Existential quantification in games and logic (towards a - - PowerPoint PPT Presentation

Existential quantification in games and logic

(towards a combinatorics of proofs)

Paul-Andr´ e Melli` es CNRS, Universit´ e Paris 7 Computational Interpretations of Proofs 29-30 November 2007

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Two totems inherited from history

Intuitionistic Logic Classical Logic

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Our ambition is to unify them...

Intuitionistic Logic Classical Logic

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and to disclose...

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and to disclose...

a series of primitive components !

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Guiding idea

A proof π : A ⊢ B is a linguistic choreography where Proponent tries to convince Opponent Opponent tries to refute Proponent which we would like to decompose in elementary particles of logic

The linear decomposition of the intuitionistic arrow

A ⇒ B = (!A) ⊸ B [1] a proof of A ⊸ B uses its hypothesis A exactly once, [2] a proof of !A is a bag containing an infinite number of proofs of A. Andreas Blass discovered this decomposition as early as 1972...

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Three primitive components of logic

[1] the negation ¬ [2] the linear conjunction ⊗ [3] the repetition modality ! Logic = Data Structure + Duality

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Three primitive components of logic

[1] the negation ¬ [2] the linear conjunction ⊗ [3] the repetition modality ! An algebraic account of games semantics.

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Negation

Every fact has two poles: Proponent and Opponent

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An early observation...

Fact: there are just as many canonical proofs

2p

• R

¬ · · · ¬ A ⊢

2q

• R

¬ · · · ¬ A as there are increasing functions [p] − → [q] between the ordinals [p] = {0 < 1 < · · · < p − 1} and [q]. Hence, the same combinatorics as augmented simplices.

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The two generators

Every increasing function is composite of faces and degeneracies: η : [0] ⊢ [1] µ : [2] ⊢ [1] Similarly, every proof is composite of the two generators: η : A ⊢ ¬¬A µ : ¬¬¬¬A ⊢ ¬¬A Categorically speaking, double negation defines a monad.

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Negation

A negation on a symmetric monoidal category A is defined as

• a functor:

¬ :

A −

→ Aop

• a bijection

ϕA,B,C between the morphisms in A ⊗ B − → ¬ C and the morphisms in A − → ¬ (B ⊗ C) natural in A, B and C.

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The reason for the double negation monad

Every negation induces a pair of adjoint functors

A

L

• ⊥

Aop

R

• where the adjunction is witnessed by the series of bijection:

A(A, ¬ B)

∼ =

A(B, ¬ A)

∼ =

Aop(¬ A, B)

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The reason for the double negation monad

The unit and multiplication of the resulting monad R ◦ L are depicted as η : Id − → R ◦ L

L R η

µ : R ◦ L ◦ R ◦ L − → R ◦ L

L L L R R R µ

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The two generators in sequent calculus A ⊢ A

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A , ¬A ⊢

1

A ⊢ ¬¬A A ⊢ A

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A , ¬A ⊢

5

¬A ⊢ ¬A

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¬A , ¬¬A ⊢

3

¬A ⊢ ¬¬¬A

2

¬¬¬¬A , ¬A ⊢

1

¬¬¬¬A ⊢ ¬¬A

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A typical proof

L L L L L R R R R R

This discloses the secret algebra of game semantics

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Cut elimination

R L

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A purely diagrammatic cut elimination

ε η L L

=

L L η ε R R

=

R R

The typical way to compose strategies in game semantics

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Revisiting the negative translation

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The algebraic point of view (in the style of Boole)

The negated elements of a Heyting algebra form a Boolean algebra.

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The algebraic point of view (in the style of Frege)

A double negation monad is commutative iff it is involutive. This amounts to the following diagrammatic equations:

L L R R R R

=

L L R R R R R R L L L L

=

R R L L L L

In that case, the negated elements form a ∗-autonomous category.

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Tensorial conjunction

Let me tell you this ⊗ this...

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Tensor vs. negation

A well-known fact: the continuation monad is strong (¬¬ A) ⊗ B − → ¬¬ (A ⊗ B) The starting point of the theory of computational effects (Moggi)

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The two generators (1)

A conjunctive strength κ+ : R(A LB) C − → R(A L(B C))

• R

C

• A

L B − → R

• A

L

• B

C

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The two generators (2)

A disjunctive strength κ− : L(R(A B) C) − → A L(R(B) C) L

• R

C

• A

B − →

• A

L

• R

C B

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Leads to 3-dimensional proof diagrams

κ+ κ+ ε B A R A B R R L L L 27

A factorization theorem

The four proofs η, ǫ, κ+ and κ− generate every proof of the logic. Moreover, every such proof X

ǫ

− → κ+ − →

ǫ

− →

ǫ

− → η − → η − → κ− − →

ǫ

− → η − →

ǫ

− → κ− − → η − → η − → Z factors uniquely as X

κ+

− → − →

ǫ

− → − →

η

− → − → κ− − → − → Z Corollary: two proofs are equal iff their associated strategies are equal.

