Does this algorithm halt? Yes

Does this algorithm halt? Yes No

REC RE

ΚΕΦΑΛΑΙΟ 0,, 2 A,,,,, A, A,,,,,,,,, A P n A P A A n N n f A B f : A B f ((a, b 1 ) f (a, b 2 ) f) b 1 = b 2 (a, b) f f(a) = b f : A B f b B a A((a, b) f) f ((a 1, b) f (a 2, b) f) a 1 = a 2 f : A B f a A b B((a, b) f) f(a) = B f : A B f dom(f) = {a A f(a) B} A

f im(f) = {b B a A(f(a) = b)} B f : A B f f 1 : im(f) dom(f) f 1 (x) = y y f(y) = x f : A B g : B C f g g f : A C g(f(x)), x dom(f) g f(x) =, χ A : A {0, 1} A 1, a A χ A (a) = 0, N Z [n] {1,..., n} n N R A A N A N A, B f : A B A N A = ℵ 0 A A < 2 A f : A 2 A B = {x A x f(x)} B A f b A f(b) = B b B b B x x! : x y : x y x y : x y {0, 1} {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} {q, w, e, r, t, y, u, i, o, p, a, s, d, f, g, h, j, k, l, z, x, c, v, b, n, m,,.,?,!} 0

Σ Σ 101010101010110 {0, 1} 1917 {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} mpla {q, w, e, r, t, y, u, i, o, p, a, s, d, f, g, h, j, k, l, z, x, c, v, b, n, m,,.,?,!} w Σ w w 101010101010110 = 15 1917 = 4 mpla = 4 ϵ 0 Σ i = {w Σ w = i} i N Σ Σ = Σ i i=1 Σ w w R w mpla R = alpm w 1 = x 1 x m w 2 = y 1 y n Σ w 1 w 2 w 1 w 2 w 1 w 2 = x 1 x m y 1 y n w, w Σ w w w 1, w 2 Σ w 1 = ϵ w 2 = ϵ w = w 1 w w 2 Σ σ : Σ [ Σ ] Σ σ {0, 1} σ(0) = 1 σ(1) = 2 ϵ, 0, 1, 00, 01, 10, 11, 000, 001, $, Σ σ($) > σ( ) $

Σ σ : Σ [ Σ ] Σ σ Σ N Σ = ℵ 0 Σ σ : Σ [ Σ ] w, w Σ w w σ w < σ w Σ σ w < w w w w w 11 < 000 Σ x Σ x k x x k k N Σ x Σ L Σ Σ {0, 1} L = {w {0, 1} w = 0 n 1 n, n N} L = {w {0, 1} w = w R } L = L = {ϵ} Σ L 1, L 2 Σ L 1 L 2 = {w Σ w 1 L 1 w 2 L 2 (w = w 1 w 2 )} Σ L Σ L 0 = {ϵ} L 1 = L L n = LL n 1 L = L i i N Σ Σ Σ {[, ],,, } Σ a, b [ab] a, b [a b] a a (01) 2 01 2 = 011 = 0101

R Σ Σ [0 1 ], [0 1 ], [[01 ]0], [01] {0, 1} Σ L : R Σ 2 Σ {a} L(x) = L(a)L(b) L(a) L(b) L(a), x =, x = a Σ, x = [ab] a, b R Σ, x = [a b] a, b R Σ, x = a a R Σ L [[0 1] 0] {0, 1} L([[0 1] 0]) = L([0 1] )L(0) = L([0 1]) {0} = (L(0) L(1)) {0} = ({0} {1}) {0} = {0, 1} {0} = {w {0, 1} w 1 {0, 1} (w = w 1 0)}

ΚΕΦΑΛΑΙΟ 1 M = (Q, Σ, Γ, δ, q 0, q, q ) Q, Σ, Γ Q q 0, q, q Q q q q 0 q, q Σ Γ Σ Γ, Γ Σ δ : Q Γ Q Γ {, } a Γ q {q, q }(δ(q, a) = ) q, q Q {q, q } a Γ(δ(q, ) = (q, a, x) x = ) M = (Q, Σ, Γ, δ, q 0, q, q ) w Σ M M Γ Γ M w w M Q M w

w q q q 0 q1 q2 δ (q 3, 0, q 4, 3, ) δ M = (Q, Σ, Γ, δ, q 0, q, q ) M w M = (Q, Σ, Γ, δ, q 0, q, q ) 100010100 q 3 100q 3 010100 (Γ Q) M = (Q, Σ, Γ, δ, q 0, q, q ) w Σ M w M(w) q 0 w M(w) w 1 qw 2 w 1, w 2 Γ q {q, q } M = (Q, Σ, Γ, δ, q 0, q, q ) a i Γ, i [n] q Q a 1 a 2 a i 1 qa i a n

q q q 0 q1 q2 q 3 q 4 q q q 0 q1 q2 q 3 q 4 δ(q, a i ) = (p, b, ) i > 1 a 1 a 2 a i 2 pa i 1 ba i+1 a n a 1 a 2 a i 1 qa i a n M a 1 a 2 a i 2 pa i 1 ba i+1 a n δ(q, a i ) = (p, b, ) i > 1 a 1 a 2 a i 1 bpa i+1 a n a 1 a 2 a i 1 qa i a n M a 1 a 2 a i 1 bpa i+1 a n M = (Q, Σ, Γ, δ, q 0, q, q ) w Σ C 1, C 2 M(w) C 1 M C 2 C 1 C 2 M M C 1 M C 2 C 1 C 2 M = (Q, Σ, Γ, δ, q 0, q, q ) w Σ M(w) C 1,..., C n C 1 C i M C i+1 i [n 1] C(C M C) C 1 M C n C 2,..., C n 1(C 1 M C 2 M M C n 1 M C n)

0/0, /, /, q q 0 1/1, q M(w) w 1, w 2 Γ q {q, q } q 0 w M w 1 qw 2 q = q M(w) q M w q = q M(w) q M w M(w) M(w) M w Σ M(w) t q M(w) t q t N M(w) M(w) M(w) t M(w) q q M(w) q M = (Q, Σ, Γ, δ, q 0, q, q ) M L(M) = {w Σ M(w) q } δ M = (Q, Σ, Γ, δ, q 0, q, q ) Q = {q 0, q, q } Σ = {0, 1} Γ = {,, 0, 1} δ = {(q 0, 0, q 0, 0, ), (q 0,, q 0,, ), (q 0,, q,, ), (q 0, 1, q, 1, )} δ L(M) = {w {0, 1} w = 0 n, n N} M = (Q, Σ, Γ, δ, q 0, q, q ) Q = {q 0, q, q }

/, q 0 q q Σ { } q q Σ = {0, 1} Γ = {,, 0, 1} δ L(M) = M = (Q, Σ, Γ, δ, q, q, q ) Q = {q, q } Σ = {0, 1} Γ = {,, 0, 1} δ L(M) = {0, 1} L Σ L M L = L(M) w L M(w) q L Σ L M w L M(w) q w L M(w) q

1/1, 1/1, 0/0, q 0 q 1 /, 0/0, /, q q 1/1, 1/1, 1/1, 0/0, 0/0, 0/0, q 0 q 1 q 2 q /, /, /, q L = {w {0, 1} 0 w } L w {0, 1} 0 w M(w) q 0 w M(w) q L = {w {0, 1} 0 w 3} L L = {w {0, 1} w = w R } L M = (Q, Σ, Γ, δ, q 0, q, q ) Q = {q 0, q 1, q 2, q 3, q 4, q 5, q, q } Σ = {0, 1} Γ = {,, 0, 1, } δ Σ {0, 1} 2 {0,1} {A, B, C} A 01 B 011 C 0111 ACAB 01011101011

/, /, 0/, 0/0, 1/1, q 1 /, /, q 2 /, 0/, q q 0 /, /, q 3 0/0, 1/1, 1/, 1/, q 4 q 5 /, 1/1, /, 0/0, 0/0, 1/1, q 2 {0,1} RE REC REC RE RE = {L {0, 1} L} REC = {L {0, 1} L} REC RE 2 {0,1} L RE REC L 2 {0,1} RE

0/, 1/, /, 1/1, /1, q 0 q 1 /, q 2 /, q q Σ { } 0/, /, 0/0, q 0 /0, q 1 /, q 2 /, q 1/1, 1/1, q q 3 /, Σ { } q 3 f : Σ Σ M = (Q, Σ, Γ, δ, q 0, q, q ) w dom(f)( q 0 w M qf(w)) q {q, q } M f id : Σ Σ id(x) = x f : {0, 1} {0, 1} f(x) = 1 f : {0, 1} {0, 1} f(x) = 0, x {0 n n N}, : Σ Σ (x) = M = (Q, Σ, Γ, δ, q 0, q ) q

