A fully complete model of propositional linear logic Paul-Andr e - - PowerPoint PPT Presentation

Asynchronous Games 4

A fully complete model of propositional linear logic

Paul-Andr´ e Melli` es CNRS, Universit´ e Paris 7 Logic in Computer Science Chicago, Wednesday 29th June 2005

1

An epic in 50 slides

Twenty-four seconds each

2

A crash course on Mazurkiewicz traces

The 2-dimensional geometry of concurrency

3

Trace semantics

Interpret any process π = a | b by the sequences of actions it performs in the course of time : ab ba

4

The synchronization tree of a process

a | b interpreted as · · ·

b

• ·

a

• ·

a

• b
• 5

The synchronization tree of a process

ab + ba interpreted as · · ·

b

• ·

a

• ·

a

• b
• 6

This is a problem

Trace semantics cannot see the difference between a | b no interference and ab + ba interference

7

Idea : replace synchronization trees...

a | b interpreted as · · ·

b

• ·

a

• ·

a

• b
• 8

... by Mazurkiewicz traces

a | b interpreted as · ·

b

• ∼

·

a

• ·

a

• b
• 9

True concurrency = homotopy

· ·

b

• ∼

·

a

• ·

a

• b
• Think of this permutation as a 2-dimensional tile.

10

Interference = holes

ab + ba interpreted as · ·

b

• ∼

·

a

• ·

a

• b
• 11

Asynchronous games

Games played on Mazurkiewicz traces

12

Game semantics

V F q

true

• false
• ∗

q

• Player in red

Opponent in blue The boolean game B

13

Traditional game semantics

true

• false
• q
• Player in red

Opponent in blue The boolean game B

14

Traditional game semantics : an interleaving semantics

false2

• true1
• q2
• q1
• true1
• false2
• q1
• q2
• The tensor product of two boolean games B1 and B2

15

Bend the branches !

false2

• true1
• q2
• q1
• true1
• false2
• q1
• q2
• 16

Tile the diagram !

false2

• ∼

true1

• q2
• ∼

true1

• false2
• ∼

q1

• true1
• q2
• ∼

q1

• false2
• q1
• q2
• 17

Tag the positions !

V ⊗ F V ⊗ q

false2

• ∼

q ⊗ F

true1

• V ⊗ ∗

q2

• ∼

q ⊗ q

true1

• false2
• ∼

∗ ⊗ F

q1

• q ⊗ ∗

true1

• q2
• ∼

∗ ⊗ q

q1

• false2
• ∗ ⊗ ∗

q1

• q2
• 18

A 2-dimensional space of interaction

V ⊗ F V ⊗ q

false2

• ∼

q ⊗ F

true1

• V ⊗ ∗

q2

• ∼

q ⊗ q

true1

• false2
• ∼

∗ ⊗ F

q1

• q ⊗ ∗

true1

• q2
• ∼

∗ ⊗ q

q1

• false2
• ∗ ⊗ ∗

q1

• q2
• 19

Asynchronous games

A 2-dimensional graph equipped with tiles of the shape · ·

n

• ∼

·

m

• ·

m

• n
• in which :
• every edge is polarized Player or Opponent
• an initial position ∗ is distinguished

20

Sequential play

A sequential play is defined as an alternated path ∗ m1 − → x1

m2

− → x2

m3

− → · · · xk−1

mk

− → xk starting by an Opponent move.

21

Strategies

A strategy is a set of sequential plays of even length, such that :

• σ contains the empty play,
• σ is closed under even-length prefix

s · m · n ∈ σ ⇒ s ∈ σ

• σ is deterministic

s · m · n1 ∈ σ and s · m · n2 ∈ σ ⇒ n1 = n2 A strategy plays according to the current play.

22

Innocence 1994

Martin Hyland, Luke Ong, Hanno Nickau An interactive characterization of λ-terms

23

Innocence : strategies with partial information

m · n

• · p
• · n
• · p
• · n
• · p
• · m · n
• · p
• The Player view s

: what the Player can remember of the play s. An innocent strategy plays according to the current Player view.

