Non-uniform Hypercoherences

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Non-uniform Hypercoherences

Pierre Boudes

Institut de Math´ematiques de Luminy

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What’s this talk about?

Denotational semantics (of λ-calculus, ^PCF, ^LL) proofs, terms → some math. structures: agents

sequentiality & determinism. No concurrency Internal interactivity of calculus

extensionality (^PER), extensional collapse (quotient) uniformity: the stage where agents interact

Exponential modalities of linear logic (! and ?) resource management

complexity issue (^ELL, ^LLL), polarities, . . .

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What’s this talk about?

Denotational semantics (of λ-calculus, ^PCF, ^LL)

proofs, terms → some math. structures: agents sequentiality & determinism. No concurrency

Internal interactivity of calculus

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What’s this talk about?

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What’s this talk about?

Internal interactivity of calculus extensionality (^PER), extensional collapse (quotient)

uniformity: the stage where agents interact Exponential modalities of linear logic (! and ?)

resource management

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What’s this talk about?

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What’s this talk about?

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What’s this talk about?

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What’s this talk about?

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What’s this talk about?

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Uniformity at a first sight

P = λb. if b then { if b then t₁ else t₂ } else { if b then t₃ else t₄ } P:Bool → Bool ∼= !Bool Bool

t_i^• ∈ {t, f}

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Uniformity at a first sight

t_i^• ∈ {t, f}

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Uniformity at a first sight

t_i^• ∈ {t, f}

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Uniformity at a first sight

t_i^• ∈ {t, f}

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Uniformity at a first sight

t_i^• ∈ {t, f}

Coherence spaces and hypercoherences semantics (set-based) are uniform

P ^• = {({t}, t ^•), ({f}, t ^•)}

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Uniformity at a first sight

t_i^• ∈ {t, f}

Relational semantics is non-uniform

P ^• = {([t, t], t1•), ([t, f], t2•), ([t, f], t3•), ([f, f], t4•)}

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Uniformity at a first sight

t_i^• ∈ {t, f}

Multiset-based coherence spaces and

hypercoherences semantics are also uniform P ^• = {([t, t], t ^•), ([f, f], t ^•)}

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Denotational semantics

Static semantics : Agents are sets of points

additives → disjoint unions, ∅

multiplicatives → product of sets, {∗}

|A B| = |A B| = |A ⊗ B| = |A| × |B|

Dynamic semantics : Agents are sets of pol. paths

multiplicatives→ interleaving of paths, {ε}, {∗^O}, {∗^P} exponentials → interleaving of paths in agents

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Denotational semantics

Static semantics: Agents are sets of points additives → disjoint unions, ∅

|A B| = |A B| = |A ⊗ B| = |A| × |B|

Dynamic semantics : Agents are sets of pol. paths

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Denotational semantics

|A B| = |A B| = |A ⊗ B| = |A| × |B|

Dynamic semantics: Agents are sets of pol. paths additives → disjoint unions, ∅

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Denotational semantics

|A B| = |A B| = |A ⊗ B| = |A| × |B|

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Denotational semantics

|A B| = |A B| = |A ⊗ B| = |A| × |B|

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Denotational semantics

|A B| = |A B| = |A ⊗ B| = |A| × |B|

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Denotational semantics

exponentials → finite sets or multisets , agents

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Denotational semantics

exponentials → finite sets or multisets, agents Dynamic semantics: Agents are sets of pol. paths

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Denotational semantics

multiplicatives→ interleaving of paths, {ε}, {∗^O}, {∗^P} exponentials → interleaving of paths

in agents

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Denotational semantics

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Denotational semantics

uniformity

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Denotational semantics

Static semantics bridge?

Dynamic semantics

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Denotational semantics

Static semantics Hypercoh.

bridge? Ext. Coll. E^HRHARD [Ehr96]

Dynamic semantics Seq. Alg.

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Denotational semantics

bridge?

PER ∼. Equality on basis types and :

f ∼_A→B g iff x ∼_A y =⇒ f(x) ∼_B f(y) (∀x, y)

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Denotational semantics

bridge?

one-way bridge abstract result

Only simple types. Not the full power of linear logic.

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Denotational semantics

bridge?

one-way bridge abstract result Only simple types. Not the full power of linear logic.

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Denotational semantics

bridge?

one-way bridge abstract result

Only simple types. Not the full power of linear logic.

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Denotational semantics

bridge?

Dynamic semantics Seq. Alg. HO non-unif.

L^AIRD [Lai01]

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Denotational semantics

Static semantics Hypercoh. non-unif.

bridge?

