Simon Castellan, LIP Soutenance de thèse 13 juillet 2017iso.mor.phis.me/publis/soutenance.pdf · A...

Structures concurrentes en sémantique des jeux

Simon Castellan, LIPSoutenance de thèse

13 juillet 2017

A Monopoly problem: the theory

Property of Albert. Property of Barnabé.

To buy both:

price = price Albert+ price Barnabé?= price Barnabé+ price Albert

Structures concurrentes en sémantique des jeux · Simon Castellan, LIP Soutenance de thèse 2 / 37

A Monopoly problem: the practice

� (to B) How much for parklane?� (B) ¿400.� (to A) How much for mayfair?� (A) Never mind the monopoly,I need money: ¿300

Total price: ¿700.

� (to A) How much for mayfair?� (A) I need money: ¿300� (to B) How much for parklane?� (B) Eh! This monopoly willcost you: ¿600

Total price: ¿900.

A Monopoly problem: the practice

� (to B) How much for parklane?� (B) ¿400.� (to A) How much for mayfair?� (A) Never mind the monopoly,I need money: ¿300

Total price: ¿700.

� (to A) How much for mayfair?� (A) I need money: ¿300� (to B) How much for parklane?� (B) Eh! This monopoly willcost you: ¿600

Total price: ¿900.

Beyond formulae : strategies

Same formula, two di�erent strategies:

Albert then Barnabé

(to A) How much for mayfair? qA

(A) I need money: ¿300

200 . . .

. . . 500

(to B) How much for park lane?

(B) Eh! ¿600

500 . . .

. . . 300

Total price: ¿900

A branch is a play of the strategy against a particular environment.

(to A) How much for mayfair? qA

(A) I need money: ¿200 200

. . . 500

(to B) How much for park lane? qB qB

(B) Eh! ¿500 500

. . . 300

Total price: ¿700 700 900

Barnabé then Albert

(to A) How much for mayfair?

(A) I need money: ¿300

200 . . . 300 . . . 500

(to B) How much for park lane?

qB qB qB

(B) Eh! ¿600

500 . . . 600 . . . 300

Total price: ¿900

700 900 800

Barnabé then Albert Albert then Barnabé

200 . . . 400 . . . 500 200 . . . 300 . . . 500

qA qA qA qB qB qB

500 . . . 300 . . . 300 500 . . . 600 . . . 400

700 700 800 700 900 900

Barnabé then Albert Albert then Barnabé

200 . . . 400 . . . 500 200 . . . 300 . . . 500

qA qA qA qB qB qB

500 . . . 300 . . . 300 500 . . . 600 . . . 400

700 700 800 700 900 900

Two kinds of interpretationsThose strategies correspond to di�erent programs:

x = askAlbert() x = askBarnabé()y = askBarnabé() y = askAlbert()tot = x + y tot = x + y

These programs have two di�erent interpretations (or semantics):

I Static interpretation via formulae

(Input / Output)

price = price Albert+ price Barnabé

I Dynamic interpretation via strategies

(Interactive process)

. . . 300 . . . 400 . . . 500

. . . qA . . . qA . . . qA

. . . 300 . . . 400 . . . 200

. . . 600 . . . 800 . . . 700

. . . 300 . . . 400 . . . 500

. . . qB . . . qB . . . qB

. . . 300 . . . 400 . . . 200

. . . 600 . . . 800 . . . 700

Two kinds of interpretationsThose strategies correspond to di�erent programs:

x = askAlbert() x = askBarnabé()y = askBarnabé() y = askAlbert()tot = x + y tot = x + y

These programs have two di�erent interpretations (or semantics):

I Static interpretation via formulae (Input / Output)

price = price Albert+ price Barnabé

I Dynamic interpretation via strategies (Interactive process)qB

. . . 300 . . . 400 . . . 500

. . . qA . . . qA . . . qA

. . . 300 . . . 400 . . . 200

. . . 600 . . . 800 . . . 700

. . . 300 . . . 400 . . . 500

. . . qB . . . qB . . . qB

. . . 300 . . . 400 . . . 200

. . . 600 . . . 800 . . . 700

Game semantics

Game semantics interpretation of more complicated formulae:

average(f )=f (

0 ) + f (

Game semantics

average(f )=f (

0 ) + f (

2against f (x)= 2× x + 3

Game semantics

average(f )=f (

0 ) + f (

2against

f (x)= 2× x + 3

Game semantics

average(f )=f (

?0 ) + f (

Game semantics

average(f )=f (0) + f (

2against f (x)= 2× 0+ 3

Game semantics

average(f )=3+ f (

2against f (x)= 3

Game semantics

average(f )=3+ f (

2against

f (x)= 2× x + 3

Game semantics

average(f )=3+ f (

Game semantics

average(f )=3+ f (

2against f (x)= 2× 1+ 3

Game semantics

average(f )=3+ 5

2against f (x)= 5

Game semantics

average(f )= 4 against f (x)= 2× x + 3

Game semantics

average(f )=f (0)+ f (1)

