1951

{0, 1}

N = N \ {0} n m M n, m N F x i = (x i 1,..., xi m) x j = (x 1 j,..., xn j ) i j M M i j x i j m n M M M M T f : F m F f(m) f M (f(x 1 1,..., x1 m),..., f(x n 1,..., xn m)) T R F M R M R x i = (x i 1,..., xi m) i {1,..., n} x j = (x 1 j,..., xn j ) j {1,..., m} f {(x, f(x)) x Dom(f)} Dom(f) f m N I {1,..., m} R F m J = {1,..., m} \ I I = {i 1,..., i s } J = {j 1,..., j t } R I = pr m I R = x j 1... x jt R = {(x i1,..., x is ) (x 1,..., x m ) R} R I = pr m I R = prm J = R J R i = R {i} i {1,..., m} R i = R {i} i {1,..., m} S S I = {R I R S} S I S i f : F m F F {0, 1} F 0, 1 {0, 1} c m x c m x (ȳ) = x ȳ F m : F m F x F m N : F 2 F { 1 x = y = 1, (x, y) = 0

: F 2 F { 0 x = y = 0, (x, y) = 1 maj : F 3 F { x x = y x = z, maj(x, y, z) = y y = z. : F 3 F x y = z, (x, y, z) = y x = z y, z x = y z.

{0, 1} {0, 1} {0, 1} m m N n f : {0, 1} n {0, 1} n N

B B pri n B n N i {1,..., n} pri n : {0, 1} n {0, 1} n i pri n (a 1,..., a n ) = a i, (a 1,..., a n ) {0, 1} n f, f 1,..., f n B f n f 1,..., f n k f(f 1,..., f n ) : {0, 1} k {0, 1} f(f 1,..., f n )(a 1,..., a k ) := f(f 1 (a 1,..., a k ),..., f n (a 1,..., a k )), (a 1,..., a k ) {0, 1} k B S S Eq S Eq = {(0, 0), (1, 1)} S R S m k R R (a 1,..., a k ) R(pr k i 1 (a 1,..., a k ),..., pr k i m (a 1,..., a k )), (a 1,..., a k ) {0, 1} k S m, k N i 1,..., i m {1,..., k} S R S m (m 1) R m S S R, R S m m R R S

S {0, 1} S Eq S {0, 1}(x) Eq(pr 2 1(x, y), pr 2 1(x, y)). S R 1, R 2 n m S R 1, R 2 (n + m) R = R 1 R 2 R(a 1,..., a n+m ) R 1 (a 1,..., a n ) R 2 (a n+1,..., a n+m ), (a 1,..., a n+m ) {0, 1} n+m S R 1 R 2 = R 1 R 2 (n + m) R 1, R 2 R 1(a 1,..., a n+m ) R 1 (pr n+m 1 (a 1,..., a n+m ),..., pr n+m n (a 1,..., a n+m )), R 2(a 1,..., a n+m ) R 2 (pr n+m n+1 (a 1,..., a n+m ),..., pr n+m n+m (a 1,..., a n+m )). R 1, R 2 S R 1 R 2 S S m N i, j {1,..., m} {0, 1} m E m i,j = {(a 1,..., a m ) a i = a j } R {0, 1} m f : {0, 1} n {0, 1} f R f R f R a 1 = (a 1 1,..., a1 m),..., a n = (a n 1,..., an m) R f(a 1,..., a n ) := (f(a 1 1,..., a n 1 ),..., f(a 1 m,..., a n m)) R. f S f R R S

B S P ol(s) = {f : {0, 1} n {0, 1} n N f R, R S} Inv(B) = {R {0, 1} m m N f R, f B} S P ol(s) m R S n N i {1,..., n} a 1,..., a n R (pr n i (a 1 1,..., a n 1 ),..., pr n i (a 1 m,..., a n m)) = (a i 1,..., a i m) = a i R. P ol(s) n f P ol(s) k f 1,..., f n P ol(s) a 1,..., a k R f i (a 1,..., a k ) R i {1,..., n} f(f 1,..., f n )(a 1,..., a k ) = f(f 1 (a 1,..., a k ),..., f n (a 1,..., a k )) R. B Inv(B) n f B a 1,..., a n Eq Eq a i = (a i 1, ai 2 ) ai 1 = ai 2 i {1,..., n} f(a 1) = f(a 2 ) (f(a 1 ), f(a 2 )) Eq f Eq Inv(B) m R Inv(B) i 1,..., i m {1,.., k} k R R (a 1,..., a k ) R(pr k i 1 (a 1,..., a k ),..., pr k i m (a 1,..., a k )). a 1,..., a n R (a 1 i 1,..., a 1 i m ),..., (a n i 1,..., a n i m ) R f R (f(a 1 i 1,..., a n i 1 ),..., f(a 1 i m,..., a n i m )) R (f(a 1 ),..., f(a k )) R f R Inv(B) m R Inv(B) (a 1 1,..., a1 m 1 ),..., (an 1,..., an m 1 ) R m a 1 m,..., a n m (a 1 1,..., a1 m),..., (a n 1,..., an m) R f R (f(a 1 ),..., f(a m )) R (f(a 1 ),..., f(a m 1 )) R m f R m Inv(B)

