SLIDE 1
Class Definition Base(s) BF all Boolean functions {and, not} R0 { f ∈ BF | f is 0-reproducing } {and, xor} R1 { f ∈ BF | f is 1-reproducing } {or, x ⊕ y ⊕ 1} R2 R1 ∩ R0 {or, x ∧ (y ⊕ z ⊕ 1)} M { f ∈ BF | f is monotonic } {and, or, c0, c1} M1 M ∩ R1 {and, or, c1} M0 M ∩ R0 {and, or, c0} M2 M ∩ R2 {and, or} Sn { f ∈ BF | f is 0-separating of degree n } {imp, dual(hn)} S0 { f ∈ BF | f is 0-separating } {imp} Sn
1
{ f ∈ BF | f is 1-separating of degree n } {x ∧ y, hn} S1 { f ∈ BF | f is 1-separating } {x ∧ y} Sn
02
Sn
0 ∩ R2
{x ∨ (y ∧ z), dual(hn)} S02 S0 ∩ R2 {x ∨ (y ∧ z)} Sn
01
Sn
0 ∩ M
{dual(hn), c1} S01 S0 ∩ M {x ∨ (y ∧ z), c1} Sn
00
Sn
0 ∩ R2 ∩ M
{x ∨ (y ∧ z), dual(hn)} S00 S0 ∩ R2 ∩ M {x ∨ (y ∧ z)} Sn
12
Sn
1 ∩ R2
{x ∧ (y ∨ z), hn} S12 S1 ∩ R2 {x ∧ (y ∨ z)} Sn
11
Sn
1 ∩ M
{hn, c0} S11 S1 ∩ M {x ∧ (y ∨ z), c0} Sn
10
Sn
1 ∩ R2 ∩ M
{x ∧ (y ∨ z), hn} S10 S1 ∩ R2 ∩ M {x ∧ (y ∨ z)} D { f | f is self-dual } {xy ∨ xz ∨ yz} D1 D ∩ R2 {xy ∨ xz ∨ yz} D2 D ∩ M {xy ∨ yz ∨ xz} L { f | f is linear} {xor, c1} L0 L ∩ R0 {xor} L1 L ∩ R1 {eq} L2 L ∩ R2 {x ⊕ y ⊕ z} L3 L ∩ D {x ⊕ y ⊕ z ⊕ c1} V { f | f is an n-ary or-function or a constant function} {or, c0, c1} V0 [{or}] ∪ [{c0}] {or, c0} V1 [{or}] ∪ [{c1}] {or, c1} V2 [{or}] {or} E { f | f is an n-ary and-function or a constant function} {and, c0, c1} E0 [{and}] ∪ [{c0}] {and, c0} E1 [{and}] ∪ [{c1}] {and, c1} E2 [{and}] {and} N [{not}] ∪ [{c0}] ∪ [{c1}] {not, c1}, {not, c0} N2 [{not}] {not} I [{id}] ∪ [{c1}] ∪ [{c0}] {id, c0, c1} I0 [{id}] ∪ [{c0}] {id, c0} I1 [{id}] ∪ [{c1}] {id, c1} I2 [{id}] {id}
Figure 1: List of all Boolean clones with bases (hn = n+1
i=1 x1 · · · xi−1xi+1 · · · xn+1 and
dual(f)(a1, . . . , an) = ¬f(a1, . . . , an)).
SLIDE 2 R1 R0 BF R2 M M1 M0 M2 S2 S3 S0 S2
02
S3
02
S02 S2
01
S3
01
S01 S2
00
S3
00
S00 S2
1
S3
1
S1 S2
12
S3
12
S12 S2
11
S3
11
S11 S2
10
S3
10
S10 D D1 D2 L L1 L0 L2 L3 V V1 V0 V2 E E0 E1 E2 I I1 I0 I2 N2 N
Figure 2: Graph of all Boolean clones.