1 n+m2

Game semanticsGame semantics interpretation of more complicated formulae:

average(f )=f (0)+ f (1)

1 n+m2

0 n+m2

left then right right then left

What if we want to compute in parallel?

q n q m

ProblemUsual game semantics only manipulates sequential plays.

q n q m

Is this sound?For which strategies is this optimization correct?

Average

func f1(x):return 2x + 3

func f2(x):if(rand()) return 2× x

0, 1, or 2

else return 0func f3(x):increment counter

depends on the impl.

return counter

Correct sequential strategies are innocent and well-bracketed.

ProblemAnd what about concurrent strategies?

Code Averagefunc f1(x):return 2x + 3 4

func f2(x):if(rand()) return 2× x

0, 1, or 2

else return 0func f3(x):increment counter

return counter

func f2(x):if(rand()) return 2× x 0, 1, or 2else return 0

func f3(x):increment counter

return counter

func f3(x):increment counter depends on the impl.return counter

Partial orders or interleavings?

To represent concurrency:

preheat

dough mold

preheat

partial order

(true concurrency)

Partial orders or interleavings?

To represent concurrency:

preheat

dough mold

preheat

partial order interleavings

(true concurrency)

Game semantics: sequential strategies

CompositionJoyal '77

Non-linear frameworkCoquand '91

Hyland, Ong / Nickau '90sAbramsky, Jagadeesan, Malacaria '90s

InnocenceHO/N '90s

Well-bracketingHO/N '90s, AJM '90s

Functional (PCF)HO/N '90s, AJM '90s

StateAbramsky, McCusker '96

Abramsky, Honda, McCusker '98

ControlCartwright, Curien, Felleisen '94

Laird '97

Prog. Lang.Murawski, Tzevelekos '09 '14

ApplicationsGhica, McCusker '03

Ghica '07Murawski, Tzevelekos '11

InnocenceHO/N '90s

Laird '97

InnocenceHO/N '90s

Laird '97

InnocenceHO/N '90s

Laird '97

Game semantics: concurrent strategies via interleavings

CompositionLaird '01

Non-linear frameworkLaird '01

Innocence?

Well-bracketingGhica, Murawski '07

Functional?

StateGM '07

Control?

Prog. Lang.GM '07

Game semantics: truly concurrent strategies

CompositionAbramsky, Melliès '99

Curien, Faggian '05 and Faggian, Piccolo '09Rideau, Winskel '11

Non-linear framework?

InnocenceMelliès-Mimram '07

(II) CCW '15

Well-bracketing?

Functional?

State?

Control?

Prog. Lang.?

Non-linear framework(I) C., Clairambault, Winskel '14 '15

(II) CCW '15

Well-bracketing?

Functional?

State?

Control?

Prog. Lang.?

(II) CCW '15

Well-bracketingC., Clairambault '17

(similar to GM '07)

Functional(II) CCW '15

State?

Control?

Prog. Lang.?

(II) CCW '15

(similar to GM '07)

StateC., Clairambault '16

via collapse to GM '07

Control?

Prog. Lang.?

(II) CCW '15

(similar to GM '07)

StateC., Clairambault '16

via collapse to GM '07

Control?