m R 1, R 2 Inv(B) a 1,..., a n R 1 R 2 (f(a 1 ),..., f(a m )) R 1 R 2 f R 1 R 2 Inv(B) X, Y X Y B [B] B B [B] = {F B F } S S S S S = {G S G} n f Dom(f) {0, 1} n Im(f) {0, 1} R {0, 1} m f n f R a 1 = (a 1 1,..., a1 m),..., a n = (a n 1,..., an m) R a 1,..., a m Dom(f) (f(a 1 ),..., f(a m )) R. S R S B S P ol(inv(b)) = [B] Inv(P ol(s)) = S Inv(B) = Inv([B]) P ol(s) = P ol( S ) B S B = [B] S = S B S B P ol(inv(b)) S Inv(P ol(s))

n f / B R Inv(B) f T 2 n n {0, 1} n 2 n {0, 1} n t i t j B (n) n B g B (n) g(t ) 2 n m M T g(t ) m = B (n) R = {g(t ) T g B (n) } R M R Inv(B) j {1,..., n} t j R t j = prj n (T ) prj n B(n) B k = 2 n r h B x 1 = (x 1 1,..., x1 k ),..., xr = (x r 1,..., xr k ) R R x 1,..., x r M g 1,..., g r B (n) x i = (g i (t 1 ),..., g i (t k )) i {1,..., r} (h(x 1 1,..., x r 1),..., h(x 1 k,..., xr k )) = = (h(g 1 (t 1 ),..., g r (t 1 )),..., h(g 1 (t k ),..., g r (t k ))) := x. B h, g 1,..., g r B h(g 1,..., g r ) B x M R h R Inv(B) f R T T R (f(t 1 ),..., f(t k )) = f(t ) R g B (n) g(t ) = f(t ) {t 1,..., t k } = {0, 1} n f = g f B m R / S f P ol(s) R S f S S n f S Dom(f) = {0, 1} Dom(f) =

g P ol(s) P ol(s) = Dom(f) {0, 1} n f f n {0, 1} n {0, 1} n f S Dom(f) = {r 1,..., r k } k N \ {0} r / {r 1,..., r k } {0, 1} n f 0 = f {(r, 0)} f 1 = f {(r, 1)} f 0, f 1 S f S m 0 R 0 S m 1 R 1 S f 0 R 0 f 1 f 1 n m i M i R i f i f i (Mi T )T / R i i {0, 1} M i {r 1,..., r k, r} r M i i {0, 1} M 1 f 1 (M T 1 )T = f(m T 1 )T f S M 0 M 1 M 0 = M 1 = r T R 0, R 1 r i {0, 1} f i (r) = i R i M i R i M i i {0, 1} j j M i j, j {1,..., m i } j j R i R i(x 1,..., x j,..., x j 1, x j +1,...x mi ) x j R i (x 1,..., x j,..., x j,..., x mi ) Eq(j, j ) S N i R i M i j f i (Ni T )T R i f i(mi T )T / R i R i R i

n (m 1 + m 2 ) N m 1 M 1 M 2 i j N r i {1,..., m 1 } j {m 1 + 1,..., m 2 } R = R 1 R 2 R S N R N N i j N R S R R (x 1,..., x i 1, x i+1,..., x j 1, x j+1,..., x m1 +m 2 x i x j R (x 1,..., x m1 +m 2 ) Eq(x i, x j ) f S N f(n T ) T R R i {0, 1} f i (N T ) T R f i (M T i )T R i P = Q. Q S,Q R S P S R / S t P \ R N R k m R R = k k f f(n T ) = t T N {0, 1} k f i, j {1,..., m} i j N R E m i,j S S E m i,j {Q S Q R} P Em i,j f f S f R S n R S f k n M R f f(m T ) T R M i, j R (x 1,..., x i,..., x j 1, x j+1,..., x n ) x j R (x 1,..., x n ) Eq(x i, x j ).

R S M R M j f(m T ) T / R R f M N n m n R R {0, 1} (m n) S P R {0, 1} (m n) t R {0, 1} (m n) (t 1,..., t n ) R (t 1,..., t n ) = f(m T ) T / R B S (L,,, ) L x, y L x y x y X, Y B X, Y S X Y X, Y X Y X, Y (B,,, ) (S,,, ) B S X, Y B [X Y ] = X Y X Y X Y X Y X, Y X Y X Y X Y = X Y B X Y [X Y ] X Y X Y [X Y ] X Y X Y = [X Y ] B

P ol(s), Inv(B) B S P ol : S B Inv : B S (B,,, ) (S,,, ) X, Y S X Y P ol(y ) P ol(x) Z, W B Z W Inv(W ) Inv(Z) X S Z B X Inv(P ol(x)) Z P ol(inv(z)) 1 2 Y X W Z 3 Inv(P ol) : S S P ol(inv) : B B X, Y S Z, W B X Inv(P ol(x)) Z P ol(inv(z)) X Y Inv(P ol(x)) Inv(P ol(y )) Z W P ol(inv(z)) P ol(inv(w )) Inv(P ol(x)) = Inv(P ol(inv(p ol(x)))) P ol(inv(z)) = P ol(inv(p ol(inv(z)))) (B,,, ) (S,,, ) P ol : S B Inv : B S 1 1 X, Y S Z, W B X Y P ol(y ) P ol(x) Z W Inv(W ) Inv(Z)

P ol(x) = P ol(y ) Inv(P ol(x)) = Inv(P ol(y )) X = Y P ol 1 1 Z B Z = P ol(inv(z)) P ol Inv

SAT S V S ϕ S S V ϕ(x 1,..., x n ) = m R i (x i 1,..., x i m i ), i=1 R i S i {1,..., m} x i j {x 1,..., x n } V i, j S ϕ ϕ SAT (S) SAT S SAT (S) SAT ( S ) S S SAT (S) P SAT ( S ) S u 1,..., u n S ϕ ϕ R(u i1,..., u ir ) R S ϕ Eq(u 1, u 2 ) Eq / S u 1, u 2

