Towards an algebra of negation (work in progress) Paul-Andr Mellis - - PowerPoint PPT Presentation

Towards an algebra of negation

(work in progress)

Paul-André Melliès CNRS, Université Paris Denis Diderot Foundational Methods in Computer Science Halifax 31 May 2008

1

Proof-knots

Aim: formulate an algebra of these logical knots

2

A proof of the drinker’s formula

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(y)0 ⊢ ∀x.A(x), B Right ⇒ ∀y.{(A(y) ⇒ ∀x.A(x)) ⇒ B} ⊢ A(y)0 ⇒ ∀x.A(x), B Axiom B ⊢ B Left ⇒ ∀y.{(A(y) ⇒ ∀x.A(x)) ⇒ B}, (A(y)0 ⇒ ∀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

3

Starting point: game semantics

Every proof of formula A initiates a dialogue where Proponent tries to convince Opponent Opponent tries to refute Proponent An interactive approach to logic and programming languages

4

Duality

Proponent Program plays the game A Opponent Environment plays the game ¬ A Negation permutes the rôles of Proponent and Opponent

5

Duality

Opponent Environment plays the game ¬ A Proponent Program plays the game A Negation permutes the rôles of Opponent and Proponent

6

A brief history of games and linear logic

1977 André Joyal A category of games and strategies 1986 Jean-Yves Girard Linear logic 1992 Andreas Blass A semantics of linear logic Samson Abramsky 1994 Radha Jagadeesan A category of history-free strategies Pasquale Malacaria 1994 Martin Hyland A category of innocent strategies Luke Ong A schism between game semantics and linear logic

7

Sequential game semantics

A proof π alternating sequences of moves A proof π Game semantics: an interleaving semantics of proofs.

8

An interleaving semantics

The boolean game B: V F q

true

• false
• ∗

question

• Player in red

Opponent in blue

9

An interleaving semantics

The tensor product of two boolean games B1 et B2:

false2

• true1
• q2
• q1
• true1
• false2
• q1
• q2
• 10

A step towards true concurrency: bend the branches!

false2

• true1
• q2
• q1
• true1
• false2
• q1
• q2
• 11

Asynchronous games: tile the diagram!

V ⊗ F V ⊗ q

false2

• ∼

q ⊗ F

true1

• V ⊗ ∗

q2

• ∼

q ⊗ q

true1

• false2
• ∼

∗ ⊗ F

q1

• q ⊗ ∗

true1

• q2
• ∼

∗ ⊗ q

q1

• false2
• ∗ ⊗ ∗

q1

• q2
• 12

Asynchronous game semantics

A proof π1 trajectories in asynchronous transition spaces A proof π2 Main result: innocent strategies are positional.

13

Illustration: the strategy (true ⊗ false)

V ⊗ F V ⊗ q

false2

• ∼

q ⊗ F

true1

• V ⊗ ∗

q2

• ∼

q ⊗ q

true1

• false2
• ∼

∗ ⊗ F

q1

• q ⊗ ∗

true1

• q2
• ∼

∗ ⊗ q

q1

• false2
• ∗ ⊗ ∗

q1

• q2
• Strategies seen as

closure operators

• n complete lattices

14

Part 1 The topological nature of negation

At the interface between topology and algebra

15

Cartesian closed categories

A cartesian category C is closed when there exists a functor ⇒ :

Cop × C

−→

C

and a natural bijection ϕA,B,C :

C(A × B , C)

• C(A , B ⇒ C)

C B A ×

• C

B A ⇒

16

The free cartesian closed category

The objects of the category free-ccc(C) are the formulas A, B ::= X | A × B | A ⇒ B | 1 where X is an object of the category C. The morphisms are the simply-typed λ-terms, modulo βη-conversion.

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The simply-typed λ-calculus

Variable x :X ⊢ x :X Abstraction Γ, x :A ⊢ P :B Γ ⊢ λx.P :A ⇒ B Application Γ ⊢ P :A ⇒ B ∆ ⊢ Q :A Γ, ∆ ⊢ PQ :B Weakening Γ ⊢ P :B Γ, x :A ⊢ P :B Contraction Γ, x :A, y :A ⊢ P :B Γ, z :A ⊢ P[x, y ← z] :B Permutation Γ, x :A, y :B, ∆ ⊢ P :C Γ, y :B, x :A, ∆ ⊢ P :C

18

Proof invariants

Every ccc D induces a proof invariant [−] modulo execution. free-ccc(C)

[−]

D

C

• Hence the prejudice that proof theory is intrinsically syntactical...