Prog. Lang.(III) C. '15 '17

I. A framework for partial-order strategies

GamesGame: partial order where each move has a polarity (Opponent,Player).

qA qB ok

yesA yesB

(N ⇒ N) ⇒ Nq

From a game A we build its dual A⊥ by reversing polarities:

deal⊥

qA qB ok

yesA yesB

qA qB ok

yesA yesB

(N ⇒ N) ⇒ Nq

deal⊥

qA qB ok

yesA yesB

qA qB ok

yesA yesB

(N ⇒ N) ⇒ Nq

deal⊥

qA qB ok

yesA yesBStructures concurrentes en sémantique des jeux · Simon Castellan, LIP Soutenance de thèse 14 / 37

(Deterministic) Strategies

dealqA qB ok

yesA yesB

σA‖B : Albert and Barnabé in parallel

Strategies on A⊥ represent counter-strategies:

deal⊥

qA qB ok

yesA yesB

dealqA qB ok

yesA yesB

σA;B : Albert then Barnabé

deal⊥

qA qB ok

yesA yesB

dealqA qB ok

yesA yesB

deal⊥

qA qB ok

yesA yesB

dealqA qB ok

yesA yesB

De�nitionA strategy on (A,≤A) is a partial order σ = (S ,≤S) such that

I S ⊆ A, and if s ≤A s ′ then s ≤S s ′ (rule-respecting)

I S only adds immediate links �_ ⊕ (courteous)

dealqA qB ok

yesA yesB

τ : deal⊥

qA qB ok

yesA yesB

Interaction of strategies

Given a strategy on A and one on A⊥, how do they interact?

σA‖B : deal σA‖B ~ τ τ : deal⊥

qA qB ok

yesA yesB

De�nitionThe interaction of (S ,≤S) and (T ,≤T ) is obtained from

(S ∩ T , (≤S ∪ ≤T )∗)

by removing events occurring in a causal loop.

qA qB ok qA qB

qA qB ok

yesA yesB

(S ∩ T , (≤S ∪ ≤T )∗)

qA qB ok qA qB

qA qB ok

yesA yesB yesA yesB yesA yesB

(S ∩ T , (≤S ∪ ≤T )∗)

qA qB ok qA qB ok qA qB ok

(S ∩ T , (≤S ∪ ≤T )∗)

σA;B : deal σA;B ~ τ τ : deal⊥

qA qB ok

yesA yesB

(S ∩ T , (≤S ∪ ≤T )∗)

Composition

Interaction is used to compute application and composition of strat.

average

: (N ⇒ N) ⇒ Nq

1 n+m2

d : N ⇒ N

2× d + 3

Composition

average

: (N ⇒ N) ⇒ Nq

1 n+m2

d : N ⇒ N

2× d + 3

Composition

average~ d : (N ⇒ N) ⇒ Nq

d : N ⇒ N

2× d + 3

Composition

average� d :

(N ⇒ N) ⇒

d : N ⇒ N

2× d + 3

Linearity & duplication

Our average strategy is not a strategy!

It is not linear:

average(f ) =f (0) + f (1)

average : (N ⇒ N) ⇒ Nq

1 n+m2

(N ⇒ N) ⇒ N

Linearity & duplication

Our average strategy is not a strategy! It is not linear:

average(f ) =f (0) + f (1)

average : (N ⇒ N) ⇒ Nq

1 n+m2

(N ⇒ N) ⇒ N

Copy indices & the game !ATo solve this problem, we play on expanded arenas.

n !N =

q0 . . . qi . . .

0j 1k . . . . . . nl . . .

A (parallel) strategy implementing d(x) := x + x becomes:

!( N ⇒ N )

(n +m)f (j ,k)

Copy indices & the game !ATo solve this problem, we play on expanded arenas.

n !N =

q0 . . . qi . . .

0j 1k . . . . . . nl . . .

A (parallel) strategy implementing d(x) := x + x becomes:

!( N ⇒ N )

q17 q43

(n +m)f (j ,k)

The cartesian-closed category CHO

We get a cartesian closed category (CCC):

Theorem (C., Clairambault, Winskel)

The following structure CHO is a CCC:

Objects Games A (which are alternating forests)

Morphisms Strategies uniform (and single-threaded) on !(A⇒ B).

(Usual sequential innocent HO games form a subcategory of CHO)

Rich semantic framework to interpret concurrent higher-orderprograms.

What was swept under the rug

I Manage to identify strategies up to choice of copy indices.

A notion of weak isomorphism

I De�ne a notion of uniformity (when a strategy is blind toOpponent indices).

Strategies become equipped with symmetry.

I Show that, on uniform strategies, weak isomorphism is acongruence.

Proof of a bipullback property of interaction.

I Representation of nondeterminism in strategies

Addition of essential events.

What was swept under the rug

I Manage to identify strategies up to choice of copy indices.

A notion of weak isomorphism

I De�ne a notion of uniformity (when a strategy is blind toOpponent indices).

Strategies become equipped with symmetry.