1/1, 0/0, 1/1, 1/1, q 0/0, 0 q 1 /, q 2 0/1, q 3 1/0, /0, q 4 /, q /, q 1/0, {0, 1} 0/0, 1/1, 0/, q 0 /, q 1 /, q 2 /, 1/, q 4 /0, q 3 /1, q q M space next : {0, 1} {0, 1} next(x) = x {0, 1} M next space : {0, 1} {0, 1, } space(x) = x M space f : {0, 1} {0, 1, } f(x) = x M space L REC M L w M w L w / L

0/0, 1/1, 0/, q 0 /, q 1 /, q 2 /, 1/, q 4 /0, q 3 /1, q 6 q 5 /, 0/0, 1/1, 0/, q 9 /0, q /, 7 q 8 /, q 10 /, 1/, q 11 /1, q q L RE M L w M w L f : Σ Σ M x M f(x) x dom(f) M(x) M x f(x)

M x f 1 (x) M 1 M 2 f 2 (f 1 (x)) f 2 f 1 y ϵ y M f f(y) f(y) = x M y x y next(y) M next f 1 f 1, f 2 : Σ Σ f 2 f 1 M 1 M 2 f 1 f 2 M f 2 (f 1 (x)) Σ M(x) f 1 (x) = f 2 (f 1 (x)) = M(x) M f 2 f 1 f : {0, 1} {0, 1} f 1 M f f M f M next y ϵ f L Σ L REC χ L L M space

M x M L 1 0 χ L L REC x M χl M L χ L ( ) L REC M L χ L ( ) M χl χ L M L M ϕ M : {0, 1} {0, 1} M f : {0, 1} {0, 1} ϕ M (x) = f(x) M 1, M(x) q ϕ M (x) = 0, M(x) q, M(x) ϕ M M 1 M 2 ϕ M1 = ϕ M2 E = (Q, Σ, Γ, δ, q 0, q ) Q, Σ, Γ Q q 0, q Q q Q ϕ M χ L(M) L(M) REC

q q 0 q1 q2 q 0 /, q /1, {1 n n N} Σ Γ Σ Γ,, Γ Σ E δ : Q Γ Q Γ {, } q, q Q a Γ(δ(q, ) = (q, a, x) x = ) E q w pop E (w) pop(w) E E E = (Q, Σ, Γ, δ, q 0, q ) E L(E) = {w Σ w Γ ( q 0 E w wq )} L(E) = {w Σ pop E (w)} L Σ E = (Q, Σ, Γ, δ, q 0, q ) L = L(E) E L(E) = {1 n n N} E L(E) = {(10) n n N}

q 0 /, /1, q q 1 /0, {(10) n n N} x M E pop(w) x = w L E E L E pop(w), w L L Σ L RE E = (Q, Σ, Γ, δ, q 0, q ) L(E) = L ( ) ( ) E L M L L L RE L Σ E = (Q, Σ, Γ, δ, q 0, q ) L(E) = L E L L Σ L REC E = (Q, Σ, Γ, δ, q 0, q ) L ( ) L REC M L E L ( ) E L M L

w ϵ w M L w w q E pop(w) next(w) w M next w L REC M M x x = x x < w w w = E pop(w) x = w L E L L REC k M k = (Q, Σ, Γ, δ, q 0, q, q ) Q, Σ, Γ Q q 0, q, q Q q q Σ Γ Σ Γ, Γ Σ δ : Q Γ k Q Γ k {, } k

1 q q q 0 q1 q2 2 k k a 1,..., a k Γ q {q, q }(δ(q, a 1,..., a k ) = ) i [k] q, q Q {q, q } a 1,..., a k, b 1,..., b k Γ x 1,..., x k {, }(δ(q, a 1,..., a k ) = (q, b 1,..., b k, x 1,..., x k ) a i = x i = ) M k M k k k M k k k k N M k = (Q, Σ, Γ, δ, q 0, q, q ) k M = (Q, Σ, Γ, δ, q 0, q, q ) Γ = Γ { } {ȧ a Γ { }} { } M k k M k M k M k M δ Q M k i N i k i i > k M k M δ Q M

0 1 0 0 0 q q q 0 q1 q2 1 1 0 1 1 0 0 0 1 0 0 0 1 1 0 1 1 0 0 q q q 0 q1 q2 k M δ M k M space δ M w = w 1 w n Σ M w 1 w n M w 1 w n

M w 1 w 2 M M w 1 w 2 M k = (Q, Σ, Γ, δ, q 0, q, q ) Q, Σ, Γ Q q 0, q, q Q q q Σ Γ Σ Γ, Γ Σ δ Q Γ Q Γ {, } a, b Γ q Q x {, }((q, a, q, b, x) δ q, a, q, b, x) δ) q Q {q, q } q Q a Γ((q,, q, a, x) δ x = ) N δ N N 0010 1 q 0 0010 N 0q 0 010 N 00q 0 10 N 001q 0 2 q 0 0010 N 0q 0 010 N 00q 0 10 N 001q 0

0/0, 1/1, q q 0 1/1, q N = (Q, Σ, Γ, δ, q 0, q, q ) N L(N) = {w Σ w 1, w 2 Γ ( q 0 w N w 1 q w 2 )} q L Σ N N L L = L(N) L Σ N N L L = L(N) w Σ N(w) f : Σ Σ N N f w Σ w dom(f) N(w) qf(w) q {q, q } w dom(f) N(w) L = {w {0, 1} w 1} N ND N N ϵ N(ϵ)

N = (Q, Σ, Γ, δ, q 0, q, q ) M w Σ (w L(N) w L(M)) w Σ N(w) k N q M(w) k w Σ w L(N) k 0 N N(w) q N(w) k 0 M(w) q w L(M) w L(N) k N N(w) q M(w) w L(M) N(w) (q, a) q Q a Γ r = Q Γ 2 δ(q, a) N(w) r r [r] N(w) k k N c {1,..., r} k N D w c {1,..., r} N(w) N D (w) c c 0 [r] N(w) c 0 N(w) c q N D (w) c N D (w) q i [ c ] c i N N D (w) q M next w {1,..., r} next(w) {1,..., r} M w N D M M M c = c 1 c k i k c i δ

w c N N D N D w 2 w c ϵ 3 w c N D M next(c) N D (w, c) q M next 3 c M {0, 1} Σ Σ Σ M = (Q, {0, 1}, {,, 0, 1}, δ, q 0, q, q ) δ M M δ δ Σ = {0, 1, q, s, d, } δ M w M(w)

Q q {0, 1} i = 2 Q q 0 Σ = q 00 00 i q 1 Σ = q00 01 q 2 Σ = q00 10 q Σ = q11 10 q Σ = q11 11 {,, 0, 1} Σ = s00 Σ = s01 0 Σ = s10 1 Σ = s11, Σ = d0 Σ = d1 (a, q, b, p, x) δ a, b {,, 0, 1} q, p Q x {, } (a, q, b, p, x) Σ = a Σ q Σ b Σ p Σ x Σ δ = {δ 1, δ 2,..., δ k } δ δ Σ = δ 1 Σ δ 2 Σ δ k Σ q00s10q00s10d1 q00s00q00s00d1 q00s01q01s01d1 q00s11q11s11q1 M Σ δ Q = n M n {0,1} δ Σ n {0,1} n

{0, 1, q, s, d, } {0, 1} 0 {0,1} = 01 1 {0,1} = 011 q {0,1} = 0111 s {0,1} = 01111 d {0,1} = 011111 {0,1} = 0111111 {0, 1} M G = { M {0, 1} M } G {0, 1} Gödel : G N Gödel(M) = M {0, 1} Gödel(M) M G REC Gödel Gödel L 2 {0,1} RE Gödel G N G = ℵ 0 2 {0,1} > ℵ 0 G 2 {0,1} L 2 {0,1} M L(M) = L {0, 1} M = (Q, Σ, Γ, δ, q 0, q, q ) w M {0, 1} M(w) 1 ( M, w ) M, w ( M, w ) δ {0, 1, q, s, d} Gödel(M) Gödel( M ) {0, 1} M, w

M M M(w) M 2 w 1 q 0 3 M(w) ( M, w ) 3 1 1 M 2 3 1 1 3 q 3 q q q 2 3 w Σ M, w M(w) q M(w) q

