Game Semantics and the Complexity of Interaction (about a ...stefano/Berardi-Complexity-Talk-December-2015... · Game Semantics and the Complexity of Interaction (about a result of

Game Semanticsand the Complexity of Interaction

(about a result of Federico Aschieri)

Stefano Berardi

C.S. Dept. Torino University (Italy)

Talk at: Workshop onEfficient and Natural Proof Systems

University of Bath (UK), December 16, 2015

Stefano Berardi Game Semantics and the Complexity of Interaction (about a result of Federico Aschieri)

Complexity of Cut-Elimination for first order ClassicalLogic

⋮

Γ,A⋮

Γ,A�cut

Γ

if r is the height of A as formula tree

if k is the height of the proof tree in Sequent Calculus

then the height of the normal form of the tree is at most

22..2k

²r+1



⋮

Γ,A⋮

Γ,A�cut

Γ




22..2k

²r+1



⋮

Γ,A⋮

Γ,A�cut

Γ




22..2k

²r+1


Complexity of Cut-Elimination is only a worst-caseanalysis

Figure : New York in the year 2015


Complexity of Cut-Elimination

Worst case height: the New World Trade Center, 546 meters


Complexity of Cut-Elimination

Better worst-case analysis


Cut-Elimination for quantifiers

⋮

Γ,A⋮

Γ,A�cut

Γ

Γ,∃x B,B[t/x](t first-order term)

Γ,∃x B

Γ,B[a/x](a eigenvariable)

Γ,∀x B

We concentrate over eliminating cuts on a quantifier becausethis process, when applied to a proof of a simply existentialformula, defines an Herbrand disjunction for it (a disjunction ofinstances of its body which is a tautology).


Cut-Elimination for quantifiers

⋮

Γ,A⋮

Γ,A�cut

Γ

Γ,∃x B,B[t/x](t first-order term)

Γ,∃x B

Γ,B[a/x](a eigenvariable)

Γ,∀x B

We concentrate over eliminating cuts on a quantifier becausethis process, when applied to a proof of a simply existentialformula, defines an Herbrand disjunction for it (a disjunction ofinstances of its body which is a tautology).


Proofs and Expansion Trees

An expansion tree is a graph structure underlying a prooftree in sequent calculus.Cut-elimination over expansion trees for A may beinterpreted as a Tarski game between a player O negatingthe validity of the formula A and a player P asserting it.A move of P is the choice of a would-be witness for anexistential quantifier, a move of O is the choice of awould-be counterexample for an universal quantifier.Player P may “backtrack”, coming back to a previousposition of the play and making a different move.We only consider terminating player, P wins if thedisjunction of all propositional formulas considered is atautology, otherwise O wins.


Expansion Trees as game strategy

∃x ∀y ∃z P(x ,y ,z)

∀y ∃z P(t3,y ,z)

∃z P(t3,a3,z)

a3

t3

∀y ∃z P(t2,y ,z)

∃z P(t2,a2,z)

a2

t2

∀y ∃z P(t1,y ,z)

∃z P(t1,a1,z)

P(t1,a1, t4)

t4

a1

t1

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)

a3

t3

∀y ∃z P(t2,y ,z)

∃z P(t2,a2,z)

a2

t2

∀y ∃z P(t1,y ,z)

∃z P(t1,a1,z)

P(t1,a1, t4)

t4

a1

t1


Winning: P(t1,a1, t4)



∃x ∀y ∃z P(x ,y ,z)

∀y ∃z P(t3,y ,z)

∃z P(t3,a3,z)

a3

t3

∀y ∃z P(t2,y ,z)

∃z P(t2,a2,z)

a2

t2

∀y ∃z P(t1,y ,z)

∃z P(t1,a1,z)

P(t1,a1, t4)

t4

a1

t1


Winning: P(t1,a1, t4)Stefano Berardi Game Semantics and the Complexity of Interaction (about a result of Federico Aschieri)