I Show that, on uniform strategies, weak isomorphism is acongruence.

Proof of a bipullback property of interaction.

I Representation of nondeterminism in strategies

Addition of essential events.

Nondeterminism via event structures

Nondeterminism can be represented via a con�ict relation:

coin : B

De�nitionAn event structure is a partial order E equipped with a binaryrelation (representing con�ict) satisfying some axioms.

Nondeterminism via event structures

Nondeterminism can be represented via a con�ict relation:

coin : B

De�nitionAn event structure is a partial order E equipped with a binaryrelation (representing con�ict) satisfying some axioms.

Hidden divergences via essential events

But, nondeterminism and composition require some care:

} coin

: B ⇒ B

σ � coin coincides with ff !

τ } σ retains more information than τ � σ (must adequacy)

σ ~ coin :

σ � coin :

σ } coin :

II. Order of evaluation and innocence

PCF and its interpretationsWe can interpret PCF in CHO:

A,B ::= N | B | proc | A⇒ B (PCF types)

M,N ::= tt | ff | ifM N1N2 | () | M;N

| x | λx .M | M N | Y | . . . (PCF terms)

Two interpretations of if in CHO:

ifs : B ⇒ N ⇒ N ⇒ N

ifp : B ⇒ N ⇒ N ⇒ N

tt ff n m

These two interpretations are indistinguishable by terms of PCF.

M,N ::= tt | ff | ifM N1N2 | () | M;N

ifs : B ⇒ N ⇒ N ⇒ N

ifp : B ⇒ N ⇒ N ⇒ N

tt ff n m

These two interpretations are indistinguishable by terms of PCF.

M,N ::= tt | ff | ifM N1N2 | () | M;N

ifs : B ⇒ N ⇒ N ⇒ N

ifp : B ⇒ N ⇒ N ⇒ N

tt ff n m

These two interpretations are indistinguishable by terms of PCF.Structures concurrentes en sémantique des jeux · Simon Castellan, LIP Soutenance de thèse 25 / 37

Sometimes parallel if just is not the same

sync : (proc ⇒ proc ⇒ proc) ⇒ proc

q q ()

() () ()

sync can distinguish both implementations of if.

( For Mε = λxλy .ifε (x ; tt) (y ; tt) tt:

syncMp converges but not syncMs .)

What is wrong?

Player merges two threads started by Opponent.

Banning such patterns gives a notion of concurrent innocence.

q q ()

() () ()

What is wrong?

q q ()

() () ()

What is wrong?

q q ()

() () ()

What is wrong?

q q ()

() () ()

What is wrong? Player merges two threads started by Opponent.

q q ()

() () ()

What is wrong? Player merges two threads started by Opponent.

Banning such patterns gives a notion of concurrent innocence.Structures concurrentes en sémantique des jeux · Simon Castellan, LIP Soutenance de thèse 26 / 37

Finite de�nability

A PCF strategy is an innocent and well-bracketed strategy.

Theorem (Finite de�nability)

If σ is PCF strategy, there exists a term M of PCF such that

JMK and σ are indistinguishable by PCF strategies.

B ⇒ proc ⇒ Bq

tt ff ()

λbλx . if b⊥ x

B ⇒ proc ⇒ Bq

tt ff ()

λbλx . if b⊥ x

B ⇒ proc ⇒ Bq

tt ff ()

λbλx . x ; if b⊥ ()

What was swept under the rug 2

I Innocence and well-bracketing:I Stability under composition. Forking lemma.

I Requires the addition of visibility (and locality). Control interferences between Player threads

I Finite de�nability:I De�ne a notion of �nite strategies. Reduced form (P-view �dag�)

I Factorisation theorem for higher-order strategies. First-order / λ-calculus

What was swept under the rug 2

I Innocence and well-bracketing:I Stability under composition. Forking lemma.