24

Innocence 2004

From amnesia to positionality

25

Consistency (backward)

σ ∋

s2

• n2
• m2
• ∼
• n1
• ∼
• m1
• m2
• s1
• ⇒

s2

• n2
• ∼

n1

• m2
• ∼
• ∼

m1

• n1
• ∼
• n2
• m1
• m2
• s1
• ∈ σ

26

Consistency (forward)

σ ∋

m2

• ∼
• n1
• ∼
• n2
• m1
• m2
• s1
• ∈ σ

⇒ σ ∋

n2

• ∼

n1

• m2
• ∼
• ∼

m1

• n1
• ∼
• n2
• m1
• m2
• s1
• ∈ σ

27

Innocent strategies are positional

Theorem [Concur 2004] Every innocent strategy σ is positional.

• Cor. An innocent strategy σ is characterized by its set of positions σ•.

28

An 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
• 29

An 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
• 30

An 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
• 31

An 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
• 32

A quite extraordinary discovery

The concurrent nature of the λ-calculus

33

A sketched history of concurrency

Automata theory

• Petri Nets
• Synchronization trees
• Mazurkiewicz traces

CCS

• π-calculus

λ-calculus

• 34

A quite different topography revealed

Automata theory

• Petri Nets
• Synchronization trees
• Mazurkiewicz traces
• CCS
• λ-calculus

π-calculus

35

What about linear logic ?

Automata theory

• Petri Nets
• Synchronization trees
• Mazurkiewicz traces
• CCS
• λ-calculus
• π-calculus

linear logic

36

A serious difficulty arises

37

There exists no game model of linear logic !

But not that far ! We have lots of game models of intuitionistic linear logic ! In all of these game models, there exists a retraction (A ⊸ ⊥) ⊸ ⊥

θA

− → A

∂A

− → (A ⊸ ⊥) ⊸ ⊥ A phenomenon of control categories — Peter Selinger 2001

38

The game (A ⊸ ⊥) ⊸ ⊥

The game (A ⊸ ⊥) ⊸ ⊥ is obtained by lifting the game A twice : x y z ∗

• x

y z ⊥

• ⊤

n

• ∗

m

• A

(A ⊸ ⊥) ⊸ ⊥

39

(A ⊸ ⊥) ⊸ ⊥

⊸

(A ⊸ ⊥) ⊸ ⊥

x ⊸ x x ⊸ ⊥

p2

• ∼

⊥ ⊸ x

p1

• x ⊸ ⊤

n2

• ∼

⊥ ⊸ ⊥

p2

• p1
• ∼

⊤ ⊸ x

n1

• ⊥ ⊸ ⊤

n2

• p1
• ∼

⊤ ⊸ ⊥

n1

• p2
• ∼

∗ ⊸ x

m1

• ⊤ ⊸ ⊤

n1

• n2
• ∼

∗ ⊸ ⊥

m1

• p2
• ∗ ⊸ ⊤

m1

• n2
• ∗

m2

• 40

The identity strategy id(A⊸⊥)⊸⊥

x ⊸ x x ⊸ ⊥ ⊥ ⊸ x

p1

• x ⊸ ⊤

⊥ ⊸ ⊥

p2

• ⊤ ⊸ x

⊥ ⊸ ⊤

n2

• ⊤ ⊸ ⊥

∗ ⊸ x ⊤ ⊸ ⊤

n1

• ∗ ⊸ ⊥

∗ ⊸ ⊤

m1

• ∗

m2

• 41

The idempotent strategy ∂A ◦ ρA

x ⊸ x x ⊸ ⊥ ⊥ ⊸ x

p1

• x ⊸ ⊤

⊥ ⊸ ⊥ ⊤ ⊸ x

n1

• ⊥ ⊸ ⊤

⊤ ⊸ ⊥ ∗ ⊸ x

m1

• ⊤ ⊸ ⊤

∗ ⊸ ⊥

p2

• ∗ ⊸ ⊤

n2

• ∗

m2

• 42

Diagnosis

The two strategies ∂A ◦ θA and id(A⊸⊥)⊸⊥ are only equal modulo “homotopy”. Idea : force the equality ∂A ◦ θA = id(A⊸⊥)⊸⊥ in the category of games. Leads to a classical model of linear logic