Expansion Tree Strategies

∀x ∃y ∀z ¬P(x ,y ,z)

∃y ∀z ¬P(b,y ,z)

∀z ¬P(b,u3,z)

¬P(b,u3,b3)

b3

u3

∀z ¬P(b,u2,z)

¬P(b,u2,b2)

b2

u2

∀z ¬P(b,u1,z)

¬P(b,u1,b1)

b1

u1

b

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)

b3

u3

∀z ¬P(b,u2,z)

¬P(b,u2,b2)

b2

u2

∀z ¬P(b,u1,z)

¬P(b,u1,b1)

b1

u1

b





∀x ∃y ∀z ¬P(x ,y ,z)

∃y ∀z ¬P(b,y ,z)

∀z ¬P(b,u3,z)

¬P(b,u3,b3)

b3

u3

∀z ¬P(b,u2,z)

¬P(b,u2,b2)

b2

u2

∀z ¬P(b,u1,z)

¬P(b,u1,b1)

b1

u1

b




Backtracking and the justification relation

We represent a backtracking of player P by an arrow fromthe move done by P and the move of O to which P isanswering.If P answer to the last move of O, then there is nobacktrackingIf P answer to some older move of O, this is backtracking.This arrow is called a justification.


Expansion Tree and the justification relation

∃x ∀y ∃z P(x ,y ,z)

∀y ∃z P(t3,y ,z)

∃z P(t3,a3,z)

a3

t3

∀y ∃z P(t2,y ,z)

∃z P(t2,a2,z)

a2

t2

∀y ∃z P(t1,y ,z)

∃z P(t1,a1,z)

P(t1,a1, t4)

t4

a1

t1


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


Expansion Tree and the justification relation

∃x ∀y ∃z P(x ,y ,z)

∀y ∃z P(t3,y ,z)

∃z P(t3,a3,z)

a3

t3

∀y ∃z P(t2,y ,z)

∃z P(t2,a2,z)

a2

t2

∀y ∃z P(t1,y ,z)

∃z P(t1,a1,z)

P(t1,a1, t4)

t4

a1

t1


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


Simple backtracking: the case of no crossing arrows

∀x ∃y ∀z ¬P(x ,y ,z)

∃y ∀z ¬P(b,y ,z)

∀z ¬P(b,u3,z)

¬P(b,u3,b3)

b3

u3

∀z ¬P(b,u2,z)

¬P(b,u2,b2)

b2

u2

∀z ¬P(b,u1,z)

¬P(b,u1,b1)

b1

u1

b

Order: b a1 u1 b1 u2 b2 u3 b3

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


Simple backtracking: the case of no crossing arrows

∀x ∃y ∀z ¬P(x ,y ,z)

∃y ∀z ¬P(b,y ,z)

∀z ¬P(b,u3,z)

¬P(b,u3,b3)

b3

u3

∀z ¬P(b,u2,z)

¬P(b,u2,b2)

b2

u2

∀z ¬P(b,u1,z)

¬P(b,u1,b1)

b1

u1

b

Order: b a1 u1 b1 u2 b2 u3 b3

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


Backtracking and Complexity of Cut-Elimination

We may stratify proofs and expansion trees according to ameasure called the level of backtracking, which depends onhow arrows are crossing each other. When backtracking level islow, the complexity of cut-elimination is much less than thedegree of the cut-formula:

⋮

Γ,A⋮

Γ,A�cut

Γ

r is the height of A as formula tree

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

22..2k

²b+1

We may transform the proof in such a way that: 1 ≤ b ≤ r − 2




⋮

Γ,A⋮

Γ,A�cut

Γr is the height of A as formula tree


22..2k

²b+1





⋮

Γ,A⋮

Γ,A�cut



22..2k

²b+1





⋮

Γ,A⋮

Γ,A�cut



22..2k

²b+1



Game Semantics with Backtracking for Logic

First introduced by Lorentzen, without cut elimination

Coquand proves cut elimination for backtracking: ASemantics of Evidence for Classical Arithmetic (1991,1995)Herbelin: correspondence between plays andcut-elimination (1995)Coquand: notion of backtracking level 1, conjecture thatlevel 1 defines a logic (1991)Berardi-de’Liguoro: notion of backtracking level n ∈ N(2009)Berardi-Coquand-Hayashi-Makoto: backtracking level n ∈ Ndefines the sub-classical arithmetic with Excluded Middleon Σ0

n-formulas (various papers 2010-2013)



First introduced by Lorentzen, without cut eliminationCoquand proves cut elimination for backtracking: ASemantics of Evidence for Classical Arithmetic (1991,1995)

Herbelin: correspondence between plays andcut-elimination (1995)Coquand: notion of backtracking level 1, conjecture thatlevel 1 defines a logic (1991)Berardi-de’Liguoro: notion of backtracking level n ∈ N(2009)Berardi-Coquand-Hayashi-Makoto: backtracking level n ∈ Ndefines the sub-classical arithmetic with Excluded Middleon Σ0




First introduced by Lorentzen, without cut eliminationCoquand proves cut elimination for backtracking: ASemantics of Evidence for Classical Arithmetic (1991,1995)Herbelin: correspondence between plays andcut-elimination (1995)

Coquand: notion of backtracking level 1, conjecture thatlevel 1 defines a logic (1991)Berardi-de’Liguoro: notion of backtracking level n ∈ N(2009)Berardi-Coquand-Hayashi-Makoto: backtracking level n ∈ Ndefines the sub-classical arithmetic with Excluded Middleon Σ0




First introduced by Lorentzen, without cut eliminationCoquand proves cut elimination for backtracking: ASemantics of Evidence for Classical Arithmetic (1991,1995)Herbelin: correspondence between plays andcut-elimination (1995)Coquand: notion of backtracking level 1, conjecture thatlevel 1 defines a logic (1991)

Berardi-de’Liguoro: notion of backtracking level n ∈ N(2009)Berardi-Coquand-Hayashi-Makoto: backtracking level n ∈ Ndefines the sub-classical arithmetic with Excluded Middleon Σ0




First introduced by Lorentzen, without cut eliminationCoquand proves cut elimination for backtracking: ASemantics of Evidence for Classical Arithmetic (1991,1995)Herbelin: correspondence between plays andcut-elimination (1995)Coquand: notion of backtracking level 1, conjecture thatlevel 1 defines a logic (1991)Berardi-de’Liguoro: notion of backtracking level n ∈ N(2009)

Berardi-Coquand-Hayashi-Makoto: backtracking level n ∈ Ndefines the sub-classical arithmetic with Excluded Middleon Σ0




First introduced by Lorentzen, without cut eliminationCoquand proves cut elimination for backtracking: ASemantics of Evidence for Classical Arithmetic (1991,1995)Herbelin: correspondence between plays andcut-elimination (1995)Coquand: notion of backtracking level 1, conjecture thatlevel 1 defines a logic (1991)Berardi-de’Liguoro: notion of backtracking level n ∈ N(2009)Berardi-Coquand-Hayashi-Makoto: backtracking level n ∈ Ndefines the sub-classical arithmetic with Excluded Middleon Σ0



Game Semantics and Semantics of ProgrammingLanguages

Hyland-Ong: Full Abstraction for PCF (2000)

Danos-Herbelin-Regnier: correspondence between playsand linear head reduction (1996)

Russ Harmer (2007) independently rediscovers1-backtracking for λ-terms (cellar strategies).

No characterization is known for λ-terms withn-bactracking to the date.




















Some Game Terminology: 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. For sake of simplicity weassume that labels alternate. An example of chess play:

∗ e4 e5 Nf3 Nf6 Nxe5

Black player replaces his previous move Nf6 with Nc6.

