Projecting games on hypercoherences

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Projecting games on hypercoherences

polarized bordered games

Pierre Boudes

IML, Marseille, France

Projecting games on hypercoherences – p.1/12

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Games and hypercoherences

Denotational semantics:

if π →_∗ π⁰ then [π] = [π⁰]

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Games and hypercoherences

if π →_∗ π⁰ then [π] = [π⁰]

Games Hypercoherences

traces of computation results of interaction (two-players plays) (points/vertices)

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Games and hypercoherences

if π →_∗ π⁰ then [π] = [π⁰]

traces of computation results of interaction (two-players plays) (points/vertices) + coherence (hyperedges)

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Games and hypercoherences

if π →_∗ π⁰ then [π] = [π⁰]

traces of computation results of interaction (two-players plays) (points/vertices) + coherence (hyperedges) Agents (proofs or programs) are:

strategies cliques

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Games and hypercoherences

if π →_∗ π⁰ then [π] = [π⁰]

traces of computation results of interaction (two-players plays) (points/vertices) + coherence (hyperedges) Agents (proofs or programs) are:

strategies cliques

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First answer: extensional collapse

Sequential algorithms Hypercoherences strategies extensional collapse cliques

Ehrhard 1999

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First answer: extensional collapse

Ehrhard 1999

algorithm function

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First answer: extensional collapse

Ehrhard 1999

algorithm function

Quotient by partial equivalence relations (Kreisel 50’):

x ∼_ι x and f ∼_σ→τ g iff x ∼_σ y =⇒ f (x) ∼_τ g(y)

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First answer: extensional collapse

Ehrhard 1999

algorithm function

➤ Extended to others games semantics by Melliès 2004 and Laird (200?).

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First answer: extensional collapse

Ehrhard 1999

algorithm function

➤ Better detailed by Melliès 2003.

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First answer: extensional collapse

Ehrhard 1999

algorithm function

➤ Better detailed by Melliès 2003.

➤ Limited to simple types (σ := ι | σ → σ).

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Another bridge: direct projection

Games plays

Hypercoherences points

time-forgetting projection

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Another bridge: direct projection

Games strategies

Hypercoherences sets of points in general, the image of a strategy is not a clique

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Another bridge: direct projection

Games strategies

Relational model relations

relational model ≈ hypercoherences without coherence time-forgetting

projection

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Another bridge: direct projection

Games strategies

➤ Timeless games, Baillot, Danos, Ehrhard & Regnier 1997: lax functor.

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Another bridge: direct projection

Logic proofs

Games strategies

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Another bridge: direct projection

Logic (linear and polarized) proofs

Games strategies

➤ Here: commutation by use of polarized bordered games for the linear subsystem of linear logic with polarities (MALLPol).

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Another bridge: direct projection

Logic (linear and polarized) proofs

Games strategies

Hypercoherences cliques

➤ Here: commutation by use of polarized bordered games for the linear subsystem of linear logic with polarities (MALLPol).

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Linear logic with polarities (Laurent)

positives P := 0 | 1 | α^⊥ | P ⊕ P | P ⊗ P | !N negatives N := > | ⊥ | α | N & N | N N | ?P

(ax.)

`α, α^⊥ (one)

` 1

` Γ

(bot)

`⊥,Γ (top)

`>,Γ

` Γ,N ` Γ,N⁰

(with)

`Γ,N & N⁰

` N,P_i (i = 1,2)

(plus)

`N,P₁ ⊕ P₂

` N,N⁰,Γ

(par)

`N N⁰,Γ

` N,P ` N⁰,P⁰

(tens.)

`N,N⁰,P ⊗ P⁰

` N,P

(der.)

`N,?P

` ?P₁, . . . , ?P_n,N

(prom.)

`?P₁, . . . ,?P_n,!N

` Γ

(weak.)

`?P,Γ

` ?P,?P,Γ

(cont.)

`?P,Γ

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Linear logic with polarities (Laurent)

positives P := 0 | 1 | α^⊥ | P ⊕ P | P ⊗ P | ↓N negatives N := > | ⊥ | α | N & N | N N | ↑P

(ax.)

`α, α^⊥ (one)

` 1

` Γ

(bot)

`⊥,Γ (top)

`>,Γ

` Γ,N ` Γ,N⁰

(with)

`Γ,N & N⁰

` N,P_i (i = 1,2)

(plus)

`N,P₁ ⊕ P₂

` N,N⁰,Γ

(par)

`N N⁰,Γ

` N,P ` N⁰,P⁰

(tens.)

`N,N⁰,P ⊗ P⁰

` N,P

(der.)

`N,↑P

` ↑P₁, . . . , ↑P_n,N

(prom.)

`↑P₁, . . . ,↑P_n,↓N

` Γ

(weak.)

`?P,Γ

` ?P,?P,Γ

(cont.)