19

However, a striking similarity with knot invariants

A tortile category is a monoidal category with

B B A A

A A

A A∗ A A∗

braiding twists duality unit duality counit The free tortile category is a category of framed tangles

20

Knot invariants

Every tortile category D induces a knot invariant free-tortile(C)

[−]

D

C

• A deep connection between algebra and topology

first noticed by Joyal and Street

21

Dialogue categories

A symmetric monoidal category C equipped with a functor ¬ :

Cop

−→

C

and a natural bijection ϕA,B,C :

C(A ⊗ B , ¬ C)

• C(A , ¬ ( B ⊗ C ) )

¬ C B A ⊗

• ¬

C B A ⊗ 22

The free dialogue category

The objects of the category free-dialogue(C) are dialogue games constructed by the grammar A, B ::= X | A ⊗ B | ¬A | 1 where X is an object of the category C. The morphisms are total and innocent strategies on dialogue games. As we will see: proofs are 3-dimensional variants of knots...

23

A presentation of logic by generators and relations

Negation defines a pair of adjoint functors

C

L

• ⊥

Cop

R

• witnessed by the series of bijection:

C(A, ¬ B)

• C(B, ¬ A)
• Cop(¬ A, B)

24

The 2-dimensional topology of adjunctions

The unit and counit of the adjunction L ⊣ R are depicted as η : Id −→ R ◦ L L R η ε : L ◦ R −→ Id R L ε Opponent move = functor R Proponent move = functor L

25

A typical proof

L L L L L R R R R R

Reveals the algebraic nature of game semantics

26

A purely diagrammatic cut elimination

R L

27

The 2-dimensional dynamic of adjunction

ε η L L

=

L L η ε R R

=

R R

Recovers the usual way to compose strategies in game semantics

28

Interlude: a combinatorial 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]. This fragment of logic has the same combinatorics as simplices.

29

The two generators of a monad

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 The unit and multiplication of the double negation monad

30

The two generators in sequent calculus A ⊢ A

2

A , ¬A ⊢

1

A ⊢ ¬¬A A ⊢ A

6

A , ¬A ⊢

5

¬A ⊢ ¬A

4

¬A , ¬¬A ⊢

3

¬A ⊢ ¬¬¬A

2

¬¬¬¬A , ¬A ⊢

1

¬¬¬¬A ⊢ ¬¬A

31

The two generators in string diagrams

The unit and multiplication of the 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

32

Part 2 Tensor and negation

An atomist approach to proof theory

33

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...

35

Four primitive components of logic

[1] the negation ¬ [2] the linear conjunction ⊗ [3] the repetition modality ! [4] the existential quantification ∃ Logic = Data Structure + Duality

36

Tensor vs. negation

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

37

Tensor vs. negation

Proofs are generated by a parametric strength κX : ¬ (X ⊗ ¬ A) ⊗ B −→ ¬ (X ⊗ ¬ (A ⊗ B)) which generalizes the usual notion of strong monad : κ : ¬¬ A ⊗ B −→ ¬¬ (A ⊗ B)

38

Proofs as 3-dimensional string diagrams

The left-to-right proof of the sequent ¬¬A ⊗ ¬¬B ⊢ ¬¬(A ⊗ B) is depicted as

κ+ κ+ ε B A R A B R R L L L 39

Tensor vs. negation – conjunctive strength

κ+ : R(A LB) C −→ R(A L(B C))

• R

C

• A

L B −→ R

• A

L

• B

C Linear distributivity in a continuation framework

40

Tensor vs. negation – disjunctive strength

κ− : L(R(A B) C) −→ A L(R(B) C) L

• R

C

• A

B −→

• A

L

• R

C B Linear distributivity in a continuation framework

41

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 they are equal as strategies.

42

Part 3 Revisiting the negative translation

A rational reconstruction of linear logic

43

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

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

44

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.

45

The continuation monad is strong

(¬¬A) ⊗ B

lst

−→ ¬¬(A ⊗ B) A ⊗ ¬¬B

rst

−→ ¬¬(A ⊗ B)

46

The continuation monad is not commutative

There are two canonical morphisms ¬¬A ⊗ ¬¬B ⇒ ¬¬(A ⊗ B) ¬¬A ⊗ ¬¬B −→ ¬¬(A ⊗ B) q q a q a a ¬¬A ⊗ ¬¬B −→ ¬¬(A ⊗ B) q q a q a a Left strict and Right strict and

47

Asynchronous games

cri

a

• aR
• ∼

aL

• qR
• ∼
• ∼

qL

• aL
• ∼
• aR
• qL
• qR
• q
• Left and

Arena game models extended to propositional linear logic by identi- fying the two strategies — hence mystifying the innocent audience.

48

Asynchronous games

aie

a

• aR
• ∼

aL qR

• ∼

∼ qL aL

• ∼

aR qL

• qR

q

• Left and

Arena game models extended to propositional linear logic by identi- fying the two strategies — hence mystifying the innocent audience.