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Repetition modality

Let me tell you ! this again and again and again...

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The algebraic laws of the modality are the generators

• !A is a comonoid:

!A → !A ⊗ !A !A → 1

• the functor ! is a comonad:

!A → A !A → !!A

• the comonad ! is monoidal:

!A ⊗ !B → !(A ⊗ B) 1 → !1 Encodes the propositional fragment of intuitionistic and classical logic

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Existential quantification

Looking for witnesses in logic

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Four primitive components of logic

[1] the negation ¬ [2] the linear conjunction ⊗ [3] the repetition modality ! [4] existential quantification ∃ A theory of witnesses – what remains of intuitionism...

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Krivine’s classical realizability

A logic of tensor, negation and existential quantification

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A logic of tensor and negation

The term formula orthogonal to the stack formula X is given by ¬ ! X because a term can duplicate or erase its stack. The stack formula orthogonal to the term formula A is given by ¬ A because the term has the control and cannot be duplicated by the stack.

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A logic of tensor and negation

Given a term formula B induced by the stack formula X as B := ¬ ! X the logical implication from a term formula A is defined as: A ⇒ B := ¬ ⊗

• !A

!X

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A logic of tensor and negation

This definition has the unexpected consequence that ∀u ∈ A, tu ∈ B does not necessarily imply that t ∈ A ⇒ B. Only the converse is true: ∀t ∈ A ⇒ B, ∀u ∈ A, tu ∈ B. Hence, Krivine’s system is not based on logical implication.

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The drinker’s formula

Suppose that you enter in a pub, and that A(x) means that the customer x does not drink. The formula ∃y(A(y) ⇒ ∀xA(x)) says that there exists a customer y (the drinker) such that if y does not drink, then nobody drinks in the pub. The formula is valid in classical logic, but not in intuitionistic logic.

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The interactive proof of the drinker’s formula

The interactive proof goes like this: (1) Proponent chooses any customer y1, (2a) Opponent looses if he cannot find a customer x which drinks, (2b) Otherwise, Opponent finds a customer x which drinks, and mentions it to Proponent, (3) Proponent reacts by changing his original witness y1 for y2 = x.

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The drinker’s formula

Krivine reformulates the formula ∃y(A(y) ⇒ ∀xA(x)) in the classically equivalent way: ∀B. ∀y.{(A(y) ⇒ ∀x.A(x)) ⇒ B} ⇒ B.

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The formal proof in classical logic

Axiom A(x0) ⊢ A(x0) Right Weakening A(x0) ⊢ ∀x.A(x), A(x0) Right ⇒ ⊢ A(x0) ⇒ ∀x.A(x), A(x0) Axiom B ⊢ B Left ⇒ (A(x0) ⇒ ∀x.A(x)) ⇒ B ⊢ A(x0), B Left ∀ ∀y.{(A(y) ⇒ ∀x.A(x)) ⇒ B} ⊢ A(x0), B Right ∀ ∀y.{(A(y) ⇒ ∀x.A(x)) ⇒ B} ⊢ ∀x.A(x), B Left Weakening ∀y.{(A(y) ⇒ ∀x.A(x)) ⇒ B}, A(y0) ⊢ ∀x.A(x), B Right ⇒ ∀y.{(A(y) ⇒ ∀x.A(x)) ⇒ B} ⊢ A(y0) ⇒ ∀x.A(x), B Axiom B ⊢ B Left ⇒ ∀y.{(A(y) ⇒ ∀x.A(x)) ⇒ B}, (A(y0) ⇒ ∀x.A(x)) ⇒ B ⊢ B, B Left ∀ ∀y.{(A(y) ⇒ ∀x.A(x)) ⇒ B}, ∀y.{(A(y) ⇒ ∀x.A(x)) ⇒ B} ⊢ B, B Contraction ∀y.{(A(y) ⇒ ∀x.A(x)) ⇒ B} ⊢ B, B Contraction ∀y.{(A(y) ⇒ ∀x.A(x)) ⇒ B} ⊢ B Right ⇒ ⊢ ∀y.{(A(y) ⇒ ∀x.A(x)) ⇒ B} ⇒ B

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The classical formula decomposed in tensorial logic

¬ ⊗

• !

