[PPT] - Hypergame semantics: Ten years later Dominic J. D. Hughes Stanford PowerPoint Presentation

SLIDE 1

Hypergame semantics: Ten years later

Dominic J. D. Hughes Stanford University

GaLoP’06, Seattle, 11 August 2006

SLIDE 2

Hypergames model/semantics

For a 2nd-order logic / polymorphic progr. language
Two principles:
(GM) Games as Moves
(UC) Uniformity by Copycat expansion
Imperial College Games Workshop 1996

LICS 97, PhD thesis (Luke Ong, Oxford)

System F (LICS 97, PhD)
affine linear (Murawski-Ong ’01, Murawski PhD ’01)
Type isos (de Lataillade, yesterday)
Here cleaner notation: no ×, transition games, Felscher

SLIDE 3

(GM) Games as Moves

Zwicker’s Hypergame H (1987)
O chooses terminating game G;
✠

❅ ❅ ❅ ❘

Chess

✁ ✁ ✁ ✁ ☛

Nf3

❆ ❆ ❆ ❆ ❯

e4

✁ ✁ ✁ ✁ ☛ ❆ ❆ ❆ ❆ ❯

Nf6 d5

O’s & X’s

❄

X

play G ( P starts )

“ Is H terminating? ”

∼ Russell’s paradox (Mirimanoff 1917)

SLIDE 4

(GM) Games as Moves

a, b → a

: ∀X . X × X → X

SLIDE 5

(GM) Games as Moves

a, b → a

: ∀X . X × X → X

at N :

7, 3 → 7 : N × N → N

SLIDE 6

(GM) Games as Moves

a, b → a

: ∀X . X × X → X

at N :

7, 3 → 7 : N × N → N

at B :

true, false → true : B × B → B

SLIDE 7

(GM) Games as Moves

π1
a, b → a

:

P

∀X . X × X → X
at N :

7, 3 → 7 : N × N → N

at B :

true, false → true : B × B → B

SLIDE 8

(GM) Games as Moves

π1
a, b → a

:

P

∀X . X × X → X
at N :

7, 3 → 7 : N × N → N

at B :

true, false → true : B × B → B

at P :

π2, π1 → π2 : P × P → P

SLIDE 9

(GM) Games as Moves

π1
a, b → a

:

P

∀X . X × X → X
at N :

7, 3 → 7 : N × N → N

at B :

true, false → true : B × B → B

at P :

π2, π1 → π2 : P × P → P

SLIDE 10

(UC) Uniformity by Copycat expansion

a, b → a

: ∀X . X × X → X Uniform

SLIDE 11

(UC) Uniformity by Copycat expansion

a, b → a

: ∀X . X × X → X Uniform

at N :

7, 3 → 7 : N × N → N

at B :

true, false → true : B × B → B

SLIDE 12

(UC) Uniformity by Copycat expansion

a, b → a

: ∀X . X × X → X Uniform

at N :

7, 3 → 7 : N × N → N

at B :

true, false → true : B × B → B



  a, b → a if X = N a, b → b

therw.

   : ∀X . X × X → X

Ad hoc Uniform

SLIDE 13

(UC) Uniformity by Copycat expansion

a, b → a

: ∀X . X × X → X Uniform

at N :

7, 3 → 7 : N × N → N

at B :

true, false → true : B × B → B



  a, b → a if X = N a, b → b

therw.

   : ∀X . X × X → X

Ad hoc Uniform

at N :

7, 3 → 7 : N × N → N

SLIDE 14

(UC) Uniformity by Copycat expansion

a, b → a

: ∀X . X × X → X Uniform

at N :

7, 3 → 7 : N × N → N

at B :

true, false → true : B × B → B



  a, b → a if X = N a, b → b

therw.

