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

Impredicativity: dynamic solution � a, b � �→ a : ∀ X . X × X → X = P

Impredicativity: dynamic solution � a, b � �→ a : ∀ X . X × X → X = P ∀ X . X × X → X

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 )

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 )

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

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

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 ) O ’s  hyper- at N : ( ∀ X.X × X → X ) × ( ∀ Y.Y × Y → Y ) → N × N → N move  n

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

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

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

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 N × N → N � P ’s at N : N × N → N × ( ∀ Y.Y × Y → Y ) → hyper- n move

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

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

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

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

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

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

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

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

Full completeness for system F • Uniform (UC) • Innocent ← Hyland-Ong • Total ← λ fragment • Compact = finite ‘spine’

Full completeness for system F • Uniform (UC) • Innocent ← Hyland-Ong • Total ← λ fragment • Compact = finite ‘spine’

Full completeness for system F • Uniform (UC) • Innocent ← Hyland-Ong • Total ← λ fragment • Compact = finite ‘spine’

Full completeness for system F • Uniform (UC) • Innocent ← Hyland-Ong • Total ← λ fragment • Compact = finite ‘spine’

Full completeness for system F • Uniform (UC) • Innocent ← Hyland-Ong • Total ← λ fragment • Compact = finite ‘spine’

Full completeness for system F • Uniform (UC) • Innocent ← Hyland-Ong • Total ← λ fragment ∀ X. X × X → X • Compact = finite ‘spine’ at ◦ : ◦ × ◦ → ◦ • •

Formal Hypergames

• Assertion : type occurrence - Lorenzen/Felscher dialogue games (1960, 1985) prenex • A → ∀ X.B ∀ X. ( A → B ) � • resolve = exhaustively instantiate leading ∀ s ∀ X. X → X ∀ Y.Y ❄ ∀ Y . ( ∀ Y.Y ) → Y N → B ❄ ( ∀ Y.Y ) → ( N → B )

• Assertion : type occurrence - Lorenzen/Felscher dialogue games (1960, 1985) prenex • A → ∀ X.B ∀ X. ( A → B ) � • resolve = exhaustively instantiate leading ∀ s ∀ X. X → X ∀ Y.Y ❄ ∀ Y . ( ∀ Y.Y ) → Y N → B ❄ ( ∀ Y.Y ) → ( N → B )

• Assertion : type occurrence - Lorenzen/Felscher dialogue games (1960, 1985) prenex • A → ∀ X.B ∀ X. ( A → B ) � • resolve = exhaustively instantiate leading ∀ s ∀ X. X → X ∀ Y.Y ❄ ∀ Y . ( ∀ Y.Y ) → Y N → B ❄ ( ∀ Y.Y ) → ( N → B )

• Assertion : type occurrence - Lorenzen/Felscher dialogue games (1960, 1985) prenex • A → ∀ X.B ∀ X. ( A → B ) � • resolve = exhaustively instantiate leading ∀ s ∀ X. X → X ∀ Y.Y ❄ ( ∀ Y.Y ) → ( ∀ Y.Y ) N → B ❄ ( ∀ Y.Y ) → ( N → B )

• Assertion : type occurrence - Lorenzen/Felscher dialogue games (1960, 1985) prenex • A → ∀ X.B ∀ X. ( A → B ) � • resolve = exhaustively instantiate leading ∀ s ∀ X. X → X ∀ Y.Y ❄ ( ∀ Y.Y ) → ( ∀ Y.Y ) � prenex N → B ❄ ( ∀ Y.Y ) → ( N → B )

• Assertion : type occurrence - Lorenzen/Felscher dialogue games (1960, 1985) prenex • A → ∀ X.B ∀ X. ( A → B ) � • resolve = exhaustively instantiate leading ∀ s ∀ X. X → X ∀ Y.Y ❄ ( ∀ Y.Y ) → ( ∀ Y.Y ) � prenex N → B ∀ Y . ( ∀ Y.Y ) → Y ❄ ( ∀ Y.Y ) → ( N → B )

• Assertion : type occurrence - Lorenzen/Felscher dialogue games (1960, 1985) prenex • A → ∀ X.B ∀ X. ( A → B ) � • resolve = exhaustively instantiate leading ∀ s ∀ X. X → X ∀ Y.Y ❄ ( ∀ Y.Y ) → ( ∀ Y.Y ) � prenex ∀ Y . ( ∀ Y.Y ) → Y N → B ❄ ( ∀ Y.Y ) → ( N → B )

• Assertion : type occurrence - Lorenzen/Felscher dialogue games (1960, 1985) prenex • A → ∀ X.B ∀ X. ( A → B ) � • resolve = exhaustively instantiate leading ∀ s ∀ X. X → X ∀ Y.Y ❄ ∀ Y . ( ∀ Y.Y ) → Y N → B ❄ ( ∀ Y.Y ) → ( N → B )

• Assertion : type occurrence - Lorenzen/Felscher dialogue games (1960, 1985) prenex • A → ∀ X.B ∀ X. ( A → B ) � • resolve = exhaustively instantiate leading ∀ s ∀ X. X → X ∀ Y.Y ❄ ∀ Y . ( ∀ Y.Y ) → Y N → B ❄ ( ∀ Y.Y ) → ( N → B )

• Assertion : type occurrence - Lorenzen/Felscher dialogue games (1960, 1985) prenex • A → ∀ X.B ∀ X. ( A → B ) � • resolve = exhaustively instantiate leading ∀ s ∀ X. X → X ∀ Y.Y ❄ ∀ Y . ( ∀ Y.Y ) → Y N → B ❄ ( ∀ Y.Y ) → ( N → B )

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

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

• Assertion : type occurrence - Lorenzen/Felscher dialogue games (1960, 1985) prenex • A → ∀ X.B ∀ X. ( A → B ) � • resolve = exhaustively instantiate leading ∀ s ∀ X. X → X ∀ Y.Y N → B ❄ ( ∀ Y.Y ) → ( N → B ) • n branches : A 1 → A 2 → . . . → A n → X N → B ❄ ( ∀ Y.Y ) → ( N → B )

Hypergame H ( A )

Hypergame H ( A ) • O resolves A

Hypergame H ( A ) ∀ X. X → X → X N • O resolves A ❄ N → N → N

Hypergame H ( A ) ∀ X. X → X → X N • O resolves A ❄ N → N → N

Hypergame H ( A ) ∀ X. X → X → X N • O resolves A ❄ N → N → N O

Hypergame H ( A ) • O resolves A • resolve opposing branch

Hypergame H ( A ) • O resolves A • resolve opposing branch A 1 → . . . → A i → . . . → A n → X O B 1 . . i . B m ❄ A i [ X 1 := B 1 , . . . , X m := B m ] P

Hypergame H ( A ) • O resolves A • resolve opposing branch A 1 → . . . → A i → . . . → A n → X O i ❄ A i [ X 1 := B 1 , . . . , X m := B m ] P

Hypergame H ( A ) • O resolves A • resolve opposing branch A 1 → . . . → A i → . . . → A n → X O B 1 . . i . B m ❄ A i [ X 1 := B 1 , . . . , X m := B m ] P

Hypergame H ( A ) • O resolves A • resolve opposing branch A 1 → . . . → A i → . . . → A n → X O B 1 . . i . B m ❄ A i [ X 1 := B 1 , . . . , X m := B m ] P