平成 28 年度 ( 第 38 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 ~8 28 月年 48 日開催月 1 日 semantics FB 1 x, y, z,... FB 1. FB (Boolean) Functional

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1 1 1.1 semantics F 1 x, y, z,... F 1. F (oolean) Functional

2 2. T F F 3. P F (not P ) F 4. P 1 P 2 F (P 1 and P 2 ) F 5. x P 1 P 2 F (let x be P 1 in P 2 ) F 6. F syntax F (let x be (T and y) in ((not x) and F)) P fv(p ) fv(x) = {x} x fv(t) = fv(f) = fv((not P )) = fv(p ), fv((p 1 and P 2 )) = fv(p 1 ) fv(p 2 ) fv((let x be P 1 in P 2 )) = fv(p 1 ) (fv(p 2 ) {x}) fv(p ) = P F (P 1 and P 2 ) P 1 P 2 (let x be P 1 in P 2 ) P 1 P 2 x P operational semantics F P V {T, F} P V 1. T T F F 2. P T (not P ) F P F (not P ) T 3. P 1 T P 2 T (P 1 and P 2 ) T P 1 T P 2 F (P 1 and P 2 ) F P 1 F (P 1 and P 2 ) F 4. P 1 V 1 P 2 [x := V 1 ] V 2 (let x be P 1 in P 2 ) V 2 P [x := V ] P x V 2

3 Z X Z X Y Z T T F F P 1 T P 2 F ((P 1 and P 2 ) F Z P T (not P ) F X Z P 1 F ((P 1 and P 2 ) F X Y Z P F (not P ) T P 1 T P 2 T ((P 1 and P 2 ) T P 1 V 1 P 2 [x := V 1 ] V 2 (let x be P 1 in P 2 ) V 2 (let x be (not F) in (x and F)) F F F T T F F (not F) T (T and F) F (let x be (not F) in (x and F)) F (let x be (not F) in (x and F)) (let x be T in (x and F)) (T and F) F 1.3 denotational semantics F P fv(p ) Γ Γ [[P ]] : {0, 1} Γ {0, 1} {0, 1} Γ ρ : Γ {0, 1} [[P ]](ρ) [[P ]] ρ [[x]] ρ = ρ(x) [[T]] ρ = 1 [[F]] ρ = 0 [[(not P )]] ρ = 1 [[P ] ρ [[(P 1 and P 2 )]] ρ = [[P 1 ] ρ [[P 2 ]] ρ [[(let x be P 1 in P 2 )]] ρ = [[P 2 ] ρ[x [[P1 ] ρ] ρ[x v] x ρ x v P fv(p ) = [[P ] ρ Γ ρ Γ = ρ [[P ]] = [[P ]] ρ {0, 1} (let x be (not F) in (x and F)) [[(let x be (not F) in (x and F))]] = [[(x and F)] [x [(not F)]]] = [[(x and F)] [x (1 [[F])] = [[(x and F)] [x (1 0)] = [[(x and F)] [x 1] = [[x]] [x 1] [[F]] [x 1] = 1 0 = 0 3

4 P V [[P ]] = [[V ]] F ( ) P P [[P ]] = [[P ]] P, P P P P P V P V P V F F 1.4 F (let x be (not F) in (x and F)) (let x be T in (x and F)) (T and F) F [[(let x be (not F) in (x and F))]] = [[(let x be T in (x and F))]] = [[(T and F)] = [[F]] 4

5 1.5 F F F P [[P ]] : {0, 1} Γ {0, 1} (fv(p ) Γ) {0, 1} X [[T]] [[F]] fv(p ) Γ P [[P ]] : X n X n = Γ Γ [[P ]] X X n X 5

6 2.2 C X 1,, X n P π i : P X i (i = 1,, n) Z f i : Z X i (i = 1,, n) f i = π i h (i = 1,, n) h : Z P X 1,, X n C X 1,, X n ( ) X 1 X n π i : X 1 X n X i (i = 1,, n) Z f i : Z X i (i = 1,, n) f i = π i h (i = 1,, n) h f 1,, f n : Z X 1 X n π i f 1,, f n = f i π 1 g,, π n g = g ( g : Z X 1 X n ) n = 0 1 Z Z 1 : Z 1 f i : X i Y i (i = 1, 2,..., n) f 1... f n : X 1... X n Y 1... Y n f 1... f n = f 1 π 1,..., f n π n C (X 1,..., X n ) X 1... X n : (Ob(C)) n Ob(C) C n C 2.3 F F C X C tt, ff : 1 X, not : X X and : X 2 X not tt = ff not ff = tt and tt, tt = tt and tt, ff = ff and ff, tt = ff and ff, ff = ff F P x 1,..., x n fv(p ) x 1,..., x n [[x 1,..., x n P ]] : X n X [[x 1,..., x n x i ] = π i [[x 1,..., x n T]] = tt [[x 1,..., x n F]] = ff [[x 1,..., x n (not P )]] = not [[x 1,..., x n P ]] [[x 1,..., x n (P 1 and P 2 )] = and [[x 1,..., x n P 1 ]], [x 1,..., x n P 2 ]] [[x 1,..., x n (let x be P 1 in P 2 )]] = [[x 1,..., x n, x P 2 ]] π 1,..., π n, [x 1,..., x n P 1 ]] 6