`?P,Γ

` N,N^⊥ ` N,Γ

(cut)

`N,Γ

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Polarized bordered games

Games:

positive A = (+, Aô, A^p, S_A) S_A ⊆ (A^p · Aô)^∗ negative A = (−, Aô, A^p, S_A) S_A ⊆ (Aô · A^p)^∗

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Polarized bordered games

Games:

positive A = (+, Aô, A^p, S_A) S_A ⊆ (A^p · Aô)^∗ negative A = (−, Aô, A^p, S_A) S_A ⊆ (Aô · A^p)^∗ Well-termination:

s, s⁰ ∈ S, s ≤ s⁰ =⇒ s = s⁰

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Polarized bordered games

Games:

positive A = (+, Aô, A^p, S_A) S_A ⊆ (A^p · Aô)^∗ negative A = (−, Aô, A^p, S_A) S_A ⊆ (Aô · A^p)^∗ Well-termination:

s, s⁰ ∈ S, s ≤ s⁰ =⇒ s = s⁰

Strategy σ ⊆ S_A s.t.:

s, s⁰ ∈ σ, s , s⁰ =⇒ length(s ∧ s⁰) is











even, if A < 0 odd, if A > 0

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Interleaving of words

➤ A ∩ B = ∅, u ∈ A^∗, v ∈ B^∗

u •_A_,_B v = {w | w A = u, w B = v}

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Interleaving of words

➤ A ∩ B = ∅, u ∈ A^∗, v ∈ B^∗

u •_A_,_B v = {w | w A = u, w B = v}

➤ E = A · A, F = B · B, u ∈ E^∗, v ∈ F^∗

u _A,B v = u •_E,F v

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Interleaving of words

➤ A ∩ B = ∅, u ∈ A^∗, v ∈ B^∗

u •_A_,_B v = {w | w A = u, w B = v}

➤ E = A · A, F = B · B, u ∈ E^∗, v ∈ F^∗

u _A,B v = u •_E,F v

a · u · a⁰ ⊗_A_,_B b · v · b⁰ = {(a, b) · w · (a⁰, b⁰) | w ∈ u _A_,_B v}

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Interpretation of MALLpol/LLpol

➤ Follows the pattern of Laurent’s polarized games 2002

➤ Orthogonal: exchanges opponant and proponent

➤ Additives: disjoint unions (S_A⊕B = S_A ] S_B)

➤ Multiplicatives: S_A⊗B = {s ⊗_A_,_B s⁰ | s ∈ S_A, s ∈ S_B}

➤ shifts: S↓A = {∗ · s · ∗⁰ | s ∈ S_A}

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Interpretation of MALLpol/LLpol

➤ Follows the pattern of Laurent’s polarized games 2002

➤ Orthogonal: exchanges opponant and proponent

➤ Additives: disjoint unions (S_A⊕B = S_A ] S_B)

➤ Multiplicatives: S_A⊗B = {s ⊗_A_,_B s⁰ | s ∈ S_A, s ∈ S_B}

➤ shifts: S↓A = {∗ · s · ∗⁰ | s ∈ S_A}

➤ exponentials: !N = ↓ ]N (some choice for ])

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Projection

MALLpol proofs

PBG strategies

projection

p([A]PBG) = [A]Rel = |[A]Hc| p([π]PBG) = [π]Rel = [π]Hc

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Projection

MALLpol proofs

PBG strategies

projection

p([A]PBG) = [A]Rel = |[A]Hc| p([π]PBG) = [π]Rel = [π]Hc

Reversibility:

reverse(S_A) S_A

reverse([π]PBG) = [π]PBG

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Projection

LLpol proofs

PBG strategies

projection

p([A]PBG) ⊆ [A]Rel ⊇ |[A]Hc| p([π]PBG) ⊆ [π]Rel ⊇ [π]Hc

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Projection

LLpol proofs

PBG strategies

Non uniform hypercoherences

cliques projection

p([A]PBG) ⊆ [A]Rel = |[A]NHc| p([π]PBG) ⊆ [π]Rel = [π]NHc

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Hypercoherences unfolding

LLpol

PBG Hypercoherences

unfolding

unfold(hypergraph) = tree

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Hypercoherences unfolding

LLpol

PBG Hypercoherences

unfolding

➤ Under some hypothesis:

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Hypercoherences unfolding

LLpol

PBG Hypercoherences

unfolding

➤ unfold(X ⊗ Y) unfold(X) ⊗ unfold(Y)

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Hypercoherences unfolding

LLpol

PBG Hypercoherences

unfolding

➤ Sequential algorithms exponential unfold(!X) ↓ ]_s unfold(X)

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Hypercoherences unfolding

LLpol

PBG Hypercoherences

unfolding

➤ Sequential algorithms exponential unfold(!X) ↓ ]_s unfold(X)

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Hypercoherences unfolding

hypercoherence a

b

c d

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Hypercoherences unfolding

hypercoherence a

b

c d

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Hypercoherences unfolding

hypercoherence a

b

c d

{a, b, c, d} {a, b, c}

{a, b} a b

{a, c} a c

d

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Future works

➤ Merging the two approaches (collapse and projection)

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Future works

➤ Bordered games on DAGs and unfolding

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Future works

➤ Going back to Laurent’s polarized games

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Future works

➤ relation with bordered games

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Future works

➤ direct projection (tech. report)

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Future works

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