43

Solution

• Assign a payoff κ(x) ∈ Z to every position x of the game.
• Consider only the winning strategies σ :

∀x, x ∈ σ• ⇒ κ(x) ≥ 0

• Call external a position x with null payoff :

κ(x) = 0

• Identify two strategies when they play the same external positions.

σ ≃ τ

44

Slogan

A strategy realizes its set of external positions. The two strategies

id(A⊸⊥)⊸⊥

and ∂A ◦ ρA are identified in this way. The left and right implementations of AND are also identified.

45

A remark for the insider

This transforms the non commutative continuation monad T : A → (A ⊸ ⊥) ⊸ ⊥ into a commutative one : TA ⊗ TB

• T(A ⊗ TB)

T 2(A ⊗ B)

• T(TA ⊗ B)

T 2(A ⊗ B) T(A ⊗ B)

Property : the kleisli category is then ∗-autonomous. A very nice observation by Hasegawa Masahito

46

Well-bracketing reduces to a winning conditiona

(B

⊸ B) ⊸ B

κ(x) q −1 q +1 q −1 true −1 (+1 ⊸ −1) ⊸ −1

47

Full completeness

• Extend payoffs to paths in order to separate the two formulas :

(A&B) ⊗ ⊤ and (A ⊗ ⊤) & (B ⊗ ⊤)

• Extend payoffs to walks in order to reject the pseudo-proof of the

sequent : ⊢ 1, 1, (⊥ & (⊥ ⊗ ⊥)) ⊗ ⊤. Needs to explore several additive slices.

• Full completeness by directed proof search.

48

Main difficulty : garbage collect !

Exponentials

The exponential factors as !A =

• n∈N

↓ A Uniformity is described thanks to a left and right group action on indices. Samson Abramsky, Radha Jagadeesan, Pasquale Malacaria 1994

49

Work in progress

• 1. Our model is not the free construction.
• 2. Compare to another very nice formulation of innocence.

(joint work with Martin Hyland and Russ Harmer)

• 3. Relax alternation and compare innocence to L-nets

(joint work with Claudia Faggian and Samuel Mimram)

• 4. Add holes to the geometry and study languages with states

(joint work with Nicolas Tabareau)

• 5. Study first-order and second-order linear logic.

50

Thank you !

I am currently writing the full paper.

51

Announcement

I organize with Anca Muscholl (LIAFA) the next session of the legendary Spring School in Theoretical Computer Science Monday 29 May — Friday 2 June 2006 Ile de R´ e, France Games : Semantics and Verification whose first School was organized by Maurice Nivat in 1973. The School meets every year since then, in a different place, and on a different subject. PhD students, postdoctoral students, young researchers wishing to learn more about games and their applications to Semantics and Verifica- tion are welcome to the Spring School.

52

Announcement

The following people have already accepted to teach at the School : – Samson Abramsky (Oxford, UK) – Martin Hyland (Cambridge, UK) – Luke Ong (Oxford, UK) – Jacques Duparc (Lausanne, Switzerland) – Eric Gr¨ adel (Aachen, Germany) – Igor Walukiewicz (Bordeaux, France) – Wieslaw Zielonka (Paris, France) – Tom Henzinger (Lausanne, Switzerland) (to be confirmed) – Sylvain Sorin (Ecole Polytechnique, Paris) (to be confirmed) Contact email : Paul-Andre.Mellies@pps.jussieu.fr

53

Left and ≃ Right and

(B ⊗

B) ⊸ B

q q true q true true (B ⊗

B) ⊸ B

q q true q true true The two strategies realize the same external positions of the game.

54