0 1 2 3 4 ϵ (ϵ, 0) (ϵ, 1) (ϵ, 2) (ϵ, 3) (ϵ, 4) 0 (0, 0) (0, 1) (0, 2) (0, 3) (0, 4) 1 (1, 0) (1, 1) (1, 2) (1, 3) (1, 4) 00 (00, 0) (00, 1) (00, 2) (00, 3) (00, 4) 01 (01, 0) (01, 1) (01, 2) (01, 3) (01, 4) np {0, 1} N M L M L (w, t) (ϵ, 0) (w, t) M L (x) x Σ (x w) t E np(w, t) M np (w, t) (w, t) x i1,..., x in pop(x ij ), j [n] E L M L M w M(w) M(w) Σ np : Σ N Σ N np M np ( ) ( ) M L L M np E L

REC RE w M 1 M M 2 w M L 1 L 2 REC RE L 1, L 2 Σ L 1, L 2 REC L 1 L 2 REC L 1, L 2 RE L 1 L 2 RE M 1, M 2 L 1, L 2 M L 1 L 2 L 1, L 2 Σ L 1, L 2 REC L 1 L 2 REC L 1, L 2 RE L 1 L 2 RE w M 1 M M 2 w M L 1 L 2 M 1, M 2 L 1, L 2 M L 1 L 2 M 1, M 2 L 1, L 2 M L 1 L 2 L Σ L REC L REC M L L M L L

t + 1 t 0 t M 1 M 1 M w M 1 (w) t t t + 1 (w, t) M 2 M 2 t M 2 (w) t M L 1 L 2 M L w M L M L L L Σ L RE L RE L REC L 1, L 2 Σ L 1, L 2 REC L 1 L 2 REC L 1, L 2 RE L 1 L 2 RE L Σ L REC L REC L RE L RE L Σ L R = {w Σ w R L} L REC L R REC L RE L R RE L Σ L p = {w L w = w R } L REC L p REC L RE L p RE

REC RE

ΚΕΦΑΛΑΙΟ 2 {0, 1} w {0, 1} n w 2 {0,1} {0, 1} N N f : N m N m 1 s(x) = x + 1 c m q (x 1,..., x m ) = q q N q p m i (x 1,..., x m ) = x i 1 i m i {0, 1} ϵ, 00, 01 {0, 1} {0, 1} N f = s m = 1

g : N n N h i : N m N 1 i n g h i, 1 i n f(x 1,..., x m ) = g(h 1 (x 1,..., x m ),..., h n (x 1,..., x m )) g : N m 1 N h : N m+1 N f(0, x 1,..., x m 1 ) = g(x 1,..., x m 1 ) f(y + 1, x 1,..., x m 1 ) = h(f(y, x 1,..., x m 1 ), y, x 1,..., x m 1 ) m = 0 c 0 q = q q N m = 1 f : N N f(0) = c 0 q f(y + 1) = h(f(y), y) h : N 2 N f : N 3 N f(x, y, z) = x + 1 plus : N 2 N plus(x, y) = x + y plus f(x, y, z) = s(p 3 1(x, y, z)) plus(0, y) = p 1 1 (y) plus(x + 1, y) = f(plus(x, y), x, y) mult : N 2 N mult(x, y) = x y mult(0, y) = c 1 0 (y) mult(x + 1, y) = plus(p 3 1 (mult(x, y), x, y), p3 3 (mult(x, y), x, y)) plus fact : N N fact(x) = x! mult fact(0) = c 0 1 fact(x + 1) = mult(fact(x), s(x)) m > 1 m = 1 x

0, x = 0 pd : N N pd(x) = x 1, pd(0) = c 0 0 pd(x + 1) = p 2 2 (pd(x), x) : N 2 N (x, y) = 0, x < y x y, (x, 0) = p 1 1 (x) (x, y + 1) = pd(p 3 1 ( (x, y), x, y)) pd x y (x, y) min : N 2 N min(x, y) = {x, y} min(x, y) = x (x y) max : N 2 N max(x, y) = {x, y} max(x, y) = plus(x, y) min(x, y) plus, min exp : N 2 N exp(x, y) = x y exp(x, 0) = c 1 1 (x) exp(x, y + 1) = mult(p 3 1 (exp(x, y), x, y), p3 3 (exp(x, y), x, y)) mult P N m m 1 χ P : N m {0, 1} {(x, y) N 2 x = y} χ = (x, y) = c 0 1 plus(x y, y x) plus (x, y) = f(p 2 2(x, y), p 2 1(x, y)) f(0, y) = p 1 1(y) f(x + 1, y) = pd(p 3 1(f(x, y), x, y))

{(x, y) N 2 x y} χ (x, y) = c 0 1 (x y) {(x, y) N 2 x < y} χ χ < (x, y) = χ (s(x), y) P N m g : N m N h : N m N m 1 f : N m N g(x 1,..., x m ) f(x 1,..., x m ) = h(x 1,..., x m ), (x 1,..., x m ) P, f(x 1,..., x m ) = χ P (x 1,..., x m ) g(x 1,..., x m ) + (1 χ P (x 1,..., x m )) h(x 1,..., x m ), 1 c m 1 (x 1,..., x m ) +, exp rm : N 2 N rm(x, y) = x y rm(0, y) = 0 x + 1, y = 0 rm(x + 1, y) = rm(x, y) + 1, rm(x, y) + 1 < y 0, = <

{(x, y) N 2 x y} χ (x, y) = 1 rm(y, x) rm qt : N 2 N qt(x, y) = x y qt(0, y) = 0 0, y = 0 qt(x + 1, y) = qt(x, y) + 1, rm(x + 1, y) = 0 qt(x, y), rm gcd : N 2 N gcd(x, y) = x y rm 0, x = 0 y = 0 gcd(y, x), x < y gcd = y, y x gcd(x, rm(x, y)), dn : N 2 N dn(x, y) = y x χ dn(0, y) = 0 dn(x + 1, y) = dn(x, y) + χ (x + 1, y) Prime = {x N x } dn 1, dn(x, x) = 2 χ Prime (x) = 0, P N m+1 m 1 y N {i y (i, x 1,..., x m ) P }, (µ i y)[(i, x 1,..., x m ) P ] = y + 1,

P N m+1 m 1 f : N m+1 N f(y, x 1,..., x m ) = (µ i y)[(i, x 1,..., x m ) P] g(y, x 1,..., x m ) = prod(y, 1 χ P (i, x 1,..., x m )) f(y, x 1,..., x m ) = sum(y, x 1,..., x m ), P N m+1 g : N m N m 1 f : N m+1 N f(z, x 1,..., x m ) = (µ i g(x 1,..., x m ))[(i, x 1,..., x m ) P] pn : N N pn(x) = x pn(0) = 2 pn(x + 1) = (µ i fact(pn(x)) + 1)[(i, x) P] P = {(x, y) N 2 y < x x Prime} fact pn P n N p n < p n! + 1 χ P (x, y) = χ < (y, x) + χ Prime (x) 1 χ < χ Prime n! + 1 p < n! + 1 p n! + 1 p n! 1 p 1, 2,..., n p n < p n! + 1 sum(y, x 1,..., x m) = g(i, x 1,..., x m) y i=0 y prod(y, x) = h(i, x) g : N m+1 N h : N 2 N i=0

g : N m+1 N m 1 y N g µ (µ i y)[g(i, x 1,..., x m )=0]={i y (g(i, x 1,..., x m )=0) ( j (y j <i g(i, x 1,..., x m )>0))} i y g(i, x 1,..., x m ) = 0 (µ i y)[g(i, x 1,..., x m ) = 0] = g j y (µ i y)[g(i, x 1,..., x m ) = 0] = i j g(i, x 1,..., x m ) = 0 f : N m N m 1 µ P N m m 1 µ χ P : N m {0, 1} p N p p + 2 tpn : N N tpn(x) = x tpn(0) = 3 tpn(x + 1) = (µ i tpn(x) + 1)[2 (χ Prime (i) + χ Prime (i + 2)) = 0] χ Prime tpn tpn