7 (let x be (not F) in (x and F)) C [[ (let x be (not F) in (x and F))]] = [[x (x and F)]] [[ (not F)]] = and [[x x]], [[x F]] not [[ F]] = and id, ff not ff = and id, ff tt = and tt, ff tt = and tt, ff = ff C Set X {0, 1} tt ff not and F ρ(x i ) = v i (i = 1, 2,..., n) [[P ]] ρ = [[x 1,..., x n P ]](v 1,..., v n ) ω- Cpo (X, ) X x 0 x 1 x 2 x 3... x i (X, ) (X, ) X (X, ) (Y, ) f : X Y 1. x 1 x 2 f(x 1 ) f(x 2 ) 2. x 0 x 1 x 2... f( x i ) = f(x i ) Cpo Cpo {0, 1} 0, 1, x y (x = X x = y) F C Cpo X {0, 1} tt ff not and Cpo F F Cpo F F Cpo C D F : C D C X 1,..., X n X 1... X n π i : X 1... X n X i i = 1,..., n F (X 1... X n ) F (π i ) : F (X 1... X n ) F (X i ) i = 1,..., n D F (X 1 ),..., F (X n ) C(F) C(F) 0, 1, 2,... C(F) m n x 1,..., x m P P n ([P 1 ],..., [P n ]) (x1,...,x m ) 7

8 1: n ([x 1 ],..., [x n ]) (x1,...,x n ) ([P 1 ],..., [P m ]) (x1,...,x l ) : l m ([Q 1 ],..., [Q n ]) (x1,...,x m) : m n ([let x 1 be P 1 in... let x m be P m in Q 1 ],..., [let x 1 be P 1 in... let x m be P m in Q n ]) (x1,...,x l ) C(F) m 1,..., m n m m n C C(F) C C F C(F) 1 X and ([x 1 and x 2 ] (x1,x 2 )) : 2 1 C F C F C(F) C F F C(F) 2.4 F σ 1 σ 2 σ 1 8

9 σ 2 σ 1 σ 2 int (int int) int 3.2 X Y X Y E ε : E X Y Z f : Z X Y h : Z E ε (h id X ) = f E Y X f : Z X Y h : Z Y X f ε (f id X ) = f (f : Z X Y ) ε (h id X ) = h (h : Z Y X ) Set X Y Y X = {f : X Y } ε(f, x) = f(x) ε : Y X X Y f : Z X Y f : Z Y X (f(z))(x) = f(z, x) C [C, Set] Cpo X = (X, ) Y = (Y, ) X Y Y X f g x X f(x) g(x) ε(f, x) = f(x) ε : Y X X Y Vect K Vect fd K λ λ F C(F) λ C(λ ) F C C(λ ) C 9

10 4 4.1 sum(x) if x = 0 then 0 else x + sum(x 1) sum sum(3) sum(2) sum 3 + (2 + (1 + sum(0))) sum(0) = (2 + (1 + 0)) 6 sum sum 0 (x=0) F (f)(x) = x+f(x 1) (x 0 f(x 1) ) () sum C ( ) : C( X, X) C(, X) f : X X, h : f h = (f (h id X )) : X f : Y X, g : X Y (f π,x, g ) = f id, (g π,y, f ) : X f : X X X f = (f (id X )) : X Cpo Cpo f : (X, ) (X, ) f( ) f 2 ( ) f 3 ( )... f n ( ) sum F sum 4.2 C = (C,, I, a, l, r) C : C 2 C C I 10

11 a,,c : ( ) C ( C), l : I r : I a, l, r ( C) ( ) C I I a, l, r a, l, r c, C = (id c,c ) (c, id C ) c 1 C, = (id c 1 C, ) (c 1, id C) c, : c θ : θ I = id I θ = c, (θ θ ) c, θ = id c = c 1 symmetric monoidal category f : m n m : 2 1 f n : 2 1 X f Y Y g Z X f Y g Z g f C g D f C g D f f g c, : c 1, : C C = C C C C = C C θ : θ 1 = 11