49

Asynchronous games

cri

a

• aR

∼ aL

• qR

∼ ∼ qL

• aL

∼ aR

• qL

qR

• q
• Right and

Arena game models extended to propositional linear logic by identi- fying the two strategies — hence mystifying the innocent audience.

50

Hence, the schism between games and linear logic

The isomorphism A

• ¬¬A

means that linear logic is static and a posteriori. Imagine that a discussion is defined by the set of its final states Linear logic originates from a model of PCF

51

Tensorial logic

tensorial logic = a logic of tensor and negation = linear logic without A ¬¬A = the very essence of polarized logic Offers a synthesis of linear logic, games and continuations Research program: recast every aspect of linear logic in this setting

52

A lax monoidal structure

Typically, the family of n-ary connectives (A1 · · · An) := ¬ ( ¬A1 ⊗ · · · ⊗ ¬Ak ) := R ( LA1 · · · LAk ) is still associative, but all together, and in the lax sense.

53

A general phenomenon of adjunctions

Given a 2-monad T and an adjunction

A

L

• ⊥

B

R

• every lax T-algebraic structure

TB

b

−→ B induces a lax T-algebraic structure TA TL −→ TB

b

−→ B

R

−→ A

54

A general phenomenon of adjunctions

Consequently, every adjunction with a monoidal category B

A

L

• ⊥

B

R

• induces a lax action of B on the category A

B × A

B×L

−→

B × B

⊗B

−→

B

R

−→

A

Enscopes the double negation monad and the arrow.

55

Distributivity laws seen as lax bimodules

The distributivity law R(A LB) C

κ

−→ R(A L(B C)) may be seen as a lax notion of bimodule: (A ⊲ B) ⊳ C −→ A ⊲ (B ⊳ C) A useful extension of the notion of strong monad (case A = ∗)

56

Part 4 A relaxed notion of Frobenius algebra

After Brian Day and Ross Street

57

Dialogue categories

A symmetric monoidal category C equipped with a functor ¬ :

Cop

−→

C

and a natural bijection ϕA,B,C :

C(A ⊗ B , ¬ C)

• C(A , ¬ ( B ⊗ C ) )

¬ C B A ⊗

• ¬

C B A ⊗ 58

A coherence law

⊗ φC φB D ¬ C B A ⊗ ⊗ D ¬ C B A ⊗ D ¬ C B A φB⊗C 59

Frobenius objects

A Frobenius object F is a monoid and a comonoid satisfying

m d

=

m d

=

m d

an alternative formulation of cobordism

60

Frobenius objects

Equivalently, a Frobenius object F is a monoid equipped with a form u : F −→ I such that the induced morphism

m u

= is the evaluation of a dual pair F ⊣ F.

61

Lax 2-adjunctions in a Gray category

L L

⇒

ε η L L η ε R R

⇒

R R

62

satisfying two coherence properties (a)

η η L L R R

is the identity 3-cell on the unit η of the 2-adjunction L ⊣ R,

63

satisfying two coherence properties (b)

ε ε L L R R

is the identity 3-cell on the counit ε of the 2-adjunction L ⊣ R.

64

Pseudo Frobenius objects

A pseudo Frobenius object in the bicategory of modules

m d

• m

d

• m

d

is the same thing as a ∗-autonomous category... when the two mod- ules m and e are functors. An observation by Brian Day and Ross Street (2003)

65

Lax Frobenius objects

Relax the self-duality equivalence

C

• Cop

into an adjunction

C

L

• ⊥

Cop

R

• this connects game semantics and quantum groups

66

Quantum groupoids

A lax monoidal adjunction in the bicategory Comod ( k-Vect ) E

⊥ f∗

• Vop ⊗ V

f ∗

• where the pseudomonoids E and Vop ⊗ V and map f are ∗-autonomous.

Vop ⊣ V ⊣ Vop A definition by Brian Day and Ross Street (2003)

67

Conclusion Logic = Data Structure + Duality

This point of view is accessible thanks to 2-dimensional algebra

68

Appendix Application to the drinker’s formula

The expressive power of resource modalities

69

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...

70

The drinker’s formula

Suppose that you enter in a pub, and that A(x) means that the cus- tomer x drinks. The formula ∃y(A(y) ⇒ ∀xA(x)) says that there exists a customer y (the drinker) such that if y drinks, then everybody drinks in the pub. The formula is valid in classical logic, but not in intuitionistic logic.

71

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 men- tions it to Proponent, (3) Proponent reacts by changing his original witness y1 for y2 = x.

72

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.

73

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

74

The classical formula decomposed in tensorial logic

¬ ⊗

• !

! ¬ B ∃y ⊗

• !

! ¬ B ⊗

• !

¬ A(y) ∃x A(x)

75

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.

76

An equivalent formula

¬ ! ¬ ∃y ⊗

• A(y)

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

77

An equivalent formula

¬ ! ¬ ∃y ⊗

• A(y)

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

78

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.

79