A(0, y) = y + 1 A(x + 1, 0) = A(x, 1) A(x + 1, y + 1) = A(x, A(x + 1, y)) f : N m N x 0, x 1,..., x n 1 n i (x i +1) 2 1 3 2 5 3 7 4 = 5402250 n i i n i 5, 5 2, 5 3 5402250 5 4 3 1 = 2 n N enc n : N n N n dec : N 2 N dec(x, y) = x y enc n dec enc n (x 0,..., x n 1 ) = pn(0) x 0+1 pn(n 1) x n 1+1 (µ i y)[pn(x) i+1 y] 1, pn(x) y dec(x, y) = y + 1, +1 0 dec (x, y) N 2 y

pn pn(x) i+1 y (pn(x) i+1, y) N {(x, y) N 2 x y} f M M M n N M(n) f(n) ( ) ( ) f : N N M = (Q, Σ, Γ, δ, q 0, q, q ) M Q = {q 0, q 1,..., q n } q n Σ = {1} Γ = {a 0, a 1,..., a k } a 0 = a 1 = a 2 = 1 M M M N M(w) M 0 1 n 1 1 (n+1) M n q n 1 1 (f(n)+1)

q i Q q i N = i a i Γ a i N = i w 1 w n Γ w 1 w n N = enc n ( w 1 N,..., w n N ), N = 0 N = 2 w 1 q i w 2 i w 1 + 1 w 1 w 2 N w 1 q i w 2 N = enc 3 (i, w 1 + 1, w 1 w 2 N ) q 0 1 1 N = enc 3 (0, 2, 1 1 N ) (n+1) (n+1) x N M(n) x cs(x) = dec(0, x) ctp(x) = dec(1, x) ctn(x) = dec(2, x) cts(x) = dec(ctp(x), ctn(x)) tr M : N 2 N tr M (x, y) = M(x) y δ tr M (x, 0) = q 0 1 1 N (x+1) δ(q i0, b 0 ) = (q j0, c 0, d 0 ) δ(q i1, b 1 ) = (q j1, c 1, d 1 ) δ(q im, b m ) = (q jm, c m, d m )

q ir, q j,r Q b r, c r Γ d r {, } r [m] ns : N N j 0 j 1 ns(x) = j m cs(x), cs(x) = i 0 cts(x) = b 0 N, cs(x) = i 1 cts(x) = b 1 N, cs(x) = i m cts(x) = b m N, ntp : N N ctp(x) + d 0 N 1 ctp(x) + d 1 N 1 ntp(x) = ctp(x) + d m N 1 ctp(x), cs(x) = i 0 cts(x) = b 0 N, cs(x) = i 1 cts(x) = b 1 N, cs(x) = i m cts(x) = b m N, nts : N N c 0 N, cs(x) = i 0 cts(x) = b 0 N c 1 N, cs(x) = i 1 cts(x) = b 1 N nts(x) = c m N, cs(x) = i m cts(x) = b m N cts(x), M x N ctn(x) ctp(x) cts(x) + 1 nts(x) + 1 ctn(x) pn(ctp(x)) cts(x)+1 pn(ctp(x)) nts(x)+1 pn ntn : N N ntn(x) = qt(ctn(x), pn(ctp(x)) cts(x)+1 ) pn(ctp(x)) nts(x)+1 qt tr M tr M tr M (x, 0) = q 0 1 1 N (x+1) tr M (x, y + 1) = enc 3 (ns(tr M (x, y)), ntp(tr M (x, y)), ntn(tr M (x, y))) cs ctp ctn cts ns ntp nts ntn tr M 0 2

M x N t tr M (x, t ) = tr M (x, t) t t M(x) t tr M (x, t + 1) tr M (x, t) M(x) tr M M(x) t N tr M (x, t + 1) = tr M (x, t) term : N N term(x) = (µ t 0)[1 χ = (tr M (x, t + 1), tr M (x, t)) = 0] M(x) χ = f(x) dec(2, tr M (x, term(x))) f(x) M TM µ M M ntp(tr M (x, t)) ctp(tr M (x, t)) tr M (x, t + 1) tr M (x, t)

ΚΕΦΑΛΑΙΟ 3 G = (V, Σ, R, S) V Σ V V Σ S V Σ R V (V Σ)V V V (V Σ)V = {w V w 1, w 2 V a V Σ (w = w 1 aw 2 )}

G = (V, Σ, R, S) (u, v) R u v G = (V, Σ, R, S) G, G V V u G v w 1, w 2 V u v u = w 1 u w 2 v = w 1 v w 2 u G v n N u 1,..., u n V u G u 1 G G u n G v u G v u v G G = (V, Σ, R, S) G L(G) = {w Σ S G w} w L(G) w G G 1, G 2 L(G 1 ) = L(G 2 ) G = (V, Σ, R, S) V = {0, 1, S} Σ = {0, 1} R = {S 0S1, S ϵ} G S G 0S1 G 00S11 G 000S111 G 000111 S 0S1 S ϵ L(G) = {0 n 1 n {0, 1} n N} G = (V, Σ, R, S) V = {A, B, C, T a, T b, T c, S} Σ Σ = {a, b, c} R = {S ABCS, S T c, CA AC, BA AB, CB BC, CT c T c c, CT c T b c, BT b T b b, BT b T a b, AT a T a a, T a ϵ} u v u v w V S 0S1 ϵ

G S G ABCS G ABCABCS G ABCABCABCS G ABCABCABCT c G ABACBCABCT c G AABCBCABCT c G AABBCCABCT c G AABBCACBCT c G AABBACCBCT c G AABABCCBCT c G AAABBCCBCT c G AAABBCBCCT c G AAABBBCCCT c G AAABBBCCT c c G AAABBBCT c cc G AAABBBT b ccc G AAABBT b bccc G AAABT b bbccc G AAAT a bbbccc G AAT a abbbccc G AT a aabbbccc G T a aaabbbccc G aaabbbccc S ABCS S ABCS S ABCS S T c CA AC BA AB CB BC CA AC CA AC BA AB BA AB CB BC CB BC CT c T c c CT c T c c CT c T b c BT b T b b BT b T b b BT b T a b AT a T a a AT a T a a AT a T a a T a ϵ L(G) = {a n b n c n {a, b, c} n 1} L Σ G ( ) L RE M = (Q, Σ, Γ, δ, q 0, q, q )

M q M q q σ Q G M = (V, Σ, R, S) V = Γ Q {S} R = {S q } {bp qa q, p Q {q σ } a, b Γ (δ(q, a) = (p, b, ))} {p b qa q, p Q {q σ } a, b Γ (δ(q, a) = (p, b, )) Γ} {q σ qa q Q a Γ (δ(q, a) = (q σ,, )) Γ} { q q σ } { q 0 ϵ, ϵ} w Σ M w 1 qw 2 M(w) w 1, w 2 Γ q Q w 1 qw 2 (Γ Q) R δ M(w) q q 0 w M(w) q S G M w L(G M ) = L(M) = L ( ) G = (V, Σ, R, S) M G = (Q, Σ, Γ, δ, q 0 q, q ) V Γ L(G) M G G M G w Σ G S R u v R M G u u v M G G M M Q Σ C 1 M C 2 C 1, C 2 M(w) G M C 2 GM C 1 C 1, C 2 (Γ Q) M G

M G q M G w G M G (w) q L(M G ) = L(G) k G = (V, Σ, R, S) f : Σ Σ x, y Σ SxS G y f(x) = y f : Σ Σ f : {1} {1} f(x) = xx G = ({1, S}, {1}, S, R) R = {S1 11S, SS ϵ} f : Σ Σ G = (V, Σ, R, S) u v R u v G = (V, Σ, R, S) G = (V, Σ, R, S) u v R w 1 Aw 2 w 1 ww 2 w, w 1, w 2 V A V u ϵ L {0, 1} CS L = {a n b n c n {a, b, c} n 1} G = (V, Σ, R, S) V = {A, B, C, T b, T c, S} Σ Σ = {a, b, c} R = {S abc, S asbc, CB CT c, CT c T b T c, T b T c T b C, T b C BC, ab ab, bb bb, bc bc, cc cc} A w 1 w 2 u L {a, b, c} {0, 1} L = {(010) n (0110) n (01110) n {0, 1} n 1} T a ϵ

w q q q 0 q1 q2 M = (Q, Σ, Γ, δ, q 0, q, q ) Γ Σ M M = (Q, Σ, Γ, δ, q 0, q, q ) c N Γ δ w w Σ c w L {0, 1} CS REC M ϵ L Σ M L {ϵ}