ϕ x j R(u i1,..., u j 1, x j, u j+1,..., u ir ) R j / S R S R(u i1,..., u j 1, u, u j+1,..., u ir ) u {u 1,..., u n } ϕ R(u i1,..., u ir ) R / S R S R(x 1,..., x r ) R (pr r k 1 (x 1,..., x r ),..., pr r k s (x 1,..., x r )), R (u k1,..., u ks ) ϕ R(u 1,..., u R ) R / S R = R 1 R 2 R 1, R 2 S R 1 (u 1,..., u r ) R 2 (u 1,..., u r ) S 1 S 2 S 1 S 2 SAT (S 1 ) P SAT (S 2 ) ϕ S 1 ϕ S 2 SAT (S 1 ) P SAT ( S 2 ) = P SAT (S 2 ) S 1 S 2 P ol(s 2 ) P ol(s 1 ) SAT (S 1 ) P SAT (S 2 ) P ol(s 2 ) P ol(s 1 ) Inv(P ol(s 1 )) Inv(P ol(s 2 )) S 1 S 2 SAT (S 1 ) SAT (S 2 ) R R 0 (0,..., 0) R 1 (1,..., 1) R

S 0 0 1 S Schaefer S S 0 1 SAT (S) NP ( ) S 0 1 ( ) [], [] II 0 = Inv([{c 0 }]) II 1 = Inv([{c 1 }]) IE 2 = Inv([{ }]) IV 2 = Inv([{ }]) ID 2 = Inv([{maj}]) IL 2 = Inv([{ }]) IN 2 = Inv([{ }]) BR = {R {0, 1} n n N} S II 0, II 1, IE 2, IV 2, ID 2, IL 2 S IN 2 S IN 2 IN 2 S = BR SAT (S) = P SAT ( S ) = SAT (BR) SAT (S) NP

S = S = IN 2 = Inv([{ }]) R NAE = {0, 1} 3 \ {(0, 0, 0), (1, 1, 1)}. P ol({r NAE }) [{ }] P ol({r NAE }) {R NAE } = Inv(P ol({r NAE }) Inv([{ }]) = IN 2. N 2 = P ol(in 2 ) P ol({r NAE }) SAT ({R NAE }) P SAT (IN 2 ) = SAT (S) SAT ({R NAE }) NP SAT (S)

{0 j, 1 j } j {1,..., m} m N {0 j, 1 j } m R {0 j, 1 j } i j j=1 α R m σ R = (i 1,..., i m ) j i j a R = m j=1 i j i j N j {1,..., m} m m m m R m 1 j 0 I j 0 1 J m (j 0 + 1) R

R I {0 j0, 1 j0 } R J j 0 j {0 j, 1 j } j {1,..., m} n n N m m f = (f 1,..., f m ) f j : {0 j, 1 j } n {0 j, 1 j } j {1,..., m} f j m

B m B (pr n i,..., pr n i ) }{{} m B n N i {1,..., n} {0 j, 1 j } f = (f 1,..., f m ), f 1 = (f1 1,..., f m), 1..., f n = (f1 n,..., f m) n B f n f 1,..., f n k f( f 1,..., f n ) := ḡ = (g 1,..., g m ) j {1,..., m} g j = f j (f 1 j,..., f n j ) : {0 j, 1 j } k {0, 1}, g j (a 1,..., a k ) := f j (f 1 j (a 1,..., a k ),..., f n j (a 1,..., a k )), (a 1,..., a k ) {0 j, 1 j } k B B B j j {1,..., m} j {1,..., m} B j k f / B j n g j B j k gj 1,..., gn j B j f = (g j (gj 1,..., gn j )) g j, gj 1,..., gn j B j ḡ, ḡ 1,...ḡ n B j g j, gj 1,..., gn j h = (h 1,..., h m ) h i = g i (gi 1,..., gn i ) i j h j = f B f = h j B j Eq j = {(0 j, 0 j ), (1 j, 1 j )} j {1,..., m} Eq j,n s,t = {(a 1,..., a n ) {0 j, 1 j } n a s = a t }

S S k 1 j=1 {0 j, 1 j } Eq k m j=k+1 {0 j, 1 j } S j {1,..., m} S j {1,..., m} R S R j 1 s N j r N k N R R (a 1,..., a s, a s + 1,..., a s+k,..., a αr ) R(a 1,..., a s, pr k i 1 (a s+1,..., a a+k ),..., pr k i r (a s+1,..., a s+k ),..., a αr ), (a 1,..., a αr ) m j+1 {0 j, 1 j } s j i 1,..., i r {s + 1,..., s + k} R S S R S (α R 1) R αr S S R, R S α R = α R R R S S R 1, R 2 S j i 1 j i2 j j {1,..., m} R 1, R 2 R = R 1 R 2 R(a 1,..., a i 1 1, a i 1 1 +1,..., a i 1 1 +i2 1, a i 1 1 +i2 1 +1,..., a i 1 1 +i2 1 +i1 2,..., a α R1 +α R2 ) R 1 (a 1,..., a i 1 1, a i 1 1 +i 2 1 +1,..., a α R1 ) R 2 (a i 1 1 +1,..., a i 1 1 +i2 1, a i 1 1 +i2 1 +i1 2 +1,..., a α R2 ), (a 1,..., a αr1 +α R2 ) m j=1 {0 j, 1 j } i1 j +i2 j S m j=1 {0 j, 1 j } m j=1 Eqj k 1 j=1 {0 j, 1 j } Eq k,n s,t m j=k+1 {0 j, 1 j } j {1,..., m} n N j {1,..., m} I j j j I j

