Contracts for Mobile Processes Giuseppe Castagna Luca Padovani Laboratoire PPS, CNRS, Universit´ e Paris Diderot Istituto di Scienze e Tecnologie dell’Informazione, Universit` a di Urbino CONCUR 2009 [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 1 / 24
Outline 1 Motivation Protocols and processes Contracts and mobile systems 2 Contracts Syntax Semantics 3 Results 4 Concluding remarks [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 2 / 24
Protocols and processes Session types • prescriptions on the use of channels u : σ, v : τ, · · · ⊢ P Contracts • overall process behavior u : Ch , v : Ch , · · · ⊢ P : T Summary • both are behavioral types • σ = projection of T on u [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 3 / 24
What session types and contracts are for Characterizing well-formed systems • the system eventually terminates • the system never deadlocks Characterizing well-typed processes • sent messages have the correct/expected type • messages sent/delivered in the right order Reasoning about processes by means of their type • refactoring processes • searching for services [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 4 / 24
A problem of abstraction Session types Contracts ? Int . ? Int . (! Real ⊕ ! Error ) a . a . ( b ⊕ c ) ?(! Bool . ! Bool ) a ? A natural candidate Contracts without channel passing ⇒ ccs Contracts with channel passing ⇒ π -calculus [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 5 / 24
A problem of abstraction Session types Contracts ? Int . ? Int . (! Real ⊕ ! Error ) a . a . ( b ⊕ c ) ?(! Bool . ! Bool ) a ? A natural candidate Contracts without channel passing ⇒ ccs Contracts with channel passing ⇒ π -calculus [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 5 / 24
An example process store ?( x ) . x ?( y : Item ) . if y is in stock then bank ! � x � else x ! � available( y ) � ⇓ contract store ?( x ) . x ? Item . ( bank ! x . 1 ⊕ x ! Date . 1 ) [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 6 / 24
An example process store ?( x ) . x ?( y : Item ) . if y is in stock then bank ! � x � else x ! � available( y ) � ⇓ contract store ?( x ) . x ? Item . ( bank ! x . 1 ⊕ x ! Date . 1 ) [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 6 / 24
An example process store ?( x ) . x ?( y : Item ) . if y is in stock then bank ! � x � else x ! � available( y ) � ⇓ contract store ?( x ) . x ? Item . ( bank ! x . 1 ⊕ x ! Date . 1 ) [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 6 / 24
An example process store ?( x ) . x ?( y : Item ) . if y is in stock then bank ! � x � else x ! � available( y ) � ⇓ contract store ?( x ) . x ? Item . ( bank ! x . 1 ⊕ x ! Date . 1 ) [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 6 / 24
Some typing rules v-send v-recv Γ ⊢ e : t Γ ⊢ P : T Γ , x : t ⊢ P : T Γ ⊢ α ! e . P : α ! t . T Γ ⊢ α ?( x : t ) . P : α ? t . T c-send c-recv Γ ⊢ P : T Γ , x : Ch ⊢ P : T Γ ⊢ α ! � β � . P : α ! β. T Γ ⊢ α ? � x � . P : α ?( x ) . T [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 7 / 24
Some typing rules v-send v-recv Γ ⊢ e : t Γ ⊢ P : T Γ , x : t ⊢ P : T Γ ⊢ α ! e . P : α ! t . T Γ ⊢ α ?( x : t ) . P : α ? t . T c-send c-recv Γ ⊢ P : T Γ , x : Ch ⊢ P : T Γ ⊢ α ! � β � . P : α ! β. T Γ ⊢ α ? � x � . P : α ?( x ) . T undecidable → decidable [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 7 / 24
Outline 1 Motivation Protocols and processes Contracts and mobile systems 2 Contracts Syntax Semantics 3 Results 4 Concluding remarks [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 8 / 24
Syntax failure, success ::= 0 | 1 π. T | T + T | T ⊕ T T | T | ( ν a ) T T ::= α ? f | α ! f | α !( a ) π x | ( x ) | a | Int | Bool | · · · f ::= Infinite behaviors = infinite terms • regularity X = c ? Int . X • boundedness X = a ?( x ) . ( c ! x . 1 | X ) [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 9 / 24