∗ e4 e5 Nf3 Nf6 Nxe5 Nc6
























Play: a justified sequence of moves. For sake of simplicity weassume that labels alternate.

An example of chess play:























Arenas: an example of backtracking with crossingarrows.

∗ 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


Views

We use white ○, black ● for any pair of opposite players.Each player may ignore the fact that the other playerbacktracks, in order to simplify his choice for a move.

View may be defined recursively by looking alternatively tothe previous move when we just moved, and to the movethe opponent is pointing to when the opponent just moved.

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

View greatly reduces the moves to be consider.

●1 ○1 ●2 ○2 ●3 ○3 ●4 ○4

View of Black player in the example above: ●1 ○2 ●3 ○4


Views

We use white ○, black ● for any pair of opposite players.Each player may ignore the fact that the other playerbacktracks, in order to simplify his choice for a move.View may be defined recursively by looking alternatively tothe previous move when we just moved, and to the movethe opponent is pointing to when the opponent just moved.

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


●1 ○1 ●2 ○2 ●3 ○3 ●4 ○4



Views


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


●1 ○1 ●2 ○2 ●3 ○3 ●4 ○4



Views


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


●1 ○1 ●2 ○2 ●3 ○3 ●4 ○4



Strategies

A strategy for Player is a set σ of even length plays closedby even 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 playsclosed by odd length prefixes.


Strategies

A strategy for Player is a set σ of even length plays closedby even 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 playsclosed by odd length prefixes.


Bounded Strategies

A strategy for player p is innocent (Coquand uses the wordcut-free) if p only backtracks to p-visible moves, and onlyuses p-visible moves to decide the next move.

Alternatively, a strategy for the White player is innocentevery play in it is of the shape

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

The part of the play from any ○i to ●i is irrelevant for theWhite strategy and may be omitted.An innocent White strategy is bounded by k ∈ N if the partof the play visible by White has length ≤ k (i.e., if in thepicture above we always have 2z ≤ k). The actual play maybe much longer.


Bounded Strategies

A strategy for player p is innocent (Coquand uses the wordcut-free) if p only backtracks to p-visible moves, and onlyuses p-visible moves to decide the next move.Alternatively, a strategy for the White player is innocentevery play in it is of the shape

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

The part of the play from any ○i to ●i is irrelevant for theWhite strategy and may be omitted.

An innocent White strategy is bounded by k ∈ N if the partof the play visible by White has length ≤ k (i.e., if in thepicture above we always have 2z ≤ k). The actual play maybe much longer.


Bounded Strategies

A strategy for player p is innocent (Coquand uses the wordcut-free) if p only backtracks to p-visible moves, and onlyuses p-visible moves to decide the next move.Alternatively, a strategy for the White player is innocentevery play in it is of the shape

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

The part of the play from any ○i to ●i is irrelevant for theWhite strategy and may be omitted.An innocent White strategy is bounded by k ∈ N if the partof the play visible by White has length ≤ k (i.e., if in thepicture above we always have 2z ≤ k). The actual play maybe much longer.


Interactions Between Strategies

Let σ and τ be respectively a strategy for Player and a strategyfor Opponent over the arena A. We denote with σ ⋆ τ theunique play between the two strategies, defined as follows:

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. We denote with σ ⋆ τ theunique play between the two strategies, defined as follows:

σ ⋆ τ

=

{s m ∣ (λ(m) = P Ô⇒ s m ∈ σ ∧ s ∈ τ)}

∪

{s m ∣ (λ(m) = O Ô⇒ s m ∈ τ ∧ s ∈ σ)}


The notion of Backtracking Level

Backtracking of level one corresponds to no crossing-edge.Crossing edges are of level ≥ 2 and correspond toretracting a previous retraction of a move, restoring someolder play:

●1 . . . ●2 . . . ○1 . . . ○2

Level 1 backtracking corresponds to monotonic learning,when we retract a move we never restore it.Backtracking of level ≥ 2 corresponds to non-monotoniclearning. We may retract a move, but later on we may findout that we were wrong in retracting it, and we restore it.