S S Ij j {1,..., m} k {1,..., m} S Ik k 1 j=1 {0 j, 1 j } Eq k m j=k+1 {0 j, 1 j } S I k Eq k Eq k S Ik R S Ik R S R = R I k R R (a 1,..., a s, a s + 1,..., a s+t,..., a αr ) R (a 1,..., a s, pr k i 1 (a s+1,..., a a+t ),..., pr k i r (a s+1,..., a s+t ),..., a α R ), s k 1 R, R t k R r R R S R = R I k R S Ik R (a 1,..., a t ) R(pr k i 1 (a 1,..., a t ),..., pr k i r (a 1,..., a t )). R S Ik R S R = R I k S R = R α R S R I k = R αr S n S ik m

R k m j i j f = (f 1,..., f m ) m f j : {0 j, 1 j } n {0 j, 1 j } j {1,..., m} f R f R f R a 1 = (a 1 1,..., a1 k ),..., an = (a n 1,..., an k ) R f(a 1,..., a n ) := (f 1 (a 1 ),..., f 1 (a i1 ), f 2 (a i1 +1),..., f 2 (a i1 +i 2 ),..., f m (a k )) R. f S f R R S 1 B S MP ol(s) = { f f R, R S} MInv(B) = {R m j=1 {0 j, 1 j } i j f R, f B} S MP ol(s) t R S m a 1,..., a n R (pr n i (a 1 ),..., pr n i (a t )) = (a i 1,..., a i t) = a i R. (pri n,..., pri n ) MP ol(s) n N i {1,..., n} }{{} m n f = (f 1,..., f m ) MP ol(s) k f 1,..., f n MP ol(s) f i = (f1 i,..., f m) i i {1,..., n} a 1,..., a k R f i (a 1,..., a k ) R i {1,..., n} f( f 1,..., f n )(a 1,..., a k ) R.

B MInv(B) X, Y X Y B [B] B B [B] = {F B F } S S S S S = {G S G} B S P ol(inv(b)) = [B] Inv(P ol(s)) = S B S MP ol(minv(b)) B MInv(MP ol(s)) S n f = (f 1,..., f m ) / B R Inv(B) f T (m 2 n ) n ((j 1) 2 n ) + 1 j 2 n {0 j, 1 j } n j {1,..., m} B (n) n B ḡ = (g 1,..., g m ) B (n) ḡ(t ) g j T {0 j, 1 j } n 2 n a M ḡ(t )

R = {ḡ(t ) T g B (n) } R M R MInv(B) d T R (prd n,..., prn d }{{} ) B (n) k = (m 2 n ) m r h B x 1 = (x 1 1,..., x1 k ),..., xr = (x r 1,..., xr k ) R R x 1,..., x r M ḡ 1,..., ḡ r B (n) x i = (ḡ i (t 1,..., t k )) i {1,..., r} t i i T (h 1 (x 1 1,..., x r 1),..., h m (x 1 k,..., xr k )) = = (h 1 (g 1 (t 1 ),..., g r (t 1 )),..., h m (g 1 (t k ),..., g r (t k ))) := x. B h, ḡ 1,..., ḡ r B h(ḡ 1,..., ḡ r ) B x M R h R Inv(B) f R T T R f(t ) R ḡ B (n) ḡ(t ) = f(t ) T f = g f B n f = (f 1,..., f m ) S j Dom(f j ) {0 j, 1 j } n f j a R / S f MP ol(s) R P = Q S,Q R S P S R / S t P \ R N R k a R R = k k f f(n T ) = t T Q.

f S f R S R (0,..., 0, i j, 0,..., 0) R Ij j {1,..., m} S B = MP ol(s) j {1,..., m} S Ij = Inv(B j ) S Ij j {1,..., m} S Ij = Inv(B j ) R S Ij R S R = R IJ S = MInv(B) f = (f 1,..., f m ) B f S f j S Ij f j R R Inv(B j ) S Ij Inv(B j ) R Inv(B j ) f B j f R f B j f B f j = f j 1 f R = {0 i, 1 i } R i=1 m i=j+1 {0 i, 1 i }, R S R = R Ij R S Ij Inv(B j ) S Ij B S 1.2

X, Y B X, Y S X Y X, Y X Y X, Y (B,,, ) (S,,, ) MP ol(s), MInv(B) B S MP ol : S B MInv : B S (B,,, ) (S,,, ) X, Y S X Y P ol(y ) P ol(x) Z, W B Z W Inv(W ) Inv(Z) X S Z B X Inv(P ol(x)) Z P ol(inv(z)) MP ol MInv MInv(MP ol) : S S MP ol(minv) : B B X, Y S Z, W B X MInv(MP ol(x)) Z MP ol(minv(z)) X Y MInv(MP ol(x)) MInv(MP ol(y )) Z W MP ol(minv(z)) MP ol(minv(w )) MInv(MP ol(x)) = MInv(MP ol(minv(mp ol(x)))) MP ol(minv(z)) = MP ol(minv(mp ol(minv(z))))

(B,,, ) (S,,, ) P ol : S B Inv : B S 1 1 X, Y S Z, W B X Y P ol(y ) P ol(x) Z W Inv(W ) Inv(Z) S V S ϕ S S V ϕ(x 1,..., x n ) = k R i (x i 1,..., x i n i ), R i S n i = α Ri i {1,..., k} k N x i j {x 1,..., x n } V i, j i=1 S ϕ(x 1,..., x n )