Syntax dynamic operators ::= 0 | 1 π. T | T + T | T ⊕ T T | T | ( ν a ) T T ::= α ? f | α ! f | α !( a ) π x | ( x ) | a | Int | Bool | · · · f ::= Infinite behaviors = infinite terms • regularity X = c ? Int . X • boundedness X = a ?( x ) . ( c ! x . 1 | X ) [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 9 / 24
Syntax systems ::= 0 | 1 π. T | T + T | T ⊕ T T | T | ( ν a ) T T ::= α ? f | α ! f | α !( a ) π x | ( x ) | a | Int | Bool | · · · f ::= Infinite behaviors = infinite terms • regularity X = c ? Int . X • boundedness X = a ?( x ) . ( c ! x . 1 | X ) [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 9 / 24
Syntax ::= 0 | 1 π. T | T + T | T ⊕ T T | T | ( ν a ) T T ::= α ? f | α ! f | α !( a ) prefixes π x | ( x ) | a | Int | Bool | · · · f ::= Infinite behaviors = infinite terms • regularity X = c ? Int . X • boundedness X = a ?( x ) . ( c ! x . 1 | X ) [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 9 / 24
Syntax ::= 0 | 1 π. T | T + T | T ⊕ T T | T | ( ν a ) T T ::= α ? f | α ! f | α !( a ) π x | ( x ) | a | Int | Bool | · · · f ::= Infinite behaviors = infinite terms patterns = sets of values and names + binders • regularity X = c ? Int . X • boundedness X = a ?( x ) . ( c ! x . 1 | X ) [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 9 / 24
Syntax ::= 0 | 1 π. T | T + T | T ⊕ T T | T | ( ν a ) T T ::= α ? f | α ! f | α !( a ) π x | ( x ) | a | Int | Bool | · · · f ::= Infinite behaviors = infinite terms • regularity X = c ? Int . X • boundedness X = a ?( x ) . ( c ! x . 1 | X ) [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 9 / 24
Labeled operational semantics � − → 1 1 m ∈ f � σ m ∈ f c ! m c ! m . T − → T c ? m c ? f . T − → T σ c ! f . T − → c ! m . T 20 ∈ Int 20 ∈ Real � ∅ Example � c ! Int . 1 | c ? Real . 1 − → c !20 . 1 | c ? Real . 1 − → 1 | 1 − → [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 10 / 24
Contracts as behavioral types Systems def = T 1 | T 2 | · · · | T n S 1 when is a system well-formed? 2 when is a process well-typed? 3 when are two types equal? [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 11 / 24
Participant satisfaction Definition ⇒ T ′ | S ′ and T ′ � T ⊳ S if T | S = − → implies • T ′ µ 1 → and S ′ µ 2 − ⇒ = • µ 1 # µ 2 ( c ! m # c ? m , � # � ) for some µ 1 and µ 2 Examples • c ! Int . 1 ⊳ c ? Real . 1 • c ! Real . 1 � ⊳ c ? Int . 1 √ c ! Real . 1 | c ? Int . 1 − → c ! 2 . 1 | c ? Int . 1 stuck [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 12 / 24
Well-formed systems def = T 1 | T 2 | · · · | T n S Definition � S is well formed if for every 1 ≤ k ≤ n T k ⊳ T i i ∈{ 1 ,..., n }\{ k } Examples • c ! Int . 1 | c ? Real . 1 is well formed • c ! Real . 1 | c ? Int . 1 is ill formed [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 13 / 24
Well-typed participant Definition T is viable if T | S is well formed for some S Example def T = c ? Int . 1 + c ? Bool . 0 def = c ? Int . 0 + c ? Bool . 1 S • T is viable • S is viable • T ⊕ S is not viable [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 14 / 24
Example: global order on channels def P = a ? � x � . b ? � y � . x !3 . x ?( z : Int ) . y ! true . 0 def P ′ = a ? � x � . b ? � y � . x !3 . y ! true . x ?( z : Int ) . 0 def Q = a !( c ) . b !( d ) . c ?( z : Int ) . d ?( z : Bool ) . c !5 . 0 a !( c ) . b !( d ) . c ?( z : Int ) . c !5 . d ?( z ′ : Bool ) . 0 def Q ′ = • deadlock because of cyclic dependency • T P | T Q ill-formed ( not viable! ) [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 15 / 24
Example: global order on channels def P = a ? � x � . b ? � y � . x !3 . x ?( z : Int ) . y ! true . 0 def P ′ = a ? � x � . b ? � y � . x !3 . y ! true . x ?( z : Int ) . 0 def Q = a !( c ) . b !( d ) . c ?( z : Int ) . d ?( z : Bool ) . c !5 . 0 a !( c ) . b !( d ) . c ?( z : Int ) . c !5 . d ?( z ′ : Bool ) . 0 def Q ′ = • imposing global order • T P | T Q ′ well-formed [ slides by L.Padovani ] Contracts for Mobile Processes (G.Castagna/L.Padovani) CONCUR 2009 15 / 24
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