   : ∀X . X × X → X

Ad hoc Uniform

at N :

7, 3 → 7 : N × N → N

at B :

true, false → false : B × B → B

SLIDE 15

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

× ◦ → ◦

SLIDE 16

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

× ◦ → ◦

SLIDE 17

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

× ◦ → ◦

SLIDE 18

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

× ◦ → ◦

SLIDE 19

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

× ◦ → ◦
at N :

N × N → N

SLIDE 20

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

× ◦ → ◦
at N :

N × N → N n

SLIDE 21

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

× ◦ → ◦
at N :

N × N → N n n

SLIDE 22

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

× ◦ → ◦
at N :

N × N → N n n 7

SLIDE 23

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

× ◦ → ◦
at N :

N × N → N n n 7 7

SLIDE 24

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

× ◦ → ◦
at N :

N × N → N n n 7 7

Convention: O = Opponent = Orange P = Player = Purple

SLIDE 25

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

× ◦ → ◦

SLIDE 26

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

× ◦ → ◦

SLIDE 27

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

× ◦ → ◦

SLIDE 28

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

× ◦ → ◦

SLIDE 29

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

× ◦ → ◦
at B :

B × B → B

SLIDE 30

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

× ◦ → ◦
at B :

B × B → B b

SLIDE 31

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

× ◦ → ◦
at B :

B × B → B b b

SLIDE 32

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

× ◦ → ◦
at B :

B × B → B b b true

SLIDE 33

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

× ◦ → ◦
at B :

B × B → B b b true true

SLIDE 34

(UC) Uniformity by Copycat expansion

∀X. X × X → X at N : N × N → N n n 3 3 at B : B × B → B Ad hoc Uniform b b true true

SLIDE 35

(UC) Uniformity by Copycat expansion

∀X. X × X → X at N→B : (N→B) × (N→B) → (N→B)

SLIDE 36

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

×
→
at N→B :

(N→B) × (N→B) → (N→B)

SLIDE 37

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

×
→
at N→B :

(N→B) × (N→B) → (N→B)

SLIDE 38

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

×
→
at N→B :

(N→B) × (N→B) → (N→B)

SLIDE 39

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

×
→
at N→B :

(N→B) × (N→B) → (N→B)

SLIDE 40

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

×
→
at N→B :

(N→B) × (N→B) → (N→B) b

SLIDE 41

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

×
→
at N→B :

(N→B) × (N→B) → (N→B) b b

SLIDE 42

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

×
→
at N→B :

(N→B) × (N→B) → (N→B) b b n

SLIDE 43

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

×
→
at N→B :

(N→B) × (N→B) → (N→B) b b n n

SLIDE 44

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

×
→
at N→B :

(N→B) × (N→B) → (N→B) b b n n 5

SLIDE 45

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

×
→
at N→B :

(N→B) × (N→B) → (N→B) b b n n 5 5

SLIDE 46

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

×
→
at N→B :

(N→B) × (N→B) → (N→B) b b n n 5 5 true

SLIDE 47

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

×
→
at N→B :

(N→B) × (N→B) → (N→B) b b n n 5 5 true true

SLIDE 48

(UC) Uniformity by Copycat expansion

∀X. X × X → X at ◦ :

×
→
at N→B :

(N→B) × (N→B) → (N→B) b b n n 5 5 true true prime? , even? → prime?

SLIDE 49

compl.

Type isos (de Lataillade, yesterday)
∩-game semantics: ¬(GM) & (U∩)
affine linear (Abramsky ’97, Newton Inst. ’95)

· · · Abramsky & Lenisa · · ·

Generic game semantics: ¬(GM) & (UC′)
system F (Abramsky-Jagadeesan ’05)

full compl. ML types

SLIDE 50

Impredicativity

π1
a, b → a

:

P

∀X . X × X → X
at N :

7, 3 → 7 : N × N → N

at P :

π2, π1 → π2 : P × P → P ML , F

{ barbers : shave non-self-shavers }
H is a move in H: “H ∈ H” (Zwicker)

SLIDE 51

Impredicativity: dynamic solution

Static approaches not fully complete for F:
Domain models

(continuity/approximation/limit)

PER models

(families/∩)

Abramsky-Jagadeesan ’05

(no 2nd-order moves)

Dynamic solution: (GM) Games as Moves

SLIDE 52

Impredicativity: dynamic solution

a, b → a : ∀X . X × X → X = P

SLIDE 53

Impredicativity: dynamic solution

a, b → a : ∀X . X × X → X = P ∀X . X × X → X

SLIDE 54

Impredicativity: dynamic solution

a, b → a : ∀X . X × X → X = P ∀X . X × X → X at P : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → (∀Z.Z ×Z→Z)

SLIDE 55

Impredicativity: dynamic solution

a, b → a : ∀X . X × X → X = P ∀X . X × X → X at P : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → (∀Z.Z ×Z→Z)