S ϕ ϕ SAT (S) SAT (S) S S 1 1.3 S SAT (S) SAT ( S ) S S SAT (S) P SAT ( S ) S u 1,..., u n S ϕ ϕ R(u i1,..., u ir ) R S ϕ k 1 {0 j, 1 j }(v j ) Eq k (u 1, u 2 ) j=1 m j=k+1 {0 j, 1 j }(v j ) / S, u 1, u 2 ϕ x j R(u i1,..., u j 1, x j, u j+1,..., u ir ) R j / S R S R(u i1,..., u j 1, u, u j+1,..., u ir ) u {u 1,..., u n }

ϕ R(v 1,..., v s, u i1,..., u ir, v s+1,..., v t ) R / S j 1 s j r m j t s R S R(y 1,..., y s, x 1,..., x r, y s+1,..., y t ) R (y 1,..., y s, pr r k 1 (x 1,..., x r ),..., pr r k l (x 1,..., x r ), y s+1,..., y t ), R (v 1,..., v s, u k1,..., u kl, v s+1,..., v t ) ϕ R(u 1,..., u r ) R / S R = R 1 R 2 R 1, R 2 S R 1 (u 1,..., u r ) R 2 (u 1,..., u r ) SAT (S) S m R 1 = {0 1, 1 1 } {0 2, 1 2 } = {(0 1, 0 2 ), (0 1, 1 2 ), (1 1, 0 2 ), (1 1, 1 2 )} (1, 1) R 2 = R 1 NAE {(0 2, 0 2 )} = ({0 1, 1 1 } 3 \ {(0 1, 0 1, 0 1 ), (1 1, 1 1, 1 1 )}) {(0 2, 0 2 )} (3, 2) SAT ({R 1 }) P SAT ({R 2 }) NP {R 1 } SAT ({R NAE }) P SAT ({R 2 })

SAT ({R }) NP R = R 1,2,3 ONE IN T HREE = {(1 1, 0 2, 0 3 ), (0 1, 1 2, 0 3 ), (0 1, 0 2, 1 3 )}. R = R ONE IN T HREE = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} R R (a 1, a 2, a 3 ) R(pr 3 i 1 (a 1, a 2, a 3 ), pr 3 i 2 (a 1, a 2, a 3 ), pr 3 i 3 (a 1, a 2, a 3 )), i 1 i 2 i 3 {1, 2, 3} R = R R SAT ({R}) = P SAT ( {R} ) SAT ({R}) SAT ( {R} ) SAT ({R }) S = {R} ϕ(u 1,..., u n ) S n = 3k k N u 1,..., u n/3 u n/3+1,..., u 2n/3 R(u i, u j, u l ) i {n/3 + 1,..., 2n/3} j {2n/3 + 1,..., 3n} l {1,..., n/3} R(u l, u i, u j ) R(u i, u i, u j ) R (u i, u j ) R (a 1, a 2 ) R(pr 3 1(a 1, a 2, a 3 ), pr 3 1(a 1, a 2, a 3 ), pr 3 2(a 1, a 2, a 3 )) R(u i, u i, u i ) R empty =

R(u, v) u, v {u 1,..., u n } u R R (u, v, u R ) R(u, v) u v S 0 0 j 1 1 j 0 1 j j {1, 2, 3} SAT ({R}) = P SAT ({ R }) P SAT ({ R }) = P SAT ({R }) SAT ({R}) NP SAT ({R }) R Rj = {0 j, 1 j } j {1, 2, 3} SAT ({R } Ij ) = SAT ({Rj }) SAT ({R }) NP S j {1,..., m} SAT (S Ij ) NP SAT (S) NP R SAT (S Ij ) S Ij R S R Ij = R S Ij ϕ(u 1,..., u n ) R (u i1,..., u ini ) ϕ v1 R,..., vr t R R R R (u i1,..., u ini ) R(v1 R,..., vr s, u i1,..., u ini, vs+1 R,..., vr t R ) S SAT (S Ij ) P SAT (S) SAT (S) NP

m = 1 S V S ϕ(u 1,..., u n ) = k R i (u i 1,..., u i n i ). i=1 u V Ju R R {R 1,..., R k } u V V ϕ(u 1,..., u n ) V ϕ(u 1,..., u n ) V ϕ(u 1,..., u n ) V = k pr J{u i 1,...,u i Ri (u i n } 1,..., u i n i ), i i=1 J {u i 1,...,u i n i } = u V {u i 1,...,ui n i } J u Ri i {1,..., k} V S = {R 1, R 2 } R 1 (3, 4) R 2 (5, 1) V = {u 1,..., u 15 } S ϕ(u 1,..., u 7 ) = R 1 (u 1, u 2, u 2, u 3, u 4, u 3, u 5 ) R 2 (u 1, u 2, u 6, u 7, u 2, u 5 ). V = {u 1,..., u 10 } ϕ V = ϕ V = {u 1, u 2, u 3, u 4 } V ϕ(u 1,..., u 7 ) V = pr {1,2,3,4,5,6} R 1 (u 1, u 2, u 2, u 3, u 4, u 3 ) pr {1,2,5} R 2 (u 1, u 2, u 2 ).

