Figure 1: List of all Boolean clones with bases ( h n = i =1 x 1 - PDF document

Class Definition Base(s) BF all Boolean functions { and , not } R 0 { f ∈ BF | f is 0-reproducing } { and , xor } R 1 { f ∈ BF | f is 1-reproducing } { or , x ⊕ y ⊕ 1 } R 2 R 1 ∩ R 0 { or , x ∧ ( y ⊕ z ⊕ 1) } M { f ∈ BF | f is monotonic } { and , or , c 0 , c 1 } M 1 M ∩ R 1 { and , or , c 1 } M 0 M ∩ R 0 { and , or , c 0 } M 2 M ∩ R 2 { and , or } S n { f ∈ BF | f is 0-separating of degree n } { imp , dual( h n ) } 0 S 0 { f ∈ BF | f is 0-separating } { imp } S n { f ∈ BF | f is 1-separating of degree n } { x ∧ y , h n } 1 { f ∈ BF | f is 1-separating } { x ∧ y } S 1 S n S n 0 ∩ R 2 { x ∨ ( y ∧ z ) , dual( h n ) } 02 S 0 ∩ R 2 { x ∨ ( y ∧ z ) } S 02 S n S n 0 ∩ M { dual( h n ) , c 1 } 01 S 01 S 0 ∩ M { x ∨ ( y ∧ z ) , c 1 } S n S n 0 ∩ R 2 ∩ M { x ∨ ( y ∧ z ) , dual( h n ) } 00 S 00 S 0 ∩ R 2 ∩ M { x ∨ ( y ∧ z ) } S n S n 1 ∩ R 2 { x ∧ ( y ∨ z ) , h n } 12 S 12 S 1 ∩ R 2 { x ∧ ( y ∨ z ) } S n S n 1 ∩ M { h n , c 0 } 11 S 11 S 1 ∩ M { x ∧ ( y ∨ z ) , c 0 } S n S n 1 ∩ R 2 ∩ M { x ∧ ( y ∨ z ) , h n } 10 S 10 S 1 ∩ R 2 ∩ M { x ∧ ( y ∨ z ) } D { f | f is self-dual } { xy ∨ xz ∨ yz } D 1 D ∩ R 2 { xy ∨ xz ∨ yz } D 2 D ∩ M { xy ∨ yz ∨ xz } L { f | f is linear } { xor , c 1 } L 0 L ∩ R 0 { xor } L ∩ R 1 { eq } L 1 L ∩ R 2 { x ⊕ y ⊕ z } L 2 L ∩ D { x ⊕ y ⊕ z ⊕ c 1 } L 3 { f | f is an n -ary or -function or a constant function } { or , c 0 , c 1 } V V 0 [ { or } ] ∪ [ { c 0 } ] { or , c 0 } V 1 [ { or } ] ∪ [ { c 1 } ] { or , c 1 } V 2 [ { or } ] { or } E { f | f is an n -ary and -function or a constant function } { and , c 0 , c 1 } E 0 [ { and } ] ∪ [ { c 0 } ] { and , c 0 } E 1 [ { and } ] ∪ [ { c 1 } ] { and , c 1 } E 2 [ { and } ] { and } N [ { not } ] ∪ [ { c 0 } ] ∪ [ { c 1 } ] { not , c 1 } , { not , c 0 } N 2 [ { not } ] { not } I [ { id } ] ∪ [ { c 1 } ] ∪ [ { c 0 } ] { id , c 0 , c 1 } I 0 [ { id } ] ∪ [ { c 0 } ] { id , c 0 } I 1 [ { id } ] ∪ [ { c 1 } ] { id , c 1 } I 2 [ { id } ] { id } � n +1 Figure 1: List of all Boolean clones with bases ( h n = i =1 x 1 · · · x i − 1 x i +1 · · · x n +1 and dual( f )( a 1 , . . . , a n ) = ¬ f ( a 1 , . . . , a n )).

BF R 1 R 0 R 2 M M 1 M 0 M 2 S 2 S 2 0 1 S 2 S 2 S 2 S 2 02 01 11 12 S 3 S 3 0 1 S 2 S 2 00 10 S 3 S 3 S 3 S 3 02 01 11 12 S 3 S 3 00 10 S 0 D S 1 S 02 S 01 D 1 S 11 S 12 S 00 D 2 S 10 V L E V 1 V 0 L 1 L 3 L 0 E 1 E 0 V 2 L 2 E 2 N N 2 I I 1 I 0 I 2 Figure 2: Graph of all Boolean clones.

