game semantics and the complexity of interactionpbl/galop2016/aschierislides.pdf · game semantics...

Game Semantics and the Complexity ofInteraction

Federico Aschieri

Institute of Discrete Mathematics and GeometryTechnische Universitat Wien

Eindhoven, 2 April 2016

Federico Aschieri Game Semantics and the Complexity of Interaction

Complexity of Cut-Elimination

Γ,A⋮

Γ,A�cut

r is the height of A as formula tree

k is the height of the proof tree

22..2k

Γ,A⋮

Γ,A�cut

22..2k

Γ,A⋮

Γ,A�cut

22..2k

Figure: New York

Worst case: World Trade Center, 546 meters

Better worst-case analysis

Γ,A⋮

Γ,A�cut

Γ,∃x B,B[t/x](t first-order term)

Γ,∃x B

Γ,B[a/x](a eigenvariable)

Γ,∀x B

Γ,A⋮

Γ,A�cut

Γ,∃x B,B[t/x](t first-order term)

Γ,∃x B

Γ,B[a/x](a eigenvariable)

Γ,∀x B

Expansion Tree Strategies

∃x ∀y ∃z P(x ,y ,z)

∀y ∃z P(t3,y ,z)

∃z P(t3,a3,z)

∀y ∃z P(t2,y ,z)

∃z P(t2,a2,z)

∀y ∃z P(t1,y ,z)

∃z P(t1,a1,z)

P(t1,a1, t4)

Order: t1 a1 t2 a2 t3 a3 t4

Winning: P(t1,a1, t4)

∃x ∀y ∃z P(x ,y ,z)

∀y ∃z P(t3,y ,z)

∃z P(t3,a3,z)

∀y ∃z P(t2,y ,z)

∃z P(t2,a2,z)

∀y ∃z P(t1,y ,z)

∃z P(t1,a1,z)

P(t1,a1, t4)

Winning: P(t1,a1, t4)

∃x ∀y ∃z P(x ,y ,z)

∀y ∃z P(t3,y ,z)

∃z P(t3,a3,z)

∀y ∃z P(t2,y ,z)

∃z P(t2,a2,z)

∀y ∃z P(t1,y ,z)

∃z P(t1,a1,z)

P(t1,a1, t4)

Winning: P(t1,a1, t4)Federico Aschieri Game Semantics and the Complexity of Interaction

∀x ∃y ∀z ¬P(x ,y ,z)

∃y ∀z ¬P(b,y ,z)

∀z ¬P(b,u3,z)

¬P(b,u3,b3)

∀z ¬P(b,u2,z)

¬P(b,u2,b2)

∀z ¬P(b,u1,z)

¬P(b,u1,b1)

Order: b u1 b1 u2 b2 u3 b3

Winning: ¬P(b,u1,b1) ∨P(b,u2,b2) ∨P(b,u3,b3)

∀x ∃y ∀z ¬P(x ,y ,z)

∃y ∀z ¬P(b,y ,z)

∀z ¬P(b,u3,z)

¬P(b,u3,b3)

∀z ¬P(b,u2,z)

¬P(b,u2,b2)

∀z ¬P(b,u1,z)

¬P(b,u1,b1)

∀x ∃y ∀z ¬P(x ,y ,z)

∃y ∀z ¬P(b,y ,z)

∀z ¬P(b,u3,z)

¬P(b,u3,b3)

∀z ¬P(b,u2,z)

¬P(b,u2,b2)

∀z ¬P(b,u1,z)

¬P(b,u1,b1)

∃x ∀y ∃z P(x ,y ,z)

∀y ∃z P(t3,y ,z)

∃z P(t3,a3,z)

∀y ∃z P(t2,y ,z)

∃z P(t2,a2,z)

∀y ∃z P(t1,y ,z)

∃z P(t1,a1,z)

P(t1,a1, t4)

∗ x ∶= t1 y ∶= a1 x ∶= t2 y ∶= a2 x ∶= t3 y ∶= a3 z ∶= t4

∃x ∀y ∃z P(x ,y ,z)

∀y ∃z P(t3,y ,z)

∃z P(t3,a3,z)

∀y ∃z P(t2,y ,z)

∃z P(t2,a2,z)

∀y ∃z P(t1,y ,z)

∃z P(t1,a1,z)

P(t1,a1, t4)