L REC CS {0, 1} G = (V, Σ, R, S) V = {v 1, v 2,..., v n } Σ n N Σ = {0, 1} R = {x 1 y 1, x 2 y 2,..., x l y l } l N R V {, ; } R V {,;} = x 1 y 1 ; x 2 y 2 ; ; x l y l V {, ; } {0, 1} 0 {0,1} = 010 1 {0,1} = 0110 {0,1} = 01110 ; {0,1} = 011110 v i {0,1} = 01 10 i [n] (i+5) G R V {,;} {0,1} G G {0,1} L CS M G LBA G M G M G L {0, 1} {0, 1} L = { G {0, 1} G G L(G)} L REC M L G G L G L(G) G L

w M w = G G G M G LBA M G G M G G M G ( G ) q M G ( G ) q L M G LBA G = (V, Σ, R, S) R (V Σ) V L {0, 1} CF L = {0 n 1 n {0, 1} n N} G M = (Q, Σ, Γ, δ, q 0, q, q ) δ Q (Σ {ϵ}) (Γ {ϵ}) Q (Γ {ϵ}) (p, a, b, q, c) δ M q a Σ {ϵ} b Γ {ϵ} p c Γ {ϵ} b a = ϵ b = ϵ M c c = ϵ M b A u A u V M

w q q q 0 q1 q2 L {0, 1} {0, 1} L {0, 1} x R {0,1} L = L(x) R G = (V, Σ, R, S) R G = (V, Σ, R, S) V = {0, 1, A, S} Σ = {0, 1} R = {S 0S, A 1A, S A, A ϵ} L(G) = {0 n 1 m {0, 1} n, m N} M = (Q, Σ, Γ, δ, q 0, q, q ) δ Q (Σ {ϵ}) Q L {0, 1} A Bu A a A ϵ A, B a u Σ

RE CS CF R 2 {0,1} RE REC CS CF R L R G L(G) = L M L(M) = L R CF CS RE R CF CF CS CS RE

ΚΕΦΑΛΑΙΟ 4 2 gcd

10 m P (x 1,..., x n ) = a i x k 1 i 1 xkn i n = 0 i=1 a i Z k 1i,..., k ni N x 1,..., x n i [m] P (x 1,..., x n ) = 0 n (z 1,..., z n ) Z n P (z 1,..., z n ) = 0 3x 2 2xy y 2 z 7 = 0 x = 1, y = 2, z = 2 P (x 1, x 2,..., x n ) = m i=1 a ix k 1 i 1 xk 2 i 2 xkn i n P (x 2,..., x n ) = m i=1 a is k 1 i x k 2 i 2 xk n i n = 0 = 0 s Z P s Σ 10

P M s P P s P s M D M D REC D = { P Σ P } S N P (x 1,..., x n+1 ) s S z 1,..., z n Z (P (s, z 1,..., z n ) = 0) D REC D REC M D L RE REC L RE P (x 1,..., x n+1 ) n N L = {s N z 1,..., z n Z (P (s, z 1,..., z n ) = 0)} M L L REC HP = { M, w Σ M(w) } RE REC HP RE HP REC H D REC RE N HP HP = {w Σ ( M w Σ (w = M, w )) M(w) }

M, w M(w) M(w) HP D M M, M M, M H M, M HP REC D( D ) H( D, D ) q D, D HP D( D ) ϕ Mk k M 1 M 2 M 3 M k M 1 ϕ M1 ( M 1 ) ϕ M1 ( M 2 ) ϕ M1 ( M 3 ) ϕ M1 ( M k ) M 2 ϕ M2 ( M 1 ) ϕ M2 ( M 2 ) ϕ M2 ( M 3 ) ϕ M2 ( M k ) M 3 ϕ M3 ( M 1 ) ϕ M3 ( M 2 ) ϕ M3 ( M 3 ) ϕ M3 ( M k ) M k ϕ Mk ( M 1 ) ϕ Mk ( M 2 ) ϕ Mk ( M 3 ) ϕ Mk ( M k ) D 1, ϕ Mk ( M k ) = ϕ D ( M k ) =, ϕ Mk ( M k ) ϕ Gödel(D) ( M Gödel(D) ) HP RE HP RE HP, HP RE HP REC HP HP

2 {0,1} HP RE HP REC REC RE Σ A x y ϕ B ϕ(x) ϕ(y) Σ A B ϕ A B A, B Σ A B A m B ϕ : Σ Σ ϕ w Σ (w A ϕ(w) B) ϕ A B L = { M, w Σ M(w) q } ϕ : Σ Σ q M, w ϕ q 1 M /, q M, w q q 2 /, M M q q 1 q 2 ϕ(x) = x

M w M ϕ ϕ(w) M B x Σ M w Σ x = M, w ϕ HP L ϕ M, w HP M(w) M(w) q M(w) q M (w) q1 M (w) q2 M (w) q ϕ( M, w ) L M, w HP M(w) M (w) ϕ( M, w ) L HP m L A, B Σ A m B B REC B RE A REC A RE ϕ A B M ϕ M B B A w A ϕ(w) B M B (ϕ(w)) q M(w) q w A ϕ(w) B M B (ϕ(w)) q M(w) q A REC A RE A, B Σ A m B A REC A RE B REC B RE HP m L HP REC L REC A, B Σ A m B B A B A ϕ ϕ x HP M w Σ (x = M, w M, w HP ) x HP ( M w Σ (x = M, w )) ( M w Σ (x = M, w M, w HP )) REC RE

L = { M, w, q Σ M(w) q} ϕ : Σ Σ ϕ( M, w ) = M, w, q L L ϕ M, w L M(w) q M, w, q L M, w L M(w) M(w) q M(w) q M, w, q L L REC L REC L ϵ = { M Σ M(ϵ) } ϕ : Σ Σ M, w ϕ x x = ϵ w w M /, M w HP L ϵ ϕ M, w HP M(w) M w (ϵ) M w L ϵ M, w HP M(w) M w (ϵ) M w L ϵ HP REC L ϵ REC L = { M Σ L(M) = ℵ 0 } ϕ : Σ Σ M ϕ ϵ M M /, q L ϵ L ϕ M L ϵ M(ϵ) w Σ (M (w) q ) L(M ) = Σ L(M ) = ℵ 0 M L M L ϵ M(ϵ) w Σ (M (w) q ) L(M ) = L(M ) = 0 M L L ϵ REC L REC

ϕ L ϵ L Σ = { M Σ L(M) = Σ } L Σ REC L = { M, w Σ M(w) 1 } ϕ : Σ Σ q M, w ϕ q Ṁ /, M, ẇ q q Ṁ ẇ M, w Σ L L ϕ M, w L M(w) q M (ẇ) q M (ẇ) 1 M, ẇ L M, w L M(w) q M (ẇ) q M (ẇ) 1 M, ẇ L L REC L REC L = { M 1, M 2 Σ L(M 1 ) = L(M 2 )} ϕ : Σ Σ M, w ϕ w w M 1 M, q q M 2 L L ϕ M, w L M(w) q x Σ (M 1 (x) q ) L(M 1 ) = Σ = L(M 2 ) M 1, M 2 L M, w L M(w) q x Σ (M 1 (x) q ) L(M 1 ) = L(M 2 ) M 1, M 2 L L REC L REC L = { M Σ L(M) = } ϕ : Σ Σ

M, w ϕ x x = w M M w x Σ ϕ ϕ(x) = M M L L ϕ M, w L M(w) q x Σ (M w (x) q ) L(M w ) = M w L M, w L M(w) q M w (w) q L(M w ) = {w} M w L L RE REC L RE L RE m 2 {0,1} A, B, C {0, 1} A m B B m C ϕ 1 A B ϕ 2 B C ϕ 2 ϕ 1 A C ϕ 2 ϕ 1 w A ϕ 1 (w) B ϕ 2 (ϕ 1 (w)) C A m C A, B Σ A m B A m B ϕ A B ϕ A B w A w A ϕ(w) B ϕ(w) B x = M M x L x L x L M, w ϕ(x) = M L {0, 1}

ΚΕΦΑΛΑΙΟ 5 L REC = { M Σ L(M) REC} ϕ : Σ Σ M, x = M, w ϕ(x) = M, M M L L REC ϕ M, w L M(w) q x Σ (M (x) ) L(M ) = REC M L REC M, w L M(w) q L(M ) = { D, y Σ D(y) q } = L REC M L REC L RE L REC RE L REC L REC ( REC) ϕ ϕ( M, w ) = L ( REC), M(w) q, M(w) q

x w x 2 M w M x x x = D, y D D, y D(y) D(y) q M P RE L RE P L P P = {L RE L = ℵ 0 } P N = {L RE L = N} P ϵ = {L RE ϵ L} REC P RE L P = { M Σ L(M) P} P RE L P REC P = P = RE ( ) P RE P {, RE} P L P M L L m L P ϕ : Σ Σ ϕ( M, w ) = M M ϕ M, w L M(w) q L(M ) = L(M L ) = L P M L P M, w L M(w) q L(M ) = P M L P L REC L P REC P P {, RE}

x w x 2 M w M x x M L M ( ) P = L P = { M Σ L(M) } = REC P = RE L P = { M Σ L(M) RE} = G REC L N = { M Σ L(M) = N} REC P N P N {ϵ} P N P N RE Σ P N L = { M Σ L(M) = L(M) = {(01) n n N} L(M) = Σ } REC P = {, {(01) n n N}, Σ } L = L P P {, RE} P RE L P RE L P L RE (L L L P) L P ( L = ℵ 0 L L ( L N L P)) F P = {w 1 w 2 w n (Σ { }) {w 1, w 2,..., w n } {L P L N}} RE Σ