BF M L V E N I Figure 3: Graph of all Boolean clones that contain all constant functions

BR II 0 II 1 II IN 2 IN IE 2 IL 2 IV 2 IE 0 IE 1 IL 0 IL 3 IL 1 IV 0 IV 1 IE IL IV IS 10 ID 2 IS 00 IS 12 IS 11 ID 1 IS 01 IS 02 IS 1 ID IS 0 IS 3 IS 3 10 00 IS 3 IS 3 IS 3 IS 3 12 11 01 02 IS 2 IS 2 10 00 IS 3 IS 3 1 0 IS 2 IS 2 IS 2 IS 2 12 11 01 02 IS 2 IS 2 1 0 IM 2 IM 0 IM 1 IM IR 2 IR 0 IR 1 IBF Figure 4: Graph of all Boolean co-clones

Cl. Or. Remark Base(s) of corresponding co-clone BF 0 { = } , {∅} { x } R 0 1 dual of R 1 R 1 1 { x } R 2 1 R 0 ∩ R 1 { x, x } , { xy } M 2 { x → y } M 1 2 M ∩ R 1 { x → y, x } , { x ∧ ( y → z ) } M 0 2 M ∩ R 0 { x → y, x } , { x ∧ ( y → z ) } M 2 2 M ∩ R 2 { x → y, x, x } , { x → y, x → y } , { xy ∧ ( u → v ) } S m { OR m } m 0 S m dual of S m { NAND m } m 1 0 { OR m | m ≥ 2 } ∩ m ≥ 2 S m ∞ S 0 0 { NAND m | m ≥ 2 } S 1 ∞ dual of S 0 S m S n { OR m , x, x } 0 ∩ R 2 m 02 { OR m | m ≥ 2 } ∪ { x, x } S 02 ∞ S 0 ∩ R 2 S m S m { OR m , x → y } m 0 ∩ M 01 { OR m | m ≥ 2 } ∪ { x → y } S 01 ∞ S 0 ∩ M S m S n { OR m , x, x, x → y } m 0 ∩ R 2 ∩ M 00 { OR m | m ≥ 2 } ∪ { x, x, x → y } S 00 ∞ S 0 ∩ R 2 ∩ M S m dual of S m { NAND m , x, x } m 12 02 { NAND m | m ≥ 2 } ∪ { x, x } S 12 ∞ dual of S 02 S m dual of S m { NAND m , x → y } m 11 01 { NAND m | m ≥ 2 } ∪ { x → y } ∞ S 11 dual of S 01 S m dual of S m { NAND m , x, x, x → y } m 10 00 { NAND m | m ≥ 2 } ∪ { x, x, x → y } ∞ S 10 dual of S 00 D 2 { x ⊕ y } D 1 2 D ∩ R 1 { x ⊕ y, x } , every R ∈ {{ ( a 1 , a 2 , a 3 ) , ( b 1 , b 2 , b 3 ) } | ∃ c ∈ { 1 , 2 } such that Σ 3 i =1 a i = Σ 3 i =1 b i = c } D 2 2 D ∩ M { x ⊕ y, x → y } , { xy ∨ xyz } { EVEN 4 } L 4 { EVEN 4 , x } , { EVEN 3 } L 0 3 L ∩ R 0 { EVEN 4 , x } , { ODD 3 } L 1 3 L ∩ R 1 { EVEN 4 , x, x } , every { EVEN n , (1) } where n ≥ 3 is odd L 2 3 L ∩ R 2 { EVEN 4 , x ⊕ y } , { ODD 4 } L 3 4 L ∩ D { x ∨ y ∨ z } V 3 V 0 3 V ∩ R 0 { x ∨ y ∨ z, x } V ∩ R 1 { x ∨ y ∨ z, x } V 1 3 V 2 3 V ∩ R 2 { x ∨ y ∨ z, x, x } E 3 dual of V { x ∨ y ∨ z } E 1 3 dual of V 0 { x ∨ y ∨ z, x } E 0 3 dual of V 1 { x ∨ y ∨ z, x } E 2 3 dual of V 2 { x ∨ y ∨ z, x, x } { DUP 3 } N 3 { DUP 3 , EVEN 4 , x ⊕ y } , { NAE 3 } N ∩ L 3 N 2 3 { EVEN 4 , x → y } I 3 L ∩ M { EVEN 4 , x → y, x } , { DUP 3 , x → y } I 0 3 L ∩ M ∩ R 0 { EVEN 4 , x → y, x } , { x ∨ ( x ⊕ z ) } L ∩ M ∩ R 1 I 1 3 { EVEN 4 , x → y, x, x } , { 1 text − IN − 3 } , { x → ( y ⊕ z ) } I 2 3 L ∩ M ∩ R 2 Figure 5: Bases for all Boolean co-clones