12 η : I ε : I = = = θ = 1, c = c 1 K K Vect fd K U V U K V K c U,V U V V U U U hom(u, K) Rel Rel id X : X X {(x, x) x X} R : X Y S : Y Z S R : X Z {(x, z) X Z y Y (x, y) R, (y, z) S} X Y X Y 1 X X Rel Tangle Tangle + (s 1,..., s m ) (s 1,..., s n) + s 1,..., s m, s 1,..., s n (+,, +) (, +, +) (, +, +) (+) Tangle C Tangle C Rel Rel Joyal, Street, Verity 12

13 2: T r X, ((k id X) f (h id X )) = k T r,(f) X h f = f h k h k T rx,x(c X X,X ) θ 1 X = id X = T rx,x(c X 1 X,X ) θ X = = T rc,c (id X C f) = id C T r,(f) X f f = T r X,(T r Y X, X(f)) = T r Y,(T r X Y, Y ((id c Y,X ) f (id c 1 f = f Y,X ))) C C C T r, X : C( X, X) C(, ) 2 f : X X T r, X (f) : f Vect fd K, = = I T ri,i X : C(X, X) C(I, I) Rel R : X X T r, X R : {(a, b) x((a, x), (b, x)) R} Tangle + Tangle + Tangle + Rel 13

14 Hyland Cpo 4.4 C C Int C Int Int C Int C C Int C (X, U) (Y, V ) C X V Y U Int C f : (X, U) (Y, V ) g : (Y, V ) (Z, W ) V W U f g X Y Z Int Girard 1980 Geometry of Interaction bramsky Int 40 Knuth Int Int 14

15 f f x g x g y x g f y y Int C Int C Cpo 5 F [3] [1, 2] [5] [4] [3] [7] [6] [10] Mac Lane [9] Leinster [8] [1],, 59 (2007), [2], -,,, [3],,,,

16 [4],,, [5],., [6],,,, [7],,,,, [8] T. Leinster, asic Category Theory, Cambridge University Press, [9] S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer-Verlag, 1998,,,, 2005 /, [10],,, C Ob(C), Ob(C) C(, ) Ob(C) id C(, ),, C Ob(C) : C(, C) C(, ) C(, C) ( g, f ) g f (h g) f = h (g f) f C(, ), g C(, C), h C(C, D) id f = f = f id f C(, ) id g f f g C(, ) Hom C (, ) Hom(, ) f C(, ) f : f Set Rel K Vect K K Vect fd K Grp Cpo 16

17 C f : g : g f = id f g = id g f f 1 = C C op Ob(C op ) C op (, ) = C(, ) C C C D C D C D Ob(C D) = ObC) Ob(D) (C D)((, ), (, )) = C(, ) D(, ) id (, ) = (id, id ) (g, g ) (f, f ) = (g f, g f ).2 C D C D F : C D F () : Ob(C) Ob(D), Ob(C) f F (f) : C(, ) D(F (), F ()) F (id ) = id F () F (g f) = F (g) F (f) C C id C : C C id C () = id C (f) = f F : C D G : D E G F : C E (G F )() = G(F ()) (G F )(f) = G(F (f)) Cat F : C D, Ob(C) f F (f) : C(, ) D(F (), F ()), Ob(C) f F (f) : C(, ) D(F (), F ()) C D Ob(C) Ob(D), Ob(C) C(, ) D(, ) C D C D, Ob(C) C(, ) = D(, ) C D X X f : X Y J(f) = {(x, y) X Y f(x) = y} J : Set Rel X X X = X { X } x y (x = X x = y) f : X Y f : X Y f(x) = x (x X), f( X ) = Y ( ) : Set Cpo F Vect fd K Vect K 17

18 .3 F, G : C D F G α : F G C D {α : F () G()} Ob(C) f : α F (f) = G(f) α α : F () G() α F : C D F id F : F F (id F ) = id F () F, G, H : C D α : F G β : G H β α : F H (β α) = β α C D [C, D] D C [C, D] F, G : C D F G F G F = G C, D G F = id C F G = id D F : C D G : D C C D G F = id C F G = id D F : C D G : D C C D.4 C, D F : C D U : D C (C, D) D(F (C), D) : C op D Set (C, D) C(C, U(D)) : C op D Set F U U F C X 1,..., X n X 1... X n X n (X, X,..., X) n : C C n C X C Y Y X f : Y 1 Y 2 f id X : Y 1 X Y 2 X ( ) X : C C Z Z X g : Z 1 Z 2 g ε : Z X 1 ZX 2 ( )X : C C J : Set Rel X 2 X Cpo X Int(Cpo) (X, 1) f : X 1 X 2 f id 1 : (X 1, 1) (X 2, 1) J : Cpo Int(Cpo) (X, U) X U 18

211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,

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