SLIDE 56

Impredicativity: dynamic solution

a, b → a : ∀X . X × X → X = P ∀X . X × X → X at P : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → (∀Z.Z ×Z→Z) at N : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → N × N → N

SLIDE 57

Impredicativity: dynamic solution

a, b → a : ∀X . X × X → X = P ∀X . X × X → X at P : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → (∀Z.Z ×Z→Z) at N : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → N × N → N n

SLIDE 58

Impredicativity: dynamic solution

a, b → a : ∀X . X × X → X = P ∀X . X × X → X at P : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → (∀Z.Z ×Z→Z) at N : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → N × N → N    O’s

hyper- move

n

SLIDE 59

Impredicativity: dynamic solution

a, b → a : ∀X . X × X → X = P ∀X . X × X → X at P : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → (∀Z.Z ×Z→Z) at N : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → N × N → N n

SLIDE 60

Impredicativity: dynamic solution

a, b → a : ∀X . X × X → X = P ∀X . X × X → X at P : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → (∀Z.Z ×Z→Z) at N : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → N × N → N n at N : N × N → N × (∀Y.Y ×Y →Y ) → N × N → N

SLIDE 61

Impredicativity: dynamic solution

a, b → a : ∀X . X × X → X = P ∀X . X × X → X at P : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → (∀Z.Z ×Z→Z) at N : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → N × N → N n at N : N × N → N × (∀Y.Y ×Y →Y ) → N × N → N n

SLIDE 62

Impredicativity: dynamic solution

a, b → a : ∀X . X × X → X = P ∀X . X × X → X at P : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → (∀Z.Z ×Z→Z) at N : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → N × N → N n at N : N × N → N × (∀Y.Y ×Y →Y ) → N × N → N P’s

hyper- move

n

SLIDE 63

Impredicativity: dynamic solution

a, b → a : ∀X . X × X → X = P ∀X . X × X → X at P : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → (∀Z.Z ×Z→Z) at N : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → N × N → N n at N : N × N → N × (∀Y.Y ×Y →Y ) → N × N → N n

SLIDE 64

Impredicativity: dynamic solution

a, b → a : ∀X . X × X → X = P ∀X . X × X → X at P : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → (∀Z.Z ×Z→Z) at N : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → N × N → N n at N : N × N → N × (∀Y.Y ×Y →Y ) → N × N → N n m

SLIDE 65

Impredicativity: dynamic solution

a, b → a : ∀X . X × X → X = P ∀X . X × X → X at P : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → (∀Z.Z ×Z→Z) at N : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → N × N → N n at N : N × N → N × (∀Y.Y ×Y →Y ) → N × N → N n m m

SLIDE 66

Impredicativity: dynamic solution

a, b → a : ∀X . X × X → X = P ∀X . X × X → X at P : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → (∀Z.Z ×Z→Z) at N : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → N × N → N n at N : N × N → N × (∀Y.Y ×Y →Y ) → N × N → N n m m 4

SLIDE 67

Impredicativity: dynamic solution

a, b → a : ∀X . X × X → X = P ∀X . X × X → X at P : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → (∀Z.Z ×Z→Z) at N : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → N × N → N n at N : N × N → N × (∀Y.Y ×Y →Y ) → N × N → N n m m 4 4

SLIDE 68

Impredicativity: dynamic solution

a, b → a : ∀X . X × X → X = P ∀X . X × X → X at P : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → (∀Z.Z ×Z→Z) at N : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → N × N → N n at N : N × N → N × (∀Y.Y ×Y →Y ) → N × N → N n m m 4 4 5

SLIDE 69

Impredicativity: dynamic solution

a, b → a : ∀X . X × X → X = P ∀X . X × X → X at P : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → (∀Z.Z ×Z→Z) at N : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → N × N → N n at N : N × N → N × (∀Y.Y ×Y →Y ) → N × N → N n m m 4 4 5 5

SLIDE 70

Impredicativity: dynamic solution

a, b → a : ∀X . X × X → X = P ∀X . X × X → X at P : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → (∀Z.Z ×Z→Z) at N : (∀X.X ×X→X) × (∀Y.Y ×Y →Y ) → N × N → N n at N : N × N → N × (∀Y.Y ×Y →Y ) → N × N → N n m m 4 4 5 5

succ

SLIDE 71

Full completeness for system F

Uniform (UC)
Innocent ← Hyland-Ong
Total

← λ fragment

Compact = finite ‘spine’