M x ND x M 1 x w x 2 w M x x M 2 M L P RE ( ) L P RE L 1 P L 2 RE L 1 L 2 L 2 P M 1, M 2 L 1, L 2 L m L P ϕ : Σ Σ M, x = M, w ϕ(x) = M 2, M ϕ M, w L M(w) q L(M ) = L(M 1 ) = L 1 P M L P M, w L M(w) q L(M ) = L(M 2 ) = L 2 P M L P L RE L P RE L P RE L P L = ℵ 0 P M L L L m L P ϕ : Σ Σ M, x = M, w ϕ(x) = M, M M ϕ L P RE x L ϕ(x) = M 2 M 2 L P L 2 P L P RE

x M L x x x M M, w M, w M(w) x M(w) q M(w) q M L P RE M, w L M(w) q L(M ) = L(M L ) = L P M L P M, w L t N (M(w) t q ) L(M ) = {x L x t} L(M ) L L(M ) N L(M ) P M L P L RE L P RE L P RE E L P E F P (Σ { }) w = w 1 w 2 w n w i Σ i [n] M w w w = w 1 w = w 2 w = w n Σ E M 1 w (Σ { }) (Σ { }) w ; M 2 (Σ {, ; }) w = w 1 ; w 2 ;... ; w n w i (Σ { }) i [n] M w1 ; M w2 ;... ; M wn (Σ {; }) M np (w, i) Σ N E w = w 1 w 2 w n F P M w w {w 1, w 2,..., w n } P i 0 N E pop( M w ) E (next(w), i 0 ) pop(w) E pop w = w 1 w 2 w n E pop M w {w 1, w 2,..., w n } P w F P E F P x L ϕ(x) = M L(M ) = L M L P w i i [n]

(w, i) (ϵ, 0) (w, i) np(w, i) E M 1 w 1 ; w 2 ;... ; w n M 2 Mw1 ; M w2 ;... ; M wn E i pop M w1, M w2,..., M wn M wij, j [k] M np (w, i) pop(w ij ) j [k] E L P RE ( ) E F P D L P L P L(D) M L P L(M) P L L(M) ( L N L P), L = {w 1,..., w n } E pop(w 1 w n ) L L(M) t N s [n] (M(w s ) t q ), D( M ) q M L(D) L(D) L P M L(D) (i, j) N s [k] (M(w s ) j q ), w 1 w k pop E i L = {w 1,..., w n } L(M) E F P L P L(M) P M L P L REC P = REC P L L P L L REC RE M np D

(i, j) (0, 0) D (i, j) np(i, j) M np M E i w 11 w 12 w 1k1 w 21 w 22 w 2k2 w l1 w l2 w lkl (i, j) r [l] s [k r ] (M(w rs ) j q ) D w M E w 1 w 2 w n w = w 1 w 2 w n M L P = {L RE L L } L P L L L RE (L L L L ) P L L w L L {w} L {w} L {w} P P E F P M L w L {w} P E pop(w) L P RE E i 10 i + 1 i i i + 1 pop S (Σ { }) 10 S i i i w 1 w 2 w n E

P = {L RE L 10} L P L 10 L RE (L L L L 10) P L 10 w 1,..., w 10 L L = {w 1,..., w 10 } L L P P E F P P L P RE L 10 = { M L(M) 10}

ΚΕΦΑΛΑΙΟ 6 A B B A A M M

M M M 1, M 2 M 1 M 2 w M 1 M 2 M 1 M 2 M 1 w M 2 M 1 (w) conc M 1 M 2 conc M 1 M 2 M 1 M 2 M 1 M 2 agm agm P w x P w w

w Q w Q P w w P w A M Q P M M conc A P M M P A A x P A A P A A A P A A P A A agm t : Σ Σ Σ R x Σ ϕ R (x) = t( R, x) A P w x w, x T t B M, x M x M A B P M M x P M M, x P M M, x P BT BT x, y x y

M, w H T P BT BT x P BT BT, x B P BT BT, x T t( P BT BT, x) x Σ t( P AT AT, x) R L REC H L 0, M(w) q t( M, w) = 1, M(w) q R w Σ L REC 0, R(w) q 0, ϕ R (w) = 1 ϕ R (w) = t( R, w) = = 1, R(w) q 1, ϕ R (w) = 0 L RE R L R M L t : Σ Σ Σ t(x, y) = ϕ M (y) t R y Σ ϕ R (y) = t( R, y) = ϕ M (y) REC R R t

R R w R, w R, w H H L R R w H M 2 M 1 R w L(R) H( R, w ) q w L(R) ( ) P RE P {, RE} L 1 P L 2 RE P M 1, M 2 L 1 L 2 H L P R L P H( R ) q L(R) = L(M 2 ) = L 2 P R L P R L P H( R ) q L(R) = L(M 1 ) = L 1 P R L P M i N i < Gödel(M) L(M i ) L(M) M i Gödel(M i ) = i L = { M Σ M } L RE

R E M R Gödel(M) > Gödel(R); M w M(w) L L = { M 1, M 2..., M n } n N L {L(M 1 ), L(M 2 ),..., L(M n )} L M 1, M 2,..., M n L RE L RE E R E M Gödel(M) > Gödel(R) L L(R) = L(M) M L } Gödel(M) Gödel(R) t : G G t : N N t (Gödel(M)) = Gödel(N) t( M ) = N t : N N i 0 N t i 0 t(i 0 ) T t M i i R i 0 w Σ ϕ R (w) = ϕ Mi0 (w) = ϕ Mt(i0 ) (w) i 0 t L = n i=1 L(M i) n > 1 L = L 1 {w} w Σ L(M 1 ) n = 1

R R Gödel(R) T i 0 t(i 0 ) M t(i0 ) M t(i0 ) w M t(i0 )(w) ϕ R (w) t i 0 N t(i 0 ) t(i 0 + 1) M i i i 0 N ϕ Mt(i0 ) = ϕ M t(i0 +1) t σ : N N σ(x) = t(t 1 (x) + 1) σ j 0 N ϕ Mj0 = ϕ Mσ(j0 ) i 0 = t 1 (j 0 ) ϕ Mt(i0 ) = ϕ M t(i0 +1) ϕ Mσ(j0 ) = ϕ M t(t 1 (j0 )+1) = ϕ M t(i0 +1) ϕ Mj0 = ϕ Mt(t 1 (j0 )) = ϕ M t(i0 ) Smn g : N m+n N x 1,..., x m N f : N n N y 1,..., y n N (g(x 1,..., x n, y 1,..., y m ) = f(y 1,..., y n )) i g S m n : N m+1 N S m n (i, x 1,..., x m ) = f M i g M f f S m n (i, x 1,..., x m ) = Gödel(M f ) M i x 1,..., x m M f

x 1,..., x m M f y 1,..., y n M i M f 1 φ Γ 1 P P φ P φ P ϕ M P φ(x, y) x N ϕ M (x) = y P φ(x, y) ϕ M (x) y P φ(x, y) M i i f i ϕ Mi φ i f i g : N 2 N 1, P k φ i (j, k) g(i, j) =, g M M χp roof P roof M TM µ M µ φ Gödel(M) = l f j N g(i, j) = f(j) f = ϕ t t(i) = MS 1 S1 (l,i) 1 (l, i) 1 i 0 N j N g(i 0, j) = ϕ MS 1 1 (l,i 0 ) (j) = ϕ M t(i0 ) (j) = ϕ M i0 (j) k φ i0 (j, k) P k φ i0 (j, k) g(i 0, j) = 1 ϕ Mi0 (j) = 1 n N ϕ Mi0 (j) n ϕ Mi0 (j) N P k φ i0 (j, k) N P N k φ i0 (j, k) n N N φ i0 (j, n) ϕ Mi0 (j) = n φ i0 f i0 P φ(j, n) N φ(j, n) n N ϕ Mi0 (j) n

i, j i j M i M TM µ M µ φ M i f i φ i k φ i (j, k) M k φ i (j, k) y 0 y + 1 k φ i (j, k), y k φ i (j, k), y M χproof 1 1 0 y y + 1 M g P k φ i0 (j, k) g(i 0, j) = ϕ Mi0 (j) = n N ϕ Mi0 (j) = n ϕ Mi0 (j) N n N N φ(j, n) ϕ Mi0 (j) n P φ(j, n) N φ(j, n) n N ϕ Mi0 (j) = n φ φ, φ N Γ 1