SLIDE 3
I V E N BF M L
Figure 3: Graph of all Boolean clones that contain all constant functions
SLIDE 4 IR0 IR1 IBF IR2 IM IM0 IM1 IM2 IS2
1
IS3
1
IS1 IS2
12
IS3
12
IS12 IS2
11
IS3
11
IS11 IS2
10
IS3
10
IS10 IS2 IS3 IS0 IS2
02
IS3
02
IS02 IS2
01
IS3
01
IS01 IS2
00
IS3
00
IS00 ID2 ID ID1 IL2 IL IL0 IL1 IL3 IE2 IE IE0 IE1 IV2 IV IV1 IV0 II0 II1 IN2 II BR IN
Figure 4: Graph of all Boolean co-clones
SLIDE 5
Cl. Or. Remark Base(s) of corresponding co-clone BF {=}, {∅} R0 1 dual of R1 {x} R1 1 {x} R2 1 R0 ∩ R1 {x, x} ,{xy} M 2 {x → y} M1 2 M ∩ R1 {x → y, x} ,{x ∧ (y → z)} M0 2 M ∩ R0 {x → y, x} ,{x ∧ (y → z)} M2 2 M ∩ R2 {x → y, x, x} ,{x → y, x → y} , {xy ∧ (u → v)} Sm m {ORm} Sm
1
m dual of Sm {NANDm} S0 ∞ ∩m≥2Sm {ORm | m ≥ 2} S1 ∞ dual of S0 {NANDm | m ≥ 2} Sm
02
m Sn
0 ∩ R2
{ORm, x, x} S02 ∞ S0 ∩ R2 {ORm | m ≥ 2} ∪ {x, x} Sm
01
m Sm
0 ∩ M
{ORm, x → y} S01 ∞ S0 ∩ M {ORm | m ≥ 2} ∪ {x → y} Sm
00
m Sn
0 ∩ R2 ∩ M
{ORm, x, x, x → y} S00 ∞ S0 ∩ R2 ∩ M {ORm | m ≥ 2} ∪ {x, x, x → y} Sm
12
m dual of Sm
02
{NANDm, x, x} S12 ∞ dual of S02 {NANDm | m ≥ 2} ∪ {x, x} Sm
11
m dual of Sm
01
{NANDm, x → y} S11 ∞ dual of S01 {NANDm | m ≥ 2} ∪ {x → y} Sm
10
m dual of Sm
00
{NANDm, x, x, x → y} S10 ∞ dual of S00 {NANDm | m ≥ 2} ∪ {x, x, x → y} D 2 {x ⊕ y} D1 2 D ∩ R1 {x ⊕ y, x}, every R ∈ {{(a1, a2, a3), (b1, b2, b3)} | ∃c ∈ {1, 2} such that Σ3
i=1ai = Σ3 i=1bi = c}
D2 2 D ∩ M {x ⊕ y, x → y} ,{xy ∨ xyz} L 4 {EVEN4} L0 3 L ∩ R0 {EVEN4, x} ,{EVEN3} L1 3 L ∩ R1 {EVEN4, x} ,{ODD3} L2 3 L ∩ R2 {EVEN4, x, x} , every {EVENn, (1)} where n ≥ 3 is odd L3 4 L ∩ D {EVEN4, x ⊕ y} ,{ODD4} V 3 {x ∨ y ∨ z} V0 3 V ∩ R0 {x ∨ y ∨ z, x} V1 3 V ∩ R1 {x ∨ y ∨ z, x} V2 3 V ∩ R2 {x ∨ y ∨ z, x, x} E 3 dual of V {x ∨ y ∨ z} E1 3 dual of V0 {x ∨ y ∨ z, x} E0 3 dual of V1 {x ∨ y ∨ z, x} E2 3 dual of V2 {x ∨ y ∨ z, x, x} N 3 {DUP3} N2 3 N ∩ L3 {DUP3, EVEN4, x ⊕ y}, {NAE3} I 3 L ∩ M {EVEN4, x → y} I0 3 L ∩ M ∩ R0 {EVEN4, x → y, x} ,{DUP3, x → y} I1 3 L ∩ M ∩ R1 {EVEN4, x → y, x} ,{x ∨ (x ⊕ z)} I2 3 L ∩ M ∩ R2 {EVEN4, x → y, x, x}, {1text−IN−3} ,{x → (y ⊕ z)}
Figure 5: Bases for all Boolean co-clones