SLIDE 72

Full completeness for system F

Uniform (UC)
Innocent ← Hyland-Ong
Total

← λ fragment

Compact = finite ‘spine’

SLIDE 73

Full completeness for system F

Uniform (UC)
Innocent ← Hyland-Ong
Total

← λ fragment

Compact = finite ‘spine’

SLIDE 74

Full completeness for system F

Uniform (UC)
Innocent ← Hyland-Ong
Total

← λ fragment

Compact = finite ‘spine’

SLIDE 75

Full completeness for system F

Uniform (UC)
Innocent ← Hyland-Ong
Total

← λ fragment

Compact = finite ‘spine’

SLIDE 76

Full completeness for system F

Uniform (UC)
Innocent ← Hyland-Ong
Total

← λ fragment

Compact = finite ‘spine’

∀X. X × X → X at ◦ : ◦ × ◦ → ◦

SLIDE 77

Formal Hypergames

SLIDE 78

Assertion: type occurrence
Lorenzen/Felscher dialogue games (1960, 1985)
A → ∀X.B

prenex

∀X.(A → B)
resolve = exhaustively instantiate leading ∀s

∀X. X → X

❄

∀Y.Y

∀Y . (∀Y.Y ) → Y

❄

N→B

(∀Y.Y ) → (N→B)

SLIDE 79

Assertion: type occurrence
Lorenzen/Felscher dialogue games (1960, 1985)
A → ∀X.B

prenex

∀X.(A → B)
resolve = exhaustively instantiate leading ∀s

∀X. X → X

❄

∀Y.Y

∀Y . (∀Y.Y ) → Y

❄

N→B

(∀Y.Y ) → (N→B)

SLIDE 80

Assertion: type occurrence
Lorenzen/Felscher dialogue games (1960, 1985)
A → ∀X.B

prenex

∀X.(A → B)
resolve = exhaustively instantiate leading ∀s

∀X. X → X

❄

∀Y.Y

∀Y . (∀Y.Y ) → Y

❄

N→B

(∀Y.Y ) → (N→B)

SLIDE 81

Assertion: type occurrence
Lorenzen/Felscher dialogue games (1960, 1985)
A → ∀X.B

prenex

∀X.(A → B)
resolve = exhaustively instantiate leading ∀s

∀X. X → X

❄

∀Y.Y

(∀Y.Y ) → (∀Y.Y )

❄

N→B

(∀Y.Y ) → (N→B)

SLIDE 82

Assertion: type occurrence
Lorenzen/Felscher dialogue games (1960, 1985)
A → ∀X.B

prenex

∀X.(A → B)
resolve = exhaustively instantiate leading ∀s

∀X. X → X

❄

∀Y.Y

(∀Y.Y ) → (∀Y.Y )

❄

N→B

prenex

(∀Y.Y ) → (N→B)

SLIDE 83

Assertion: type occurrence
Lorenzen/Felscher dialogue games (1960, 1985)
A → ∀X.B

prenex

∀X.(A → B)
resolve = exhaustively instantiate leading ∀s

∀X. X → X

❄

∀Y.Y

(∀Y.Y ) → (∀Y.Y )

❄

N→B

prenex

∀Y . (∀Y.Y ) → Y (∀Y.Y ) → (N→B)

SLIDE 84

Assertion: type occurrence
Lorenzen/Felscher dialogue games (1960, 1985)
A → ∀X.B

prenex

∀X.(A → B)
resolve = exhaustively instantiate leading ∀s

∀X. X → X

❄

∀Y.Y

(∀Y.Y ) → (∀Y.Y )

❄

N→B

prenex

∀Y . (∀Y.Y ) → Y (∀Y.Y ) → (N→B)

SLIDE 85

Assertion: type occurrence
Lorenzen/Felscher dialogue games (1960, 1985)
A → ∀X.B

prenex

∀X.(A → B)
resolve = exhaustively instantiate leading ∀s

∀X. X → X

❄

∀Y.Y

∀Y . (∀Y.Y ) → Y

❄

N→B

(∀Y.Y ) → (N→B)