pr {1,2,5} R 2 (u 1, u 2, u 2 ) x 3 x 4 x 6 R 2 (u 1, u 2, x 3, x 4, u 2, x 6 ). S ϕ(u 1,..., u n ) s s 2 s n V {u 1,..., u n } s 1 u V ϕ(u 1,..., u n ) V ϕ(u 1,..., u n ) V {u} S ϕ(u 1,..., u n ) s s t 2 t s S ϕ(u 1,..., u n ) s 2 s n u {u 1,..., u n } \ V S = {R 1, R 2 } R 1 = {(0 1, 1 1, 1 1, 0 2, 1 2, 1 3 ), (1 1, 0 1, 1 1, 0 2, 1 2, 0 3 ), (1 1, 1 1, 1 1, 0 2, 1 2, 0 3 )}, R 2 = {(0 1, 1 1, 0 2, 0 3, 0 3 ), (1 1, 1 1, 0 2, 0 3, 0 3 ), (1 1, 0 1, 0 2, 1 3, 1 3 )}. S ϕ(u 1, u 2, u 3, u 4, u 5 ) = R 1 (u 1, u 2, u 1, u 3, u 4, u 5 ) R 2 (u 2, u 1, u 3, u 5, u 5 ). ϕ(u 1, u 2, u 3, u 4, u 5 ) 5 V = {u 2, u 3, u 4, u 5 } V ϕ V (u 1, u 2, u 3, u 4, u 5 ) = x 1 x 3 R 1 (x 1, u 2, x 3, u 3, u 4, u 5 ) x 2 R 2 (u 2, x 2, u 3, u 5, u 5 ). (u 2, u 3, u 4, u 5 ) = (1 1, 0 2, 1 2, 1 3 ) ϕ V (u 1, u 2, u 3, u 4, u 5 ) ϕ V {u1 } (u 1, u 2, u 3, u 4, u 5 ).

(u 1, u 2, u 3, u 4, u 5 ) = (1 1, 0 1, 0 2, 1 2, 0 3 ) 5 S = {R 1, R 2, R 3 } R 1 = {(0 1, 1 1, 1 1, 0 2, 1 2, 1 3 ), (1 1, 0 1, 1 1, 0 2, 1 2, 0 3 ), (1 1, 1 1, 1 1, 0 2, 1 2, 0 3 )}, R 2 = {(0 1, 1 1, 0 2, 0 3, 0 3 ), (1 1, 1 1, 0 2, 0 3, 0 3 ), (1 1, 0 1, 0 2, 1 3, 1 3 )}, S ϕ(u 1, u 2, u 3, u 4, u 5 ) = R 3 = {(0 1, 0 2, 1 2, 0 3 ), (1 1, 0 2, 1 2, 0 3 ). R 1 (u 1, u 2, u 1, u 3, u 4, u 5 ) R 2 (u 2, u 1, u 3, u 5, u 5 ) R 3 (u 2, u 3, u 4, u 5 ). ϕ(u 1, u 2, u 3, u 4, u 5 ) 5 R 3 u 1 s s s s 1 s s s s = 2, 3 s r n R m j=1 {0 j, 1 j } i j R r r n ā = (a 1,..., a n ) m j=1 {0 j, 1 j } i j pr I (ā) pr I R, I {1,..., n} I r R r j R Ij

R = {(0 1, 1 1, 1 1, 0 2, 1 2 ), (1 1, 0 1, 1 1, 1 2, 0 2 ), (1 1, 1 1, 0 1, 1 2, 1 2 )}, 2 1 2 R = R {(1 1, 1 1, 1 1, 0 2, 0 2 )}, 2 2 ā = {(1 1, 1 1, 1 1, 0 2, 0 2 )} / R pr {i} ā = 1 1 R i = {0 1, 1 1 } i {1, 2, 3} pr i,j ā = {1 1, 1 1 } R i,j = {(0 1, 1 1 ), (1 1, 0 1 ), (1 1, 1 1 )} i, j {1, 2, 3} R 2 2 2 R 4,5 R 2 2 R 4,5 = {0 2, 1 2 } 2 2 R I 1 1 1 ā = (a 1, a 2, a 3 ) a 1 = a 3 = 0 1 a 2 {0 1, 1 1 } pr 1,3 ā = {0 1, 0 1 } / R 1,3 ā R 0 1 R 2 ā = (1 1, 1 1, 1 1, 0 2, 1 2 ) / R pr {1,2,3} ā R I 1 pr {4,5} ā R 2 I {{1, 4}, {1, 5}, {2, 4}, {2, 5}, {3, 4}, {3, 5}} pr {i,j} ā R i,j i, j {1, 2, 3, 4, 5}

S maj = (maj,..., maj) MP ol(s) }{{} m R S 2 S ϕ 3 f = (f 1,..., f m ) f j = maj S Ij n R S n = 2 (n 1) n 3 ā = (a 1,..., a n ) pr I ā R I I {1,..., n} I 2 ā R ā j = prj n(a 1, a 2,..., a n ) j {1, 2, 3} ā j R j j {1, 2, 3} x 1, x 2, x 3 R x 1 = (x 1, a 2,..., a n ), x 2 = (a 1, x 2, a 3,..., a n ) x 3 = (a 1, a 2, x 3, a 4,..., a n ) R maj R maj(x 1, x 2, x 3 ) = ā R S ϕ(u 1,..., u n ) 3 s {1,..., n} ϕ(u 1,..., u n ) s V = {v 1,..., v s 1 } {u 1,..., u n } v {u 1,..., u n } \ {v 1,..., v s 1 } l ϕ V ϕ V {v} k ϕ V {v} (u 1,..., u n ) = R i (v1, i..., vn i i ), R i ϕ(u 1,..., u n ) vj i V {v} j {1,..., n i } i {1,..., k} S ϕ (W ) W = k i=1 W i {v } W i s 1 i {1,..., k} f i : V W i 1 1 i=1

i {1,..., k} f i f i (v) = v S ϕ (W ) ϕ (W ) = k R i (f i (v1), i..., f i (vn i i )). i=1 (k (s 1)) R ϕ (W ) k i=1 f i(v ) R S R 2 l(v ) = (l(v 1 ),..., l(v s 1 )) l(v j ) v j V l t l(v ) k t / R l (t, l ) R ϕ (W ) R R v l ϕ V {v} I {1,..., k(s 1)} I 2 pr I t R I I I 2 pr I t / R I R ϕ(u 1,..., u n ) 3