Shifting to non-uniformity might improve the bridge

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Denotational semantics

Static semantics Hypercoh. non-unif.

bridge?

Shifting to non-uniformity might improve the bridge

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Uniform spaces

Coherence space X = (|X|, ^__X): graph

{[a, a] | a ∈ |X|} ⊆ ^__X ⊆ M_{₂_}(|X|)

Hypercoherence X = (|X|, ^__X): hypergraph {{a} | a ∈ |X|} ⊆ ^__X ⊆ P_fin^∗ (|X|)

Reflexivity

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Uniform spaces

{[a, a] | a ∈ |X|} ⊆ ^__X ⊆ M_{₂_}(|X|) Hypercoherence X = (|X|, ^__X): hypergraph

{{a} | a ∈ |X|} ⊆ ^__X ⊆ P_fin^∗ (|X|)

Reflexivity

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Uniform spaces

{{a} | a ∈ |X|} ⊆ ^__X ⊆ P_fin^∗ (|X|) Reflexivity

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Uniform spaces

Powers

A Power P is a functor from the category of sets and

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Uniform spaces

A Power P is a functor from the category of sets and inclusions to itself.

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Uniform spaces

{{a} | a ∈ |X|} ⊆ ^__X ⊆ P_fin^∗ (|X|) Reflexivity : ∪_a∈|X_|P ({a}) ⊆ ^__X

A Power P is a functor from the category of sets and

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Uniform spaces

{{a} | a ∈ |X|} ⊆ ^__X ⊆ P_fin^∗ (|X|) Reflexivity : ∪_a∈|X_|P ({a}) ⊆ ^__X

Orthogonal : X^⊥

|X^⊥| = |X| and ^_ = _^

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Uniform spaces

{{a} | a ∈ |X|} ⊆ ^__X ⊆ P_fin^∗ (|X|) Reflexivity : ∪_a∈|X_|P ({a}) ⊆ ^__X ^_

Orthogonal : X^⊥

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Uniform spaces

{{a} | a ∈ |X|} ⊆ ^__X ⊆ P_fin^∗ (|X|)

Reflexivity : ∪_a∈|X_|P ({a}) ⊆ ^__X ^_ _^

P (|X|) Orthogonal : X^⊥

|X^⊥| = |X| and ^_ = _^

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Uniform spaces

{{a} | a ∈ |X|} ⊆ ^__X ⊆ P_fin^∗ (|X|)

P (|X|)

^

_ Orthogonal : X^⊥

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Uniform spaces

{{a} | a ∈ |X|} ⊆ ^__X ⊆ P_fin^∗ (|X|)

P (|X|)

^

_ Agent x ⊆ |X|, P (x) ⊆ ^__X

Counter-agent y ⊆ |X|, P (y) ⊆ _^_X Determinism ](x ∩ y) ≤ 1

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Uniform spaces

{{a} | a ∈ |X|} ⊆ ^__X ⊆ P_fin^∗ (|X|)

P (|X|)

^

_ Agent x ⊆ |X|, P (x) ⊆ ^__X

Determinism ](x ∩ y) ≤ 1

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Uniform spaces

{{a} | a ∈ |X|} ⊆ ^__X ⊆ P_fin^∗ (|X|)

P (|X|)

^

_ Agent x ⊆ |X|, P (x) ⊆ ^__X

Counter-agent y ⊆ |X|, P (y) ⊆ _^_X

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Non-uniform spaces

P -coherence space (|X|, ^__X, _^_X) ^_ _^

P (|X|) N^X

^

_

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Non-uniform spaces

P (|X|) N^X

^ No longer reflexive. Neutrality. _

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Non-uniform spaces

P (|X|) N^X

eg, for the power M_{₂_}, one can have both a ^_ b and a _^ b with a 6= b and also c _ c, d ^ d.

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Non-uniform spaces

P (|X|) N^X

In [BE01], B^UCCIARELLI & E^HRHARD have set a general framework where each phase-valued provability semantics of an indexed linear logic gives a non-uniform denotational semantics of linear logic.

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Non-uniform spaces

P (|X|) N^X

In [BE01], B^UCCIARELLI & E^HRHARD have set a general framework where each phase-valued provability semantics of an indexed linear logic gives a non-uniform denotational semantics of linear logic.