SLIDE 86

Assertion: type occurrence
Lorenzen/Felscher dialogue games (1960, 1985)
A → ∀X.B

prenex

∀X.(A → B)
resolve = exhaustively instantiate leading ∀s

∀X. X → X

❄

∀Y.Y

∀Y . (∀Y.Y ) → Y

❄

N→B

(∀Y.Y ) → (N→B)

SLIDE 87

Assertion: type occurrence
Lorenzen/Felscher dialogue games (1960, 1985)
A → ∀X.B

prenex

∀X.(A → B)
resolve = exhaustively instantiate leading ∀s

∀X. X → X

❄

∀Y.Y

∀Y . (∀Y.Y ) → Y

❄

N→B

(∀Y.Y ) → (N→B)

SLIDE 88

Assertion: type occurrence
Lorenzen/Felscher dialogue games (1960, 1985)
A → ∀X.B

prenex

∀X.(A → B)
resolve = exhaustively instantiate leading ∀s

∀X. X → X

❄

∀Y.Y

∀Y . (∀Y.Y ) → Y

❄

N→B

❄

N→B

(∀Y.Y ) → (N→B) (∀Y.Y ) → (N→B)

SLIDE 89

Assertion: type occurrence
Lorenzen/Felscher dialogue games (1960, 1985)
A → ∀X.B

prenex

∀X.(A → B)
resolve = exhaustively instantiate leading ∀s

∀X. X → X

❄

∀Y.Y N→B

(∀Y.Y ) → (N→B)

❄

N→B

(∀Y.Y ) → (N→B)

SLIDE 90

Assertion: type occurrence
Lorenzen/Felscher dialogue games (1960, 1985)
A → ∀X.B

prenex

∀X.(A → B)
resolve = exhaustively instantiate leading ∀s

∀X. X → X

❄

∀Y.Y N→B

(∀Y.Y ) → (N→B)

❄

N→B

(∀Y.Y ) → (N→B)

n branches: A1 → A2 → . . . → An → X

SLIDE 91

Hypergame H(A)

SLIDE 92

Hypergame H(A)

O resolves A

SLIDE 93

Hypergame H(A)

O resolves A

∀X. X → X → X

❄ N

N → N → N

SLIDE 94

Hypergame H(A)

O resolves A

∀X. X → X → X

❄ N

N → N → N

SLIDE 95

Hypergame H(A)

O resolves A

∀X. X → X → X

❄ N

N → N → N O

SLIDE 96

Hypergame H(A)

O resolves A
resolve opposing branch

SLIDE 97

Hypergame H(A)

O resolves A
resolve opposing branch

A1→. . .→Ai→. . .→An→X O

❄ i B1 . . . Bm

Ai [X1 :=B1, . . . , Xm :=Bm] P

SLIDE 98

Hypergame H(A)

O resolves A
resolve opposing branch

A1→. . .→Ai→. . .→An→X O

❄ i

Ai [X1 :=B1, . . . , Xm :=Bm] P

SLIDE 99

Hypergame H(A)

O resolves A
resolve opposing branch

A1→. . .→Ai→. . .→An→X O

❄ i B1 . . . Bm

Ai [X1 :=B1, . . . , Xm :=Bm] P

SLIDE 100

Hypergame H(A)

O resolves A
resolve opposing branch

A1→. . .→Ai→. . .→An→X O

❄ i B1 . . . Bm

Ai [X1 :=B1, . . . , Xm :=Bm] P

SLIDE 101

Hypergame H(A)

O resolves A
resolve opposing branch

A1→. . .→Ai→. . .→An→X P

❄ i B1 . . . Bm

Ai [X1 :=B1, . . . , Xm :=Bm] O

SLIDE 102

Hypergame H(A)

O resolves A
resolve opposing branch

A1→. . .→Ai→. . .→An→X O

❄ i B1 . . . Bm

Ai [X1 :=B1, . . . , Xm :=Bm] P

SLIDE 103

Hypergame H(A)

O resolves A
resolve opposing branch

A1→. . .→Ai→. . .→An→X O . . . · · ·

P

· · ·

O

. . . · · ·

P

· · ·

O

. . .