R 1, R 2 (i 1,..., i m ) (j 1,..., j m ) V R 1 (u 1,..., u s ) R 2 (v 1,..., v t ) u i, v j V i {1,..., n} j {1,..., m} s, t N {w 1,..., w n } = {u 1,..., u s, v 1,..., v t } n N R 1 (u 1,..., u s ) R 2 (v 1,..., v t ) R 1 R 2 (w 1,..., w n ) = pr k1,...,k l (R 1 R 2 ) k l j=k 1 Eq n w j (w 1,..., w n ), Eqw n j n w j w k1,..., w kl w 1,..., w n R 1, R 2 S R 1 R 2 S R 1 R 2 R 1 (u 1, u 2, u 1, u 5, u 4 ) R 2 (u 3, u 1, u 4, u 6 ) R 1 = {(1 1, 0 1, 1 1, 1 2, 0 2 ), (0 1, 0 1, 1 1, 0 2, 0 2 )}, R 2 = {(0 1, 1 1, 0 2, 1 2 ), (0 1, 0 1, 1 2, 0 2 )}. R 1 R 2 (u 1, u 2, u 3, u 4, u 5, u 6 ) = {(1 1, 0 1, 0 1, 0 2, 1 2, 1 2 )} S m j {1,..., m} f j = (f j 1,..., f j m) MP ol(s) f j j {cn 0 j, c n 1 j,,, maj, }, f j f j j P = n N SAT (S) 1 S j {1,..., m} f j = (f j 1,..., f m) j MP ol(s) f j j {,, maj} n N SAT (S) P {, } H = {j {1,..., m} f j j = } S

S ϕ(u 1,..., u n ) = k R i (u i 1,..., u i n i ). i=1 3, R r (u r 1,..., ur n r ) R s (u s 1,..., us n s ) R t (u t 1,..., ut n t ) r, s, t {1,..., k} (R r R s R t ) {u r 1,...,u r nr }, (R r R s R t ) {u s 1,...,u s ns }, (R r R s R t ) {u t 1,...,u t n t }. S ϕ (u 1,..., u n ) D(u) u ϕ (u 1,..., u n ) u {u 1,..., u n } u ϕ (u 1,..., u n ) P ol(s Ij ) j H P ol( S Ij ) P ol(d(u)) u H D(u) = D(u) H t = (t 1,..., t n ) t i = (D(u)) u H t R R ϕ (u 1,..., u n ) t 1, t 2 R v 1, v 2 t 1 i = (D(v 1 )) t 2 j = (D(v 2 )) i j H v 1, v 2 {u 1,..., u n } i, j R t 1 1,2 = (t1 1,..., t1 i,..., t2 j,..., t1 n) t 2 1,2 = (t2 1,..., t1 i,..., t2 j,..., t2 n) R t R

D(u) 2 S j {1,..., m} f j = (f j 1,..., f j m) MP ol(s) f j j {,, maj, } f j f j j = SAT (S) P S ϕ(u 1,..., u n ) A = {j {1,..., m} f j j = } ḡ = (g 1,..., g m ) g j = j A ϕ V (u 1,..., u n ) V A ϕ V (u 1,..., u n ) ( n 3) 3 S j {1,..., m} f j = (f j 1,..., f j m) MP ol(s) f j j {cn 0 j, c n 1 j,,, maj, } n N SAT (S) P C = {j {1,..., m} f j j {cn 0 j, c n 1 j }, n N} j C g := f j f j j ( x) := c j x {0 j, 1 j } n c j {0 j, 1 j } i {1,..., m} \ {j} f j i f j i S Ii (c, c,..., c) S Ij f j i f j i ( x) = c i {0 i, 1 i } x {0 i, 1 i } n

S j {1,..., m} f j = (f j 1,..., f m) j MP ol(s) f j j / {c 0 j, c 1j,,, maj, } SAT (S) NP P ol(s Ij ) = P ol(inv(mp ol(s) j )) = MP ol(s) j. c 0j, c 1j,,, maj, / P ol(s Ij ) SAT (S Ij ) NP SAT (S Ij ) SAT (S) SAT (S) NP m = 1 C (S) S S S S S ϕ(u 1,..., u n ) S ϕ(u 1,..., u n, u) u 0 j S ϕ (u 1,..., u n, u) = ϕ(u 1,..., u n, u) {0 j }(u).

