~~~~~~~~}}\)
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\(q_j\!\)
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| \(\operatorname{Conc}^0\)
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\(\xrightarrow{\operatorname{~~~~~~~~~~}}\)
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\(\operatorname{Node}^0\)
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\(\xrightarrow{\operatorname{~~~~~~~~~~}}\)
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\(\underline{1}\)
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| \(\operatorname{Conc}^k_j s_j\)
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\(\xrightarrow{\operatorname{~~~~~~~~~~}}\)
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\(\operatorname{Node}^k_j C_j\)
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\(\xrightarrow{\operatorname{~~~~~~~~~~}}\)
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\(\operatorname{Conj}^k_j q_j\)
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| \(\operatorname{Surc}^0\)
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\(\xrightarrow{\operatorname{~~~~~~~~~~}}\)
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\(\operatorname{Lobe}^0\)
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\(\xrightarrow{\operatorname{~~~~~~~~~~}}\)
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\(\underline{0}\)
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| \(\operatorname{Surc}^k_j s_j\)
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\(\xrightarrow{\operatorname{~~~~~~~~~~}}\)
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\(\operatorname{Lobe}^k_j C_j\)
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\(\xrightarrow{\operatorname{~~~~~~~~~~}}\)
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\(\operatorname{Surj}^k_j q_j\)
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Table 14.2 Semantic Translation : Equational Form
| \(\downharpoonleft \operatorname{Sentence} \downharpoonright\)
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\(\stackrel{\operatorname{Parse}}{=}\)
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\(\downharpoonleft \operatorname{Graph} \downharpoonright\)
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\(\stackrel{\operatorname{Denotation}}{=}\)
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\(\operatorname{Proposition}\)
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| \(\downharpoonleft s_j \downharpoonright\)
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\(=\!\)
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\(\downharpoonleft C_j \downharpoonright\)
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\(=\!\)
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\(q_j\!\)
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| \(\downharpoonleft \operatorname{Conc}^0 \downharpoonright\)
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\(=\!\)
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\(\downharpoonleft \operatorname{Node}^0 \downharpoonright\)
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\(=\!\)
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\(\underline{1}\)
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| \(\downharpoonleft \operatorname{Conc}^k_j s_j \downharpoonright\)
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\(=\!\)
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\(\downharpoonleft \operatorname{Node}^k_j C_j \downharpoonright\)
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\(=\!\)
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\(\operatorname{Conj}^k_j q_j\)
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| \(\downharpoonleft \operatorname{Surc}^0 \downharpoonright\)
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\(=\!\)
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\(\downharpoonleft \operatorname{Lobe}^0 \downharpoonright\)
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\(=\!\)
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\(\underline{0}\)
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| \(\downharpoonleft \operatorname{Surc}^k_j s_j \downharpoonright\)
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\(=\!\)
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\(\downharpoonleft \operatorname{Lobe}^k_j C_j \downharpoonright\)
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\(=\!\)
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\(\operatorname{Surj}^k_j q_j\)
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Table Stuff
Table 15. Boolean Functions on Zero Variables
| \(F\!\)
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\(F\!\)
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\(F()\!\)
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\(F\!\)
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| \(\underline{0}\)
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\(F_0^{(0)}\!\)
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\(\underline{0}\)
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\((~)\)
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| \(\underline{1}\)
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\(F_1^{(0)}\!\)
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\(\underline{1}\)
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\(((~))\)
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Table 16. Boolean Functions on One Variable
| \(F\!\)
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\(F\!\)
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\(F(x)\!\)
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\(F\!\)
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\(F(\underline{1})\)
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\(F(\underline{0})\)
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| \(F_0^{(1)}\!\)
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\(F_{00}^{(1)}\!\)
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\(\underline{0}\)
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\(\underline{0}\)
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\((~)\)
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| \(F_1^{(1)}\!\)
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\(F_{01}^{(1)}\!\)
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\(\underline{0}\)
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\(\underline{1}\)
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\((x)\!\)
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| \(F_2^{(1)}\!\)
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\(F_{10}^{(1)}\!\)
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\(\underline{1}\)
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\(\underline{0}\)
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\(x\!\)
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| \(F_3^{(1)}\!\)
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\(F_{11}^{(1)}\!\)
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\(\underline{1}\)
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\(\underline{1}\)
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\(((~))\)
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Table 17. Boolean Functions on Two Variables
| \(F\!\)
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\(F\!\)
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\(F(x, y)\!\)
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\(F\!\)
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\(F(\underline{1}, \underline{1})\)
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\(F(\underline{1}, \underline{0})\)
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\(F(\underline{0}, \underline{1})\)
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\(F(\underline{0}, \underline{0})\)
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| \(F_{0}^{(2)}\!\)