I Requires the addition of visibility (and locality). Control interferences between Player threads

I Finite de�nability:I De�ne a notion of �nite strategies. Reduced form (P-view �dag�)

I Factorisation theorem for higher-order strategies. First-order / λ-calculus

III. What lies beyond PCF?

What real concurrent programs are made of

while(1) { while(1) {while (canWrite == 0) ; while (canRead == 0) ;x = produce(); x = value;value := x; consume(x);canRead := 1; canWrite := 0; canRead := 0; canWrite := 1;

loops, conditionals, function calls, shared memory.

while(1) { while(1) {while (canWrite == 0) ; while (canRead == 0) ;x = produce(); x = value;value := x; consume(x);canRead := 1; canWrite := 0; canRead := 0; canWrite := 1;

loops, conditionals, function calls︸︷︷︸in PCF

, shared memory.

value := 1; f ← canRead;canRead := 1; x ← value ;

Expectation: f = 1 implies x = 1

loops, conditionals, function calls, shared memory

value := 1; f ← canRead;canRead := 1; x ← value ;

Expectation: f = 1 implies x = 1

loops, conditionals, function calls, shared memory

Running concurrent programs on a processor

value:= 1;

canRead:= 1;

f ← canRead;

v ← value;

value: 0

canRead: 0

memory

Several executions, but never f = 1 and v = 0.

value:= 1;

canRead:= 1;

f ← canRead;

v ← value;

value: 0

canRead: 0

memory

write 1 to value

canRead?

value:= 1;

canRead:= 1;

f ← canRead;

v ← value;

value: 1

canRead: 0

memory

canRead?

value:= 1;

canRead:= 1;

f ← canRead;

v ← value;

value: 1

canRead: 0

memory

write 1 to canRead

canRead?

value:= 1;

canRead:= 1;

f ← canRead;

v ← value;

value: 1

canRead: 1

memory

canRead?

value:= 1;

canRead:= 1;

f ← canRead;

v ← value;

value: 1

canRead: 1

memory

canRead=1

value:= 1;

canRead:= 1;

f ← canRead;

v ← value;

value: 1

canRead: 1

memory

value?

value:= 1;

canRead:= 1;

f ← canRead;

v ← value;

value: 1

canRead: 1

memory

value=1

value:= 1;

canRead:= 1;

f ← canRead;

v ← value;

value: 1

canRead: 1

memory

Weak memory models

On some processors, we get instead:

value:= 1;

canRead:= 1;

f ← canRead;

v ← value;

value: 0

canRead: 0

memory

Outcome depends on the architecture (TSO, PSO, ARM). . .

Goal: Model denotationally such complex reorderings.

Weak memory models

value:= 1;

canRead:= 1;

f ← canRead;

v ← value;

value: 0

canRead: 0

memory

write 1 to value

write 1 to canRead

canRead?

Weak memory models

value:= 1;

canRead:= 1;

f ← canRead;

v ← value;

value: 0

canRead: 1

memory

write 1 to value

canRead?

Weak memory models

value:= 1;

canRead:= 1;

f ← canRead;

v ← value;

value: 0

canRead: 1

memory

canRead=1

write 1 to value

Weak memory models

value:= 1;

canRead:= 1;

f ← canRead;

v ← value;

value: 0

canRead: 1

memory

write 1 to value

value?

Weak memory models

value:= 1;

canRead:= 1;

f ← canRead;

v ← value;

value: 0

canRead: 1

memory

value=0

write 1 to value

Weak memory models

value:= 1;

canRead:= 1;

f ← canRead;

v ← value;

value: 0

canRead: 1

memory

write 1 to value

Weak memory models

value:= 1;

canRead:= 1;

f ← canRead;

v ← value;

value: 1

canRead: 1

memory

Weak memory models

value:= 1;

canRead:= 1;

f ← canRead;

v ← value;

value: 1

canRead: 1

memory

Weak memory models

value:= 1;

canRead:= 1;

f ← canRead;

v ← value;

value: 1

canRead: 1

memory

Taking a closer look

x:= 1;

y:= 1;

r ← z;

The behaviour of the thread corresponds to a strategy:

mem ⇒ proc

Cx :=1 Cy :=1

x:= 1;

y:= 1;

r ← z;

z: ?write 1 to x

write 1 to y

mem ⇒ proc

Cx :=1 Cy :=1

x:= 1;

y:= 1;

r ← z;

z: ?ok

mem ⇒ proc

Cx :=1 Cy :=1

x:= 1;

y:= 1;

r ← z;

z: ?z ?

mem ⇒ proc

Cx :=1 Cy :=1

x:= 1;

y:= 1;

r ← z;

mem ⇒ proc

Cx :=1 Cy :=1

x:= 1;

y:= 1;

r ← z;

mem ⇒ proc

Cx :=1 Cy :=1

()Structures concurrentes en sémantique des jeux · Simon Castellan, LIP Soutenance de thèse 33 / 37

Modelling weak memory models

An old motto (Reynolds, implemented in game semantics by Abramsky and McCusker)

The behaviour of imperative programs can be described as the

interaction of a (deterministic) pure functional program interacting

with a memory.