S S j {1,..., m} {0 j } {1 j } {0 j, 1 j } S S S c S m S c = {{0 j }, {1 j }, {0 j, 1 j }} S. j=1 n f = (f 1,..., f m ) f j {1,..., m} (a 1,..., a n ) {0 j, 1 j } n f j (a 1,..., a n ) {a 1,..., a n }. f j {1,..., m} f j (c j,..., c j ) = c j c j {0 j, 1 j } 2 S f MP ol(s c ) f f MP ol(s). SAT C (S) S ϕ ϕ SAT (S c ) S c ϕ ϕ S j {1,..., m} f j = (f j 1,..., f j m) MP ol(s c ) f j j {,, maj, }, SAT (S c ) P SAT (S c ) NP

1, 2 4,, maj,

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n n N m j {0 j, 1 j } R m {0 j, 1 j } i j, j=1 i j N j {1,..., m} R j i {1,..., α R } R i = 2 R AGG(R) n f = (f 1,..., f m ) n N f MP ol(r) f R S AGG(S) R S R S f c R 3.4 f f {{0 j }, {1 j }, {0 j, 1 j } j {1,..., m}}. f AGG(S) f MP ol(s c ) AGG(S) MP ol(s c )

R f n f i, j {1,..., m} 1 1 g ij : {0 j, 1 j } {0 j, 1 j } n (x 1,..., x n ) {0 i, 1 i } n f j (g ij (x 1 ),..., g ij (x n )) = g ij (f i (x 1,..., x n )). R S 0 j 0 1 j 1 j {1,..., m} f i f j i, j {1,..., m} 0 1 R f = (maj,..., maj) }{{} m ḡ = (,..., ) AGG(R) }{{} f, ḡ m maj, R (maj, ) AGG(R) (maj, ) 1 1 g : {0 1, 1 1 } {0 2, 1 2 } g(0 1 ) = 0 2 g(1 1 ) = 1 2 (g(0 1 ), g(1 1 ), g(0 1 )) = 1 2 0 2 = g(0 1 ) = g(maj(0 1, 1 1, 0 1 )).

R f = (,..., ) }{{} m ḡ = (,..., ) AGG(R) }{{} f, ḡ m f ḡ 1 1 g : {0 j, 1 j } {0 j, 1 j } g(0 j ) = 1 j g(1 j ) = 0 j f j (g(0 j ), g(1 j )) = (1 j, 0 j ) = 0 j 1 j = g(0 j ) = g( (0 j, 1 j )). R f n f d {1,..., n} j {1,..., m} f j = pr n d. R S R f AGG(R) S f AGG(S) R S R = m j=1 {0 j, 1 j }

P OSS(S) S S P OSS(S) SAT (S C ) S S S S n n = 2 (n 1) n 3 n f = (f 1,..., f m ) i, j {1,..., m} g : {0 i, 1 i } {0 j, 1 j } 1 1 x 1 i,..., xn i {0 i, 1 i } f j (g(x 1 i ),..., g(x n i )) g(f i (x 1 i,..., x n i )). n 3 x k i {0 i, 1 i } k {1,..., n} s t {1,..., n} x s i = xt i h = (h 1,..., h m ) h j (a 1 j,..., a s j,..., a t 1 j, a t+1 j,..., a n j ) = f j (a 1 j,..., a n j ),

a 1 j,..., an j {0 j, 1 j } j {1,..., m} (n 1) h S h j (g(x 1 i ),..., g(x n i )) g(h i (x 1 i,..., x n i )). 1 1 S S (,..., ) AGG(S) }{{} m (maj,..., maj) AGG(S) }{{} m S ( ) ( ), maj / AGG(S) S S j {1,..., m} AGG(S) j j {1,..., m} maj AGG(S) Ij AGG(S) (,..., ) }{{} m AGG(S) (maj,..., maj) }{{} m AGG(S) AGG(S) Ij j {1,..., m} S S S ( ) ( ) (,..., ) AGG(S) }{{} m (maj,..., maj) }{{} m AGG(S) S f = (f 1,..., f m ) ḡ = (g 1,..., g m ) g j (x 1, x 2, x 3 ) = f j (x 1, x 2 ) j {1,..., m} ḡ

R = {(1 1, 0 1, 0 1, 1 2, 0 2, 0 2 ), (0 1, 1 1, 0 1, 0 2, 1 2, 0 2 ), (0 1, 0 1, 1 1, 0 2, 0 2, 1 2 )}. R (f, g) R f {,, maj, } g {,, maj, } (pr1 2, pr2 2 ), (pr2 2, pr2 1 ) / AGG(R) R S SAT (S C ) P S j {1,..., m} f j = (f j 1,..., f j m) f j {,,, maj} S S S SAT (S c ) NP R = {(1 1, 0 1, 0 1, 0 2 ), (0 1, 1 1, 0 1, 0 2 ), (0 1, 0 1, 1 1, 0 2 )} = R ONE IN T HREE { 0 2 }, S = {R} SAT (S I1 ) NP SAT (S) (pr1 2, ) AGG(S) S

S (, pr 2 1, pr2 2,, pr2 1 ) m = 5 SAT (Sc ) P R j {1,..., m} n N n f j = (f j 1,..., f m) j AGG(R) f j j prn d d {1,..., n} S j {1,..., m} n N n f j = (f j 1,..., f m) j AGG(S) f j j prn d d {1,..., n} R = {(0 1, 0 1, 0 2, 0 2, 1 3 ), (0 1, 0 1, 1 2, 0 2, 1 3 )} R (,, ) AGG(R)

R = R 1 NAE {(0 2, 0 2 )} R 1 NAE = {0 1, 1 1 } 3 \ {(0 1, 0 1, 0 1 ), (1 1, 1 1, 1 1 )} R (pr1 2, pr2 2 ) AGG(R) R SAT (R NAE ) NP (f, g) AGG(R) f {,, maj, } R S S SAT (S c ) P. ( ) j {1,..., m} f j = (f 1,..., f m ) AGG(S) f j j {,, maj, } S ( ) AGG(S) j j {1,..., m} n N n g AGG j g prd n d {1,..., n} {,, maj, } AGG(S j ) f j = (f j 1,..., f m) j AGG(S) f j j {,, maj, } j {1,..., m} SAT (S c ) P