The notion of Backtracking Level

Backtracking of level one corresponds to no crossing-edge.Crossing edges are of level ≥ 2 and correspond toretracting a previous retraction of a move, restoring someolder play:

●1 . . . ●2 . . . ○1 . . . ○2

Level 1 backtracking corresponds to monotonic learning,when we retract a move we never restore it.Backtracking of level ≥ 2 corresponds to non-monotoniclearning. We may retract a move, but later on we may findout that we were wrong in retracting it, and we restore it.


Backtracking Level: the notion of active edge

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

1 The two red arrows in the right-hand-side are crossed byno arrow and therefore are active.

2 The black arrow is crossed by some active arrow andtherefore inactive.

3 The red arrow in the left-hand-side is only crossed byinactive arrows and therefore it is active.





●1 ○1 ●2 ○2 ●3 ○3 ●4 ○4 ●5








●1 ○1 ●2 ○2 ●3 ○3 ●4 ○4 ●5





Backtracking Level: the notion of inactivation

An edge e1 is inactived by an edge e2

e1◁e2

if e1 is active immediately before e2 is played

●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. Thisre-activates the edge from ○2.




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



Finite Backtracking Level

Backtracking level of a player p in play in the finite case: thelength of the longest chain of edges

e1◁e2◁ . . . ◁en

such that en is played by p (for an infinite play the definition ismore complex).

Backtracking level of a finite 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 for a strategy

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

Above: an example of backtracking level 2


Backtracking Level for a strategy

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

Above: an example of backtracking level 2


Backtracking Level of an interaction

The backtracking level of a play interaction of two strategies isthe maximum of the level of backtracking of the two strategies,and each player only sees his backtracking.

Theorem (Maximum level)Suppose 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.


Minimum Backtracking Theorem, level 1

The minimum of the backtracking level of the two playersprovides a bound the number of level of exponential of thenormalization procedure.

Theorem (For level 1)Suppose 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 interprets classical proofs which useExcluded Middle for formulas with 1 quantifier.


Minimum Backtracking Theorem, level 1



2k(log k)⋅2

Backtracking level 1 interprets classical proofs which useExcluded Middle for formulas with 1 quantifier.


Minimum Backtracking Theorem, for level 2



22k(log k)⋅2

Backtracking level 2 interprets classical proofs which useExcluded Middle for formulas with 2 quantifiers.


Minimum Backtracking Theorem, for level 2



22k(log k)⋅2

Backtracking level 2 interprets classical proofs which useExcluded Middle for formulas with 2 quantifiers.


Minimum Backtracking Theorem

Theorem (For any level and any degree of cut formula)

Suppose 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

1 Federico Aschieri, Game Semantics and the Geometry ofBacktracking: a New Complexity Analysis of Interaction,Preprint on Arxiv, November 2015,http://arxiv.org/abs/1511.06260.

2 Stefano Berardi, Ugo deLiguoro, Toward the Interpretationof non-Constructive Reasoning as non-MonotonicLearning, Information and Computation, vol. 207, n. 1, pp.6381, 2009.

3 Thierry Coquand, A Semantics of Evidence for ClassicalArithmetic, Journal of Symbolic Logic, vol. 60, n. 1, pp.325-337, 1995.

4 Russ Harmer. Cellular strategies and innocent interaction.2007. Hal, archives-ouvertes.fr.https://hal.archives-ouvertes.fr/hal-00150353


http://arxiv.org/abs/1511.06260

https://hal.archives-ouvertes.fr/hal-00150353

Documents

Game Semantics and the Complexity of Interaction (about a ...stefano/Berardi-Complexity-Talk-December-2015... · Game Semantics and the Complexity of Interaction (about a result of