ΚΕΦΑΛΑΙΟ 7 HP {0, 1} RE B A B A L Σ w Σ 1 s L 0 w L

1 q 0 q1 q q q 2 q q q? 2 1 0 0 1 1 1 0 0 0 0 3 L Σ w L 1 w L 0 w / L L Σ L Σ ϵ Σ 1 0 L Σ L L M L = (Q, Σ, Γ, δ, q 0, q, q ) 3 L Q q? q q M L q? 2 i 3 i 1 q 0 q A, B Σ M A B w B M A (w) q A, B Σ M A B B A A, B Σ M A B w B M A (w) q

M HP M, w HP 1 0 M(w) M(w) q M(w) q M HP L w B M A (w) q A, B Σ M A B B A A Σ f : Σ Σ M A f w dom(f)( q 0 w M A qf(w)) q {q, q } A Σ f : Σ Σ M A f f A L HP M HP δ {0, 1} L Σ L L M L w M L (w) L M L (w) q? HP 0

L Σ M L w Σ L M L, w M L (w) q M L (w) q {0, 1} A {0, 1} A RE A = {L {0, 1} A L} REC A = {L {0, 1} A L} A {0, 1} A REC RE A = RE REC A = REC M A A L RE A M A L A M A M A L M L L L RE RE A RE L RE M L M A L M L q? M A L L L REA RE RE A A {0, 1} RE RE A RE REC A HP RE REC HP L RE M L M HP L L REC HP L {0, 1} REC HP M L ML A ML A w M L M A w 1 0 ML A A M A

M L M HP M L w M L, w M L, w HP 1 0 M L (w) M L (w) q M L (w) q M HP HP 2 = { M HP, w {0, 1} M HP (w) } HP 2 REC HP HP 2 REC HP H HP D HP D HP M HP M HP, M HP M HP, M HP H HP M HP, M HP D HP ( D HP ) H HP ( D HP, D HP ) q D HP, D HP HP 2 D HP ( D HP ) HP 2 REC HP HP A {0, 1} L REC A L REC A C co C = {L {0, 1} L C} A {0, 1} RE A co RE A = REC A

t + 1 t 0 t M A L M A L M A w A M A L (w) t t t + 1 (w, t) M A L M A L t A M A L (w) t M A L L RE A co RE A L REC A L RE A L REC A L RE A L RE A co RE A L RE A co RE A ML A M A L L L M A L L REC A A, B, C {0, 1} A RE B B REC C A RE C A REC B B REC C A REC C M B A M C B B M B M C C A A, B, C {0, 1} A m B B REC C B RE C A REC C A RE C C 2 {0,1} C M B M B w M C w

RE C = A C RE A REC C = A C REC A REC REC = REC RE REC = RE n 1 Σ 0 1 = RE 0 1 = REC Σ 0 n+1 = RE Σ0 n 0 n+1 = REC Σ0 n Π 0 n = co Σ 0 n {0, 1} n 2 Σ 0 n+1 Σ0 n Σ 0 n 1 Σ0 n n 1 HP 1 = HP HP n+1 = { M HPn, w {0, 1} M HPn (w) } HP n HP n = {0w {0, 1} w HP n } {1w {0, 1} w HP n } n 1 HP n REC HP n+1 n n = 1 HP 1 REC HP 2 HP REC HP HP RE HP n REC HP n+1 HP n+1 m HP n+2 HP n+2 REC HP n+2 HP n+1 REC HP n+2 M HP n+1 HP n HP n ϕ : {0, 1} {0, 1}

M HPn 1 M HPn 1, w M HP n 1 (w) HP n 1 M HP n 1 (w) M HP n 1 M HPn M HPn+1 x M HPn, w ϕ x M HPn HP n+1 M HP n (x) M HP n+1 HP n, w ϕ M HP n, w HP n+1 M HP n (w) M HP n+1 (w) M HP n+1, w HP n+2 ϕ HP n+1 HP n+2 n 1 HP n+1 REC HPn n 1 HP n Σ 0 n n 2 M HP n 1 HP n n = 1 HP 1 Σ 0 1 HP RE n 1 Σ 0 n REC HP n n n = 1 Σ 0 1 RECHP 1 Σ 0 n REC HP n RE REC HP L Σ 0 n+1 A Σ0 n L RE A A REC HP n L RE HP n M HPn L M HP n+1 L L REC HPn+1 HP 2 REC HP 1 HP n+1 HP n HP n REC HP n+1 HP n HP n HP n+1

w M HPn L M HPn L M HPn L, w M HPn L, w HP n+1 1 0 HP n+1 M HPn+1 M HP n L (w) q M HP n L (w) M HP n L (w) q M HP n+1 x 1 x = 0w HP n 0 w M HPn w HP n 0 1 M HPn HP n HP n L {0, 1} n 1 L Σ 0 n L 0 n L RE HP n 1 L REC HP n 1 L Σ 0 n A Σ 0 n 1 L REA Σ 0 n 1 RECHP n 1 A REC HP n 1 L REC HP n 1 n 1 Σ 0 n Σ 0 n+1 Π0 n Π 0 n+1 Σ 0 n Π 0 n 0 n+1 Σ 0 n Π 0 n = 0 n 0 n Σ 0 n 0 n Π 0 n HP n+1 Σ 0 n+1 Σ0 n REC HP n HP n+1 REC HPn HP n+1 Σ 0 n HP n+1 Π 0 n+1 HP n+1 Σ 0 n+1 HP n+1 Π 0 n HP n+1 Σ 0 n HP n HP n REC HP n M HP n HP n HP n 0 n+1

w 0 w 0w MHP A n M A M A HP n HP n HP n HP n Σ 0 n A Σ 0 n 1 M A M A HP n HP n HP n Σ 0 n 1 HP n HP n Π 0 n HP n HP n Σ 0 n HP n HP n = {0w {0, 1} w HP n } {1w {0, 1} w HP n } = {0w {0, 1} w HP n } {1w {0, 1} w HP n } = ({x {0, 1} w {0, 1} (x = 1w)} {0w {0, 1} w HP n }) {1w {0, 1} w HP n } = {1w {0, 1} w HP n } {0w {0, 1} w HP n } HP n HP n 0 1 0 1 HP n HP n Σ 0 n L 0 n A Σ 0 n 1 L RECA L REC A L REC A L RE A L Σ 0 n L REC A A L RE A L Σ 0 L Σ 0 n Π 0 n n L Σ 0 n Π 0 n L, L Σ 0 n L, L RE HP n 1 L REC HP n 1 HP n 1 Σ 0 n 1 L 0 n HP n 0 n HP n REC HP n 1 A, B {0, 1} A B A T B A REC B L T HP

2 {0,1} Σ 0 3 Π 0 3 HP 3 0 HP 3 3 HP 2 HP 2 Σ 0 2 Π 0 2 HP 2 0 HP 2 2 HP 1 HP 1 Σ 0 1 = RE Π 0 1 = co RE 0 1 = REC HP 1 HP 1 L HP L co RE L RE L = {w {0, 1} ( M (w = M )) (L(M) )} M L L Π 0 1 L 0 2 L REC HP L T HP w w = M M L M (x, t) (ϵ, 0) (x, t) M(y) y x t y x (M(y) t q ) np(x, t) y x (M(y) t q ) M (x, t) np L M np

w L M L 0 1 M L L w M ϕ ϕ(w) B M B 1 0 M B L {0, 1} L M L HP HP HP RE A, B {0, 1} A m B A T B ϕ A B M ϕ M B A w A ϕ(w) B M B (w) q w A ϕ(w) B M B (w) q A T B T C 2 {0,1} B {0, 1} B C { m, T } A C A B B C B C n 1 HP n Σ 0 n T Σ 0 n T HP n Π 0 n T HP n Π 0 n T n 1 HP n HP n 0 n+1 T HP n HP n 0 n+1 0 n+1 T L 0 n+1 L RECHPn L T HP n

w 0 w 0w HP n HP n M HPn HP n 1 0 M HPn HP n M HPn 1 w M HPn 1 L w M HP n 1 M HP n HP n HP n HP n T HP n HP n T L T HP n HP n n 1 HP n Σ 0 n m HP n Π 0 n m HP n Σ 0 n HP n Σ 0 n m L Σ 0 n L RE HP n 1 M HP n 1 L ϕ : {0, 1} {0, 1} ϕ(w) = M HP n 1, w M HP n 1 ϕ w L M HP n 1 L (w) q M HP n 1 (w) M HP n 1, w HP n w L M HP n 1 L (w) q M HP n 1 (w) M HP n 1, w HP n L m HP n HP n Σ 0 n HP n Π 0 n HP n Π 0 n m L Π 0 n L Σ 0 n L m HP n L m HP n n N n n N n R (x 1,..., x n ) R R(x 1,..., x n )