∗ x ∶= t1 y ∶= a1 x ∶= t2 y ∶= a2 x ∶= t3 y ∶= a3 z ∶= t4

∀x ∃y ∀z ¬P(x ,y ,z)

∃y ∀z ¬P(b,y ,z)

∀z ¬P(b,u3,z)

¬P(b,u3,b3)

∀z ¬P(b,u2,z)

¬P(b,u2,b2)

∀z ¬P(b,u1,z)

¬P(b,u1,b1)

Order: b a1 u1 b1 u2 b2 u3 b3

∗ x ∶= b y ∶= u1 z ∶= b1 y ∶= u2 z ∶= b2 y ∶= u3

∀x ∃y ∀z ¬P(x ,y ,z)

∃y ∀z ¬P(b,y ,z)

∀z ¬P(b,u3,z)

¬P(b,u3,b3)

∀z ¬P(b,u2,z)

¬P(b,u2,b2)

∀z ¬P(b,u1,z)

¬P(b,u1,b1)

Order: b a1 u1 b1 u2 b2 u3 b3

∗ x ∶= b y ∶= u1 z ∶= b1 y ∶= u2 z ∶= b2 y ∶= u3

Γ,A⋮

Γ,A�cut

b is minimum among the backtracking levels of the expansiontrees for A and A�

22..2k

1 ≤ b ≤ r !

Γ,A⋮

Γ,A�cut

22..2k

1 ≤ b ≤ r !

Γ,A⋮

Γ,A�cut

22..2k

1 ≤ b ≤ r !

Γ,A⋮

Γ,A�cut

22..2k

1 ≤ b ≤ r !

Game Semantics and Logic

Coquand: A Semantics of Evidence for Classical Arithmetic(1991, 1995)

Herbelin: correspondence between plays and cut-elimination(1995)

Coquand: notion of backtracking level 1 (1991)

Berardi-de’Liguoro: notion of backtracking level n ∈ N (2009)

Game Semantics and Computer Science

Hyland-Ong: Full Abstraction for PCF (2000)

Danos-Herbelin-Regnier: Correspondence between plays andlinear head reduction (1996)

Clairambault: Complexity analysis of interaction, re-obtainingby game semantics the known bounds (2011)

Arenas

An arena is a structure A = (M,⊢, λ, I).

M: set of moves. ⊢: binary justification relation between moves.

λ: the turn function M → {Opponent ,Player}

Play: a justified sequence of moves with alternating labels

∗ e4 e5 Nf3 Nf6 Nxe5

∗ e4 e5 Nf3 Nf6 Nxe5 Nc6

Arenas

∗ e4 e5 Nf3 Nf6 Nxe5 Nc6 e5 d6

The plays on the board are now:

∗1.e4 e5 2.Nf3 Nf6 3.Nxe5 d6

∗1.e4 e5 2.Nf3 Nc6 3.e5

● ○z . . . ●z . . . ○3 . . . ●3 ○2 . . . ●2 ○1 . . . ●1

●1 ○1 ●2 ○2 ●3 ○3 ●4 ○4

View: ●1 ○2 ●3 ○4

● ○z . . . ●z . . . ○3 . . . ●3 ○2 . . . ●2 ○1 . . . ●1

●1 ○1 ●2 ○2 ●3 ○3 ●4 ○4

View: ●1 ○2 ●3 ○4

● ○z . . . ●z . . . ○3 . . . ●3 ○2 . . . ●2 ○1 . . . ●1

●1 ○1 ●2 ○2 ●3 ○3 ●4 ○4

View: ●1 ○2 ●3 ○4

Strategies

A strategy for Player is a set σ of even length plays closed byeven length prefixes.s a ∈ σ means that σ suggests to play a as next move in theplay s.

A strategy for Opponent is a set τ of odd length plays closed byodd length prefixes.

Strategies

A strategy for Player is a set σ of even length plays closed byeven length prefixes.s a ∈ σ means that σ suggests to play a as next move in theplay s.

A strategy for Opponent is a set τ of odd length plays closed byodd length prefixes.

Bounded Strategies

A strategy is bounded by k ∈ N if every play in it is of the shape

● ○z . . . ●z . . . ○i . . . ●i . . . ○2 . . . ●2 ○1 . . . ●1 ○

with 2z ≤ k

Interactions Between Strategies

Let σ and τ be respectively a strategy for Player and a strategyfor Opponent over the arena A.