→ a class of non-uniform coherence semantics which

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Non-uniform spaces

M_K-coherence space (|X|, ^__X, _^_X) ^_ _^

P (|X|) N^X

^

_

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Non-uniform spaces

P (|X|) N^X

^

_

MALL model: standard pattern

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Non-uniform spaces

P (|X|) N^X

^

_

MALL model: standard pattern

eg, |X Y | = |X| × |Y | and, for s ∈ M_K(|X Y |):

s ∈ ^__X _Y iff

(π₁(s) ∈ ^__X =⇒ π₂(s) ∈ ^__Y π1(s) ∈ _^_X ⇐= π2(s) ∈ _^_Y s ∈ N iff π (s) ∈ N and π (s) ∈ N

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Non-uniform spaces

P (|X|) N^X

^

_

MALL model: standard pattern for ^MALL neutrality is reflexivity

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Non-uniform spaces

P (|X|) N^X

^

_

MALL model: standard pattern for ^MALL neutrality is reflexivity exponentials are responsible for neutrality6=reflexivity

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Non-uniform spaces

P (|X|) N^X

^

_

the exponentials badly behaved:

no determinism

no related uniform semantics

for M _\{_0,1_}, non sequential ag. (som. like // or)

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Non-uniform spaces

P (|X|) N^X

^

_

no determinism no related uniform semantics

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Non-uniform spaces

P (|X|) N^X

^

_

no determinism

no related uniform semantics for M _\{_0,1_}, non sequential ag. (som. like // or)

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Non-uniform spaces

P (|X|) N^X

^

_

no determinism

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Non-uniform spaces

P (|X|) N^X

^

_

no determinism

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The exponentials

Web: |!X| = M_fin(|X|)

Coherence. For each [x_i | i ∈ I] ∈ M_K(|!X|) we set:

[x_i | i ∈ I] ∈ ^_!X iff ∃[a_i | i ∈ I] ∈ ^_X, ∀i ∈ I, a_i ∈ x_i [x_i | i ∈ I] ∈ N^!X iff [x_i | i ∈ I] ∈/ ^_!X and ∃(a^j_i)^j∈J_i∈I ,

∀i ∈ I, [a^j_i | j ∈ J] = x_i and

∀j ∈ J, [a^j_i | i ∈ I] ∈ N^X

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The exponentials

Web: |!X| = M_fin(|X|) Coherence. For each [x_i | i ∈ I] ∈ M_K(|!X|) we set:

∀i ∈ I, [a^j_i | j ∈ J] = x_i and

∀j ∈ J, [a^j_i | i ∈ I] ∈ N^X

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The exponentials

Web: |!X| = M_fin(|X|)

This web is the same as for relational model, so the semantics is non-uniform. A uniform web would have been {x ∈ M_fin(|X|) | supp(x) ∈ agent(X)}.

[x_i | i ∈ I] ∈ ^_!X iff ∃[a_i | i ∈ I] ∈ ^_X, ∀i ∈ I, a_i ∈ x_i [x_i | i ∈ I] ∈ N^!X iff [x_i | i ∈ I] ∈/ ^_!X and ∃(a^j_i)^j_i∈I^∈^J,

∀i ∈ I, [a^j_i | j ∈ J] = x_i and

∀j ∈ J, [a^j_i | i ∈ I] ∈ N^X

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The exponentials

Web: |!X| = M_fin(|X|)

∀i ∈ I, [a^j_i | j ∈ J] = x_i and

∀j ∈ J, [a^j_i | i ∈ I] ∈ N^X

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The exponentials

Web: |!X| = M_fin(|X|)

[x_i | i ∈ I] ∈ ^_!X iff ∃[a_i | i ∈ I] ∈ ^_X, ∀i ∈ I, a_i ∈ x_i [x_i | i ∈ I] ∈ N^!X iff [x_i | i ∈ I] ∈/ ^_!X and ∃(a^j_i)^j_i∈I^∈J,

∀i ∈ I, [a^j_i | j ∈ J] = x_i and

∀j ∈ J, [a^j_i | i ∈ I] ∈ N^X

x₁ x₂ x..._n

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The exponentials

Web: |!X| = M_fin(|X|)

[x_i | i ∈ I] ∈ ^_!X iff ∃[a_i | i ∈ I] ∈ ^_X, ∀i ∈ I, a_i ∈ x_i [x_i | i ∈ I] ∈ N^!X iff [x_i | i ∈ I] ∈/ ^_!X and ∃(a^j_i)^j_i∈I^∈J,

∀i ∈ I, [a^j_i | j ∈ J] = x_i and

∀j ∈ J, [a^j_i | i ∈ I] ∈ N^X

x₁ x₂ x..._n

a₁ a₂ a_n

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The exponentials

Web: |!X| = M_fin(|X|)

∀i ∈ I, [a^j_i | j ∈ J] = x_i and

∀j ∈ J, [a^j_i | i ∈ I] ∈ N^X

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The exponentials

Web: |!X| = M_fin(|X|)

∀i ∈ I, [a^j_i | j ∈ J] = x_i and

∀j ∈ J, [a^j_i | i ∈ I] ∈ N^X

x₁ x₂

...