n R L R = { x 1,..., x n N R(x 1,..., x n )} L N L Σ 0 n (n + 1) R L = {x N y 1 y 2 y n R(x, y 1,..., y n )} =, n, L Π 0 n (n + 1) R L = {x N y 1 y 2 y n R(x, y 1,..., y n )} =, n, n n 1 L Π 0 1 L = { M N L(M) = } = { M N y t (M(y) t q )} = {x N y t (x = M M(y) t q )} pair : N 2 N pair(i, j) = i+j+1 2 + i L = {x N z (x = M pair 1 (z) = (y, t) M(y) t q )} L = { x, z N x = M z = pair 1 (y, t) M(y) t q } M pair 1 pair 1 L Π 0 1 L N Σ 0 2 Σ 0 2 L N = { M N L(M) = N} = { M N n y ( y > n y L(M))} = { M N n y t ( y n M(y) t q )} = {x N n z (x = M pair 1 (z) = (y, t) ( y n M(y) t q ))}

x, z z x x = M M M L M pair 1 (y, t) M(y) t M(y) t q x, n, z z x n x = M M M L M pair 1 (y, t) y n (y, t) M(y) t M(y) t q M L L = { x, n, z N x = M pair 1 (z) = (y, t) ( y n M(y) t q )} L N Σ 0 2 L Σ 0 2 R L = {x N y z R(x, y, z)} M R L R ϕ : N N ϕ(x) = M x M x ϕ x L y 0 z (M R (x, y 0, z) q ) w y 0 M x (w) M R (x, y 0, z) q z L(M x ) {w N w y 0 } L(M x ) N M x L N x L y z (M R (x, y, z) q ) w ( y w z (M R (x, y, z) q )) w (M x (w) q ) L(M x ) = N L(M x ) = ℵ 0 M x L N n 1 T n = { φ N N φ = y 1 y 2 y n ψ(y 1,..., y n ) N φ} ψ Γ 1 =, n φ N, φ pair 1 pair

x M R M x x M R w y w z (M R (x, y, z) q ) M x n 1 T n Σ 0 n m L Σ 0 n (n + 1) R, n L = {x N y 1 y 2 y n R(x, y 1,..., y n )} =, R P φ R (x, y 1,..., y n ) ϕ : N N ϕ(x) = φ x N ϕ φ x = y 1 y 2 y n φ R (x, y 1,..., y n ) x L y 1 y 2 y n R(x, y 1,..., y n ) N y 1 y 2 y n φ R (x, y 1,..., y n ) N φ x φ x N T n L m T n T ruth = { φ N N φ N φ} T ruth T ruth n N T ruth Σ 0 n T n+1 m T ruth T n+1 Σ 0 n+1 m HP n+1 m T n+1 m HP n+1 m T ruth HP n+1 T T ruth HP n+1 REC T ruth REC Σ0 n = 0 n φ y 1 y 2 y n+1 ψ(y 1,..., y n+1) x x

ΠΑΡΑΡΤΗΜΑ A Γ 1 M(Γ 1 ) = {x 0, x 1,...},,, (, ) n 0 {P i i I} n n 0 {f i i I} n f ni 0 Γ 1 Γ 1 ft 1,..., t n t 1,..., t n f n Γ 1 Γ 1 O(Γ 1 ) ft 1,..., t n f(t 1,..., t n )

Γ 1 Γ 1 t 1 t 2 t 1, t 2 O(Γ 1 ) Rt 1,..., t n t 1,..., t n O(Γ 1 ) R n Γ 1 ( φ), (φ ψ), ( xφ) φ, ψ Γ 1 x Γ 1 Γ 1 T (Γ 1 ) ( xφ), (φ ψ), (φ ψ) (φ ψ) ( ( x( φ))), ( (φ ( ψ))), (( φ) ψ) ((φ ψ) (ψ φ)) φ T (Γ 1 ) x x φ φ x φ φ ( ψ) x ψ φ (χ ψ) x χ ψ φ ( yψ) x ψ y x φ φ Γ 1 φ(x 1,..., x n ) x 1,..., x n φ Γ 1 A Γ 1 A n P n P A A n n f A : A n A c c A A Γ 1 A v : M(Γ 1 ) A v v : O(Γ 1 ) A v(c) = c A c Γ 1 v(f(t 1,..., t n )) = f A ( v(t 1 ),..., v(t n )) f n Γ 1 t 1,..., t n O(Γ 1 ) t 1t 2 t 1 t 2 Rt 1,..., t n R(t 1,..., t n )

Γ 1 A v x v(x a)(y) a A a (x a)(y) = v(y), y = x, Γ 1 A v φ T (Γ 1 ) A φ v A φ[v] φ t 1 t 2 t 1, t 2 O(Γ 1 ) v(t 1 ) = v(t 2 ) φ R(t 1,..., t n ) ( v(t 1 ),..., v(t n )) R A φ ( χ) A χ[v] φ (χ ψ) A χ[v] A ψ[v] φ ( xχ) a A A χ[v(x a)] T T (Γ 1 ) φ A v A φ[v] A, v φ A v T T T A v T φ T φ T φ A A T (Γ 1 ) K κ K κ 2 T (Γ 1) T (Γ 1 ) (B, φ) κ φ κ B A A = M M A = (A, K) T T (Γ 1 ) φ φ T A T A φ τ 1,..., τ n τ n = φ τ i φ

τ i A T τ i κ K S {t 1,..., t i 1 } x t φ Γ 1 φ x t φ x t x t φ x t φ t φ x 1 x n φ x i n 0 A 1 = (A 1, K 1 ) φ (ψ φ) (φ (ψ χ)) ((φ ψ) (φ χ)) ( φ ψ) (( φ ψ) φ) xφ φ x t x t φ x(φ ψ) ( xφ xψ) φ xφ x φ x x x y (φ φ ) φ φ φ x y K 1 φ φ ψ ψ T A1 φ T φ φ φ T φ T φ T φ T φ T φ T φ

Γ 1 Γ 1 N = N N = S N = + N = N = 0 N n n N n = n P Γ 1 x 1 (x 1 ) x 1 x 2 (x 1 = x 2 x 1 = x 2 ) x 1 (x 1 = x 1 ) x 1 x 2 (x 1 x 2 = (x 1 x 2 ) ) x 1 (x 1 = ) x 1 x 2 (x 1 x 2 = x 1 x 2 x 2 ) x 1 x n (φ(x 1,..., x n, ) x 0 (φ(x 1,..., x n, x 0 ) φ(x 1,..., x n, x 0 )) x 0 φ(x 1,..., x n, x 0 )) T T (Γ 1 ) R Nn n 1 R T φ(x 1,..., x n ) m 1,..., m n N (m 1,..., m n ) R T φ(m 1,..., m n ) (m 1,..., m n ) R T φ(m 1,..., m n ) P f : N n N T φ(x 1,..., x n+1 ) m 1,..., m n+1 N f(m 1,..., m n) = m n+1 T φ(m 1,..., m n+1) f(m 1,..., m n) m n+1 T φ(m 1,..., m n+1)

Γ 1 Γ 1 x i N = 3 i+1, i = 0, 1,... N = 5 N = 7 N = 11 ( N = 13 ) N = 17 N = 19 N = 23 N = 27 N = 29 N = 31 φ = a 1 a n T (Γ 1 ) a 1,..., a n {,,, (, ),,,,, } M(Γ 1 ) φ φ N = enc n ( a 1 N,..., a n N ) enc n φ N φ φ 1,..., φ n Γ 1 φ 1,..., φ n N = enc n ( φ 1 N,..., φ n N ) P M µ φ f φ M µ φ P roof N 2 (x, y) P roof y P x P roof n R P n x Γ 1 x x

R L R REC L R P φ R (y 1,..., y n, x) φ R (y 1,..., y n, 1) R

Σ { } REC RE Σ { } Σ { } q 3 {0, 1} M space f 2 f 1 f 1 χ L L REC L χ L {1 n n N} {(10) n n N} L E L REC M L E k k

N D M np {0, 1} N E L M L M L 1 L 2 M L 1 L 2 M L 1 L 2 M L L L M G LBA M D REC HP REC RE ϕ A B M M M L P RE M L P RE E L P RE D M L E H L t M f M g M HP L M HP M A L L RE A co RE A M HP n 1 M HP n+1

M HP n HP n HP n MHP A n HP n L M np M L L M B M HPn HP n M HP n 1 M L M x