σ ⋆ τ

{s m ∣ (λ(m) = P Ô⇒ s m ∈ σ ∧ s ∈ τ)}

{s m ∣ (λ(m) = O Ô⇒ s m ∈ τ ∧ s ∈ σ)}

Interactions Between Strategies

Let σ and τ be respectively a strategy for Player and a strategyfor Opponent over the arena A.

σ ⋆ τ

{s m ∣ (λ(m) = P Ô⇒ s m ∈ σ ∧ s ∈ τ)}

{s m ∣ (λ(m) = O Ô⇒ s m ∈ τ ∧ s ∈ σ)}

Berardi-de’Liguoro Backtracking Level

Crossing edges:

●1 . . . ●2 . . . ○1 . . . ○2

Mind change about a mind change

Crossing edges:

●1 . . . ●2 . . . ○1 . . . ○2

Mind change about a mind change

An edge is active if all edges crossing it are inactive

An edge is inactive if it is crossed by an active edge

●1 ○1 ●2 ○2 ●3 ○3 ●4 ○4 ●5

An edge e1 is inactived by an edge e2

e1◁e2

if e1 is active immediately before e2 is played and e1 is crossedby e2

●1 ○1 ●2 ○2 ●3 ○3 ●4

●1 ○1 ●2 ○2 ●3 ○3 ●4 ○4

The edge from ○3 is inactivated by the edge from ○4

e1◁e2

●1 ○1 ●2 ○2 ●3 ○3 ●4

●1 ○1 ●2 ○2 ●3 ○3 ●4 ○4

e1◁e2

●1 ○1 ●2 ○2 ●3 ○3 ●4

●1 ○1 ●2 ○2 ●3 ○3 ●4 ○4

Backtracking level of a player p in play: the length of the longestchain of edges

e1◁e2◁ . . . ◁en

such that en is played by p

Backtracking level of a play: the maximum among thebacktracking levels of the players

●1 ○1 ●2 ○2 ●3 ○3 ●4 ○4 ●5 ○5

Backtracking level 2

e1◁e2◁ . . . ◁en

●1 ○1 ●2 ○2 ●3 ○3 ●4 ○4 ●5 ○5

e1◁e2◁ . . . ◁en

●1 ○1 ●2 ○2 ●3 ○3 ●4 ○4 ●5 ○5

Backtracking level of a strategy σ: the greatest among thebacktracking levels of the views of the plays in σ.

∗ x ∶= t1 y ∶= a1 x ∶= t2 y ∶= a2 x ∶= t3 y ∶= a3 z ∶= t4

Backtracking level of a strategy σ: the greatest among thebacktracking levels of the views of the plays in σ.

∗ x ∶= t1 y ∶= a1 x ∶= t2 y ∶= a2 x ∶= t3 y ∶= a3 z ∶= t4

Backtracking Levels Interaction

TheoremSuppose σ is a Player bounded strategy of backtracking level n.Suppose τ is an Opponent bounded strategy of backtrackinglevel m.Then for every s ∈ σ ⋆ τ , Player has in s backtracking level lessthan or equal to n and Opponent has in s backtracking levelless than or equal to m.

Min Backtracking Theorem

TheoremSuppose σ is a Player strategy bounded by k and ofbacktracking level 1.Suppose τ is an Opponent strategy bounded by k and ofbacktracking level m.Then for every s ∈ σ ⋆ τ , the length of s is less than

2k(log k)⋅2

Backtracking level 1 : Excluded Middle for formulas with 1quantifier.

2k(log k)⋅2

Backtracking level 1 : Excluded Middle for formulas with 1quantifier.

22k(log k)⋅2

Backtracking level 2 : Excluded Middle for formulas with 2quantifiers.

22k(log k)⋅2

Backtracking level 2 : Excluded Middle for formulas with 2quantifiers.

TheoremSuppose σ is a Player strategy bounded by k and ofbacktracking level n.Suppose τ is an Opponent strategy bounded by k and ofbacktracking level m.Suppose b = min(n,m,d −2), where d is the depth of the arena.Then for every s ∈ σ ⋆ τ , the length of s is less than

22..2k(log k)⋅2

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶b+1

References

F. Aschieri, Game Semantics and the Geometry ofBacktracking: a New Complexity Analysis of Interaction,Preprint on Arxiv, November 2015.

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