Individual programs are interpreted as parallel strategies

JtK : mem⇒ proc

In particularJt; uK = seq JtK JuK

seq : mem⇒ (mem⇒ proc)⇒ (mem⇒ proc)⇒ proc

The behaviour of concurrent programs can be described as the

interaction of (deterministic) pure functional threads interacting

with a (nondeterministic) memory.

JtK : mem⇒ proc

seq : (mem⇒ proc)⇒ (mem⇒ proc)⇒ (mem⇒ proc)

JtK : mem⇒ proc

Thread semantics via game semantics

Implementation of seq (here for PSO):

mem ⇒ (mem ⇒ proc) ⇒ ( mem ⇒ proc) ⇒ proc

Cx :=1 () Cx :=2 Cy :=3 ()

Cx :=1 ()

Cx :=2

Cy :=3 ok

Cx :=1

Cx :=2 Cy :=3

Cx :=1

Cx :=2

Cy :=3 ok

Cx :=1 ()

Cx :=2 Cy :=3

Cx :=1

Cx :=2

Cy :=3 ok

Cx :=1 ()

Cx :=2 Cy :=3

Cx :=1 ()

Cx :=2

Cy :=3 ok

Cx :=1 () Cx :=2

Cy :=3

Cx :=1 ()

Cx :=2

Cy :=3 ok

Cx :=1 () Cx :=2

Cy :=3

Cx :=1 ()

Cx :=2

Cy :=3

Cx :=1 () Cx :=2 Cy :=3 ()

Cx :=1 ()

Cx :=2

Cy :=3

Cx :=1 () Cx :=2 Cy :=3 ()

Cx :=1 ()

Cx :=2

Cy :=3 ok

Final modelDepends on the architecture A:I Thread operations: seqA, readA, writeA implement the

reorderings speci�c to A.

I Memory (representing caches, barriers, . . . ) is represented by

mA : mem

Executions on A of t1 ‖ . . . ‖ tn are represented by the interaction:

(Jt1KA ‖ . . . ‖ JtnKA)~mA.

Cx :=1 R?x cy :=1 R?y

ok 0 ok 0

R?x Cx :=1 R?y Cy :=1

1 ok 1 ok

TheoremTraces generated by this

model correspond to

operational traces (on

Final modelDepends on the architecture A:I Thread operations: seqA, readA, writeA implement the

mA : mem

Cx :=1 R?x cy :=1 R?y

ok 0 ok 0

R?x Cx :=1 R?y Cy :=1

1 ok 1 ok

model correspond to

Final model

Depends on the architecture A:I Thread operations: seqA, readA, writeA implement the

mA : mem

Cx :=1 Rx=0 Cy :=1 Ry=0

Rx=1 Cx :=1 Ry=1 Cy :=1

model correspond to

Perspectives

I Describe �nitely non-innocent strategies.

Represent e�ciently operations on them.

I Which language corresponds to innocent concurrent strategies?

Concurrent control operators (eg. fork).

I Weaker architectures (eg. ARM) and software speci�cations.

How to handle speculation, complex barriers speci�cation.

Simon Castellan, LIP Soutenance de thèse 13 juillet 2017iso.mor.phis.me/publis/soutenance.pdf · A...

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castellan-75-1993By Christiane Jaeger, Aziz Nourmamode and Alain Castellan Laboratoire de Photophysique et Photochimie Moléculaire, CNRS URA 348, Université Bordeaux I, F-33405 Talence

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Solution Castellan - Cap1

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Introduction - Princeton Universityassets.press.princeton.edu/chapters/s8767.pdf · Castellan Barbadori Acciaiuol FIGURE 1.1 Network showing ﬁfteenth-century Florentine marriages

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castellan-publicatio.monsite-orange.fr · Created Date: 6/27/2016 8:51:23 PM

Digital Vision for Cyber Security - Home - Atos · 2018-12-03 · Digital Vision for Cyber Security Adrian Gregory Chief Executive Officer, Atos UK & Ireland Pierre Barnabé Executive

LSVbouyer/files/bouyer-hdr-soutenance.pdf · Reachability analysis in timed automata Improving further [BBFL03] Behrmann, Bouyer, Fleury, Larsen. Static Guard Analysis in Timed Automata