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The exponentials

Web: |!X| = M_fin(|X|)

∀i ∈ I, [a^j_i | j ∈ J] = x_i and

∀j ∈ J, [a^j_i | i ∈ I] ∈ N^X x₁ x₂ x..._n

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Properties

Non-uniform model of linear logic (logical forgetful functor).

Co-free ⊗-comonoid. (so maximality).

Determinism: N^X ⊆ ∪_a∈|X_|M_K({a}) Stage of interaction: neutral web

|X|_N_,K = {a ∈ |X| | M_K({a}) ⊆ N^X} and functor N_K of restriction to the neutral web.

|!X|_N_,K = {x ∈ M_fin(|X|_N_,K) | supp(x) ∈ agent(X)}

u! = N_K!. A new class of uniform models related to the non-uniform ones in a very comfortable way.

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Properties

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Properties

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Properties

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Properties

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Properties

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Properties

!. A new class of uniform models related

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Semantics

[π]_M

K = [π]_R and for the related uniform semantics : [π]_u,M

K = N_K([π]_R).

K = {2} → coherence spaces. [π]_COH = N_{2}([π]_R) K = \{0, 1} → multicoherences. Generalize

hypercoherences to multiplicities aware coherence relations.

Forget multiplicities → non-unif. hypercoherences ( !

nuh = S!). Usual (multiset-based) uniform semantics. !

mh = N !

nuh.

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Semantics

[π]_M

K = N_K([π]_R).

mh = N !

nuh.

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Semantics

[π]_M

K = N_K([π]_R).

mh = N !

nuh.

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Semantics

[π]_M

K = N_K([π]_R).

mh = N !

nuh.

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Semantics

[π]_M

K = N_K([π]_R).

mh = N !

nuh.

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Extensionality

Multiset-based models aren’t extensional There is no set based non-uniform model

Extensional collapses (using a M^ELLIÈS result, [Mel01]):

m.-based coh. sp./ ∼ = s.-based coh. sp.

m.-based hypercoh./ ∼ = s.-based hypercoh.

non-unif.MK-coh./ ∼ = unif. M_K-coh./ ∼ non-unif. coh. sp./ ∼ = s.-based coh. sp.

→ s.-based multicoherences (direct description).

non-unif. hypercoh./ ∼ = s.-based hypercoh.

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Extensionality

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Extensionality

Multiset-based models aren’t extensional There is no set based non-uniform model Extensional collapses (using a M^ELLIÈS result,

[Mel01]):

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Extensionality

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Extensionality

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Extensionality

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Extensionality

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Extensionality

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Extensionality

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Extensionality

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Sequentiality

Non-uniform semantics : non-sequential, eg, {([t], t), ([f], t), ([t, f], t)}

is an agent of type !Bool Bool.

Multicoherence semantics: sequentiality at first order. Every finite agent of type

!(Bool & . . . & Bool) Bool is definable in ^PCF. Multicoherences 6= hypercoherences → two

(extensionally) different notions of higher order

sequentiality. (Contrarily to what was expected, eg in [Lon02]).

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Sequentiality

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Sequentiality

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Sequentiality

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References

[BE01] A. Bucciarelli and T. Ehrhard. On phase semantics and denotational semantics: the

exponentials. APAL, 109(3):205–241, 2001.

[Ehr96] Thomas Ehrhard. Projecting sequential

algorithms on strongly stable functions. APAL, 77, 1996.

[Lai01] J. Laird. Games and sequential algorithms.

Available by http, 2001.

[Lon02] J.R. Longley. The sequentially realizable functionals. APAL, 117(1-3):1–93, 2002.

[Mel01] Paul-André Melliès. Extensional collapse and coercions. Manuscript, December 2001.

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References

[BE01] Antonio Bucciarelli and Thomas Ehrhard. On phase semantics and denotational semantics:

the exponentials. Annals of Pure and Applied Logic, 109(3):205–241, 2001.

[Ehr96] Thomas Ehrhard. Projecting sequential

algorithms on strongly stable functions. Annals of Pure and Applied Logic, 77, 1996.

[Lai01] J. Laird. Games and sequential algorithms.

Available by http, 2001.

[Lon02] J.R. Longley. The sequentially realizable

functionals. Annals of Pure and Applied Logic, 117(1-3):1–93, 2002.

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