❄

i B1 . . . Bm Ai [X1 :=B1, . . . , Xm :=Bm] P

SLIDE 104

Hypergame H(A)

O resolves A
resolve opposing branch

A1→. . .→Ai→. . .→An→X O . . . · · ·

P

· · ·

O

. . . · · ·

P

· · ·

O

. . . i B1 . . . Bm

✑ ✑ ✰

Ai [X1 :=B1, . . . , Xm :=Bm] P

SLIDE 105

Hypergame H(A)

O resolves A
resolve opposing branch

A1→. . .→Ai→. . .→An→X O . . . · · ·

P

· · ·

O

. . . · · ·

P

· · ·

O

. . . i B1 . . . Bm

✑ ✑ ✰

Ai [X1 :=B1, . . . , Xm :=Bm] P

SLIDE 106

Hypergame H(A)

O resolves A
resolve opposing branch

A1→. . .→Ai→. . .→An→X O . . . · · ·

P

· · ·

O

. . . · · ·

P

· · ·

O

. . . i B1 . . . Bm

✑ ✑ ✰

Ai [X1 :=B1, . . . , Xm :=Bm] P

SLIDE 107

∀X. X →X

SLIDE 108

∀X. X →X

❄ ∀Y.Y N

(∀Y.Y )→N O

SLIDE 109

∀X. X →X

❄ ∀Y.Y N

(∀Y.Y )→N O

∀X. X → X ❄

∀Y.Y

(∀Y.Y ) → (∀Y.Y ) ❄

N→B

prenex

∀Y . (∀Y.Y ) → Y ❄

N

(∀Y.Y ) → N

SLIDE 110

∀X. X →X

❄ ∀Y.Y N

(∀Y.Y )→N O

SLIDE 111

∀X. X →X

❄ ∀Y.Y N

(∀Y.Y )→N O

❄ 1 (∀Z. Z→ Z)→R→B

(∀Z. Z→ Z)→R→B P

SLIDE 112

∀X. X →X

❄ ∀Y.Y N

(∀Y.Y )→N O

❄ 1 (∀Z. Z→ Z)→R→B

(∀Z. Z→ Z)→R→B P

❄ 2

R

O

SLIDE 113

∀X. X →X

❄ ∀Y.Y N

(∀Y.Y )→N O

❄ 1 (∀Z. Z→ Z)→R→B

(∀Z. Z→ Z)→R→B P

❄ 2

R

O 1 C→C ❍ ❥ C→C P

SLIDE 114

∀X. X →X

❄ ∀Y.Y N

(∀Y.Y )→N O

❄ 1 (∀Z. Z→ Z)→R→B

(∀Z. Z→ Z)→R→B P

❄ 2

R

O 1 C→C ❍ ❥ C→C P 1 N→N ✑ ✰

(N→N) → (N→N) O

SLIDE 115

Hypergame H(A)

O resolves A
resolve opposing branch

A1→. . .→Ai→. . .→An→X O . . . · · ·

P

· · ·

O

. . . · · ·

P

· · ·

O

. . . i B1 . . . Bm

✑ ✑ ✰

Ai [X1 :=B1, . . . , Xm :=Bm] P

SLIDE 116

Composition

2nd-order extension of Hyland-Ong (λ fragment):

Interaction = Hyland-Ong interaction

Originally (LICS’97, PhD) abstract machine style:

(uniform) view ↔ (uniform) view [Danos-Herbelin-Regnier, LICS’96; Nickau PhD ’96]

This tutorial:
back to Hyland-Ong style (all info shared)
Transition (system) games

SLIDE 117

Transition game LTS(A)

States: TYPES(F ∪ {⋆})
Transitions:

A1→. . .→An→X i

− →

B1 · · · Bm

Ai [X1:=B1, . . . , Xm :=Bm]

Initial state:

A → ⋆

SLIDE 118

∀X. X →X

❄ ∀Y.Y N

(∀Y.Y )→N O

❄ 1 (∀Z. Z→ Z)→R→B

(∀Z. Z→ Z)→R→B P

❄ 2

R

O 1 C→C ❍ ❥ C→C P 1 N→N ✑ ✰

(N→N) → (N→N) O

SLIDE 119

(∀X. X →X) → ⋆

❄ 1 ∀Y.Y N

(∀Y.Y )→N O

❄ 1 (∀Z. Z→ Z)→R→B

(∀Z. Z→ Z)→R→B P

❄ 2

R

O 1 C→C ❍ ❥ C→C P 1 N→N ✑ ✰

(N→N) → (N→N) O

SLIDE 120

(∀X. X →X) → ⋆

❄ 1 ∀Y.Y N

(∀Y.Y )→N

❄ 1 (∀Z. Z→ Z)→R→B

(∀Z. Z→ Z)→R→B

❄ 2

R

1 C→C ❍ ❥ C→C 1 N→N ✑ ✰

(N→N) → (N→N)