! ¬ B ∃y ⊗

• !

! ¬ B ⊗

• !

¬ A(y) ∃x A(x)

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An equivalent formula

¬ ! ¬ ∃y ⊗

• A(y)

¬ ∃x A(x) The repetition modality ! gives an explicit account of the fact that Proponent is allowed to change her witness.

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An equivalent formula

¬ ! ¬ ∃y ⊗

• A(y)

¬ ∃x A(x) The formula explicates the exact rules of the game (and space of logical interaction)

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An equivalent formula

¬ ! ¬ ∃y ⊗

• A(y)

¬ ∃x A(x) The formula is valid in tensorial logic.

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The intuitionistic formula decomposed in tensorial logic

∃y ⊗

• A(y)

¬ ∃x A(x) This formula is not valid in tensorial logic, because Proponent has to choose the witness y immediately – and cannot change its mind later.

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G¨

• del’s Dialectica interpretation

As revisited by Martin Hyland and Valeria de Paiva

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G¨

• del’s original Dialectica interpretation

φD ≡ ∃u∀x φD(u, x) ψD ≡ ∃v∀y ψD(v, y) (φ ∧ ψ)D ≡ ∃uv∀xy φD(u, x) ∧ ψD(x, y) (φ ∨ ψ)D ≡ ∃cuv∀xy (c = 0 ∧ φD(u, x)) ∨ (c = 0 ∧ ψD(x, y)) (φ ⇒ ψ)D ≡ ∃xv∀uy φD(u, xuy) ⇒ ψD(vu, y) (∀z φ)D ≡ ∃u∀zx φD(uz, x, z) (∃z φ)D ≡ ∃zu∀x φD(u, x, z).

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Dialectica interpretation of linear logic

φL ≡ ∃u∀x φL(u, x) ψL ≡ ∃v∀y ψL(v, y) (φ ⊗ ψ)L ≡ ∃uv∀xy (φ ⊗ ψ)L ≡ ∃uv∀xy φL(u, xv) ∧ ψL(v, yu) (!φ)L ≡ ∃u∀x φL(u, xu) (∃xφ)L ≡ ∃uz∀x φL(u, xz, z) (¬ φ)L ≡ ∃x∀u ¬φL(u, x). Here, a recent variant by Masaru Shirahata

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The fundamental observation

G¨

• del’s Dialectica interpretation commutes to the tensorial negation.

φL ≡ ∃u∀x φL(u, x). (¬ φ)L ≡ ∃x∀u ¬φL(u, x). A contemporary perspective on the Dialectica interpretation.

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Illustration

Linear decomposition of the intuitionistic implication φL ≡ ∃u∀x φL(u, x) ψL ≡ ∃v∀y ψL(v, y) The linear implication is defined as follows φ ⊸ ψ := ¬(φ ⊗ ¬ψ) Now, (φ ⊸ ψ)L ≡ ∃xv∀uy φL(u, xy) ⇒ ψL(y, vu) ((!φ) ⊸ ψ)L ≡ ∃xv∀uy φL(u, xyu) ⇒ ψL(y, vu)

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Existential quantifiers as left adjoints

An algebraic point of view on quantification by Bill Lawvere

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The Dialectica interpretation induces a fibration

E

F

• B

Every element X of the basis B induces a category FX called its fiber. So, a fibration may be seen alternatively as a contravariant pseudo-functor F :

B

− →

Cat.

Typically, F associates to X ∈ Set its powerset ℘(X).

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Existential quantifiers

Every projection map X × Y − → X in the basis B induces a fresh variable functor newy : FX − → FX×Y Every term t of sort Y induces a map X − → X × Y and thus a substitution functor [y := t] : FX×Y − → FX

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An easy diagrammatic equality

=

FX FX FX new(y) y := t FX×Y

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Existential quantification is left adjoint to substitution

FX(∃y.P, Q) ∼ = FX×Y (P, newy.Q) In particular, the associated monad indicates that: P ⊢X×Y newy.∃y.P for every predicate in the fiber FX×Y .

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The introduction law of existential quantification

y := t ∃y new(y) ∃y y := t FX×Y FX FX×Y FX

Similar diagrammatic analysis of the Beck-Chevalley conditions.

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Current and future works

• Existential quantification in programming languages – eg. the logical

status of modules in ML, investigated by Montagu and R´ emy (INRIA)

• Dynamic linking interpreted by Hilbert’s notation εx.A,
• General study of modalities in algebra and string diagrams,
• Axiom of Choice and bar induction in games (Coquand, Krivine)
• A clean semantics of assembly code and computational effects,
• Connections to algebraic geometry and model theory.

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