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\(F_{0000}^{(2)}\!\)
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\(\underline{0}\)
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\(\underline{0}\)
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\(\underline{0}\)
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\(\underline{0}\)
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\((~)\)
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| \(F_{1}^{(2)}\!\)
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\(F_{0001}^{(2)}\!\)
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\(\underline{0}\)
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\(\underline{0}\)
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\(\underline{0}\)
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\(\underline{1}\)
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\((x)(y)\!\)
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| \(F_{2}^{(2)}\!\)
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\(F_{0010}^{(2)}\!\)
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\(\underline{0}\)
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\(\underline{0}\)
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\(\underline{1}\)
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\(\underline{0}\)
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\((x) y\!\)
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| \(F_{3}^{(2)}\!\)
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\(F_{0011}^{(2)}\!\)
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\(\underline{0}\)
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\(\underline{0}\)
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\(\underline{1}\)
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\(\underline{1}\)
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\((x)\!\)
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| \(F_{4}^{(2)}\!\)
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\(F_{0100}^{(2)}\!\)
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\(\underline{0}\)
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\(\underline{1}\)
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\(\underline{0}\)
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\(\underline{0}\)
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\(x (y)\!\)
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| \(F_{5}^{(2)}\!\)
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\(F_{0101}^{(2)}\!\)
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\(\underline{0}\)
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\(\underline{1}\)
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\(\underline{0}\)
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\(\underline{1}\)
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\((y)\!\)
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| \(F_{6}^{(2)}\!\)
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\(F_{0110}^{(2)}\!\)
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\(\underline{0}\)
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\(\underline{1}\)
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\(\underline{1}\)
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\(\underline{0}\)
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\((x, y)\!\)
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| \(F_{7}^{(2)}\!\)
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\(F_{0111}^{(2)}\!\)
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\(\underline{0}\)
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\(\underline{1}\)
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\(\underline{1}\)
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\(\underline{1}\)
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\((x y)\!\)
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| \(F_{8}^{(2)}\!\)
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\(F_{1000}^{(2)}\!\)
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\(\underline{1}\)
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\(\underline{0}\)
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\(\underline{0}\)
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\(\underline{0}\)
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\(x y\!\)
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| \(F_{9}^{(2)}\!\)
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\(F_{1001}^{(2)}\!\)
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\(\underline{1}\)
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\(\underline{0}\)
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\(\underline{0}\)
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\(\underline{1}\)
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\(((x, y))\!\)
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| \(F_{10}^{(2)}\!\)
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\(F_{1010}^{(2)}\!\)
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\(\underline{1}\)
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\(\underline{0}\)
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\(\underline{1}\)
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\(\underline{0}\)
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\(y\!\)
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| \(F_{11}^{(2)}\!\)
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\(F_{1011}^{(2)}\!\)
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\(\underline{1}\)
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\(\underline{0}\)
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\(\underline{1}\)
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\(\underline{1}\)
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\((x (y))\!\)
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| \(F_{12}^{(2)}\!\)
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\(F_{1100}^{(2)}\!\)
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\(\underline{1}\)
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\(\underline{1}\)
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\(\underline{0}\)
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\(\underline{0}\)
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\(x\!\)
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| \(F_{13}^{(2)}\!\)
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\(F_{1101}^{(2)}\!\)
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\(\underline{1}\)
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\(\underline{1}\)
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\(\underline{0}\)
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\(\underline{1}\)
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\(((x)y)\!\)
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| \(F_{14}^{(2)}\!\)
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\(F_{1110}^{(2)}\!\)
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\(\underline{1}\)
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\(\underline{1}\)
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\(\underline{1}\)
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\(\underline{0}\)
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\(((x)(y))\!\)
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| \(F_{15}^{(2)}\!\)
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\(F_{1111}^{(2)}\!\)
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\(\underline{1}\)
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\(\underline{1}\)
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\(\underline{1}\)
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\(\underline{1}\)
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\(((~))\)
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