SLIDE 121

part of

LTS(∀X. X →X) (∀X. X →X) → ⋆

❄ 1 ∀Y.Y N

(∀Y.Y )→N

❄ 1 (∀Z. Z→ Z)→R→B

(∀Z. Z→ Z)→R→B

❄ 2

R

1 C→C ❍ ❥ C→C 1 N→N ✑ ✰

(N→N) → (N→N)

SLIDE 122

Loops in LTS(A) (∀X. X)→Y

1 (∀X. X)→Y

SLIDE 123

Transition game G(A)

Hyland-Ong arena Ar(LTS) := Tracesne(LTS)
Hyland-Ong game G(LTS)
G(A) := G(LTS(A)) ∼

= H(A)

Arena isos:

Ar(A→B) ∼ = Ar(A) ⇒ Ar(B) Ar(∀X.A) ∼ =

Types B

Ar(A[B/X])

SLIDE 124

Transition game G(A)

Hyland-Ong arena Ar(LTS) := Tracesne(LTS)
Hyland-Ong game G(LTS)

(−VISIBILITY)

G(A) := G(LTS(A)) ∼

= H(A)

Arena isos:

Ar(A→B) ∼ = Ar(A) ⇒ Ar(B) Ar(∀X.A) ∼ =

Types B

Ar(A[B/X])

SLIDE 125

Hypergame composition Ar(B →C) ∼ = Ar(B) ⇒ Ar(C) Composition = Hyland-Ong composition

SLIDE 126

Uniformity & Full Completeness

SLIDE 127

Uniformity precondition

COPYCAT Condition [LICS’97, PhD]:

X is rightmost var in O’s assertion ⇓ X is rightmost var in P’s assertion Strengthens (D10) in Felscher’85 ← Lorenz(en)

SLIDE 128

Uniformity precondition

COPYCAT Condition [LICS’97, PhD]:

X is rightmost var in O’s assertion ⇓ X is rightmost var in P’s assertion Strengthens (D10) in

Note: henceforth read N, B, R, . . . as type variables

Felscher’85 ← Lorenz(en)

SLIDE 129

∀X. X →X

❄ ∀Y.Y N

(∀Y.Y )→N O

❄ 1 (∀Z. Z→ Z)→R→B

(∀Z. Z→ Z)→R→B P

❄ 2

R

O 1 C→C ❍ ❥ C→C P 1 N→N ✑ ✰

(N→N) → (N→N) O

SLIDE 130

∀X. X →X

❄ ∀Y.Y N

(∀Y.Y )→N O

❄ 1 (∀Z. Z→ Z)→R→N

(∀Z. Z→ Z)→R→N P

❄ 2

R

O 1 R→R ❍ ❥ R→R P 1 N→N ✑ ✰

(N→N) → (N→N) O

SLIDE 131

COPYCAT → (UC) Uniformity by Copycat expansion

Suppose
play p satisfies COPYCAT
X free, first imported by O
Copycat expansion p(X → A) : copycat A+ ↔ A−

∼ Felscher’s skeleton expansion

Strategy uniform if:
satisfies COPYCAT
closed under copycat expansions

SLIDE 132

Copycat expansion ∼ η expansion

ΛX.λf X.f : ∀X.X → X (η-exp. β-normal)

❄ N→N

λf N→N.f : (N → N) → (N → N) (not η-exp.)

❄ η

λf N→N.λxN.fx : (N → N) → (N → N)

SLIDE 133

Copycat expansion ∼ η expansion

ΛX.λf X.f : ∀X.X → X (η-exp. β-normal)

❄ N→N

λf N→N.f : (N → N) → (N → N) (not η-exp.)

❄ η

λf N→N.λxN.fx : (N → N) → (N → N) (η-exp.)

SLIDE 134

Full completeness

An innocent uniform strategy is compact if it is generated from a finite set of views by copycat expansion THEOREM: System F terms

nto