A Compositional Encoding for the Asynchronous Pi-Calculus into the Join-Calculus

A Compositional Encoding of the Asynchronousπ-Calculus into the Join-Calculus

Stephan Mennicke† Tobias Prehn‡ Tsvetelina Yonova-Karbe‡

Institute for Programming and Reactive Systems, TU Braunschweig†

Institute for Software Engineering and Theoretical Computer Science, TU Berlin‡

September 3, 2012YR-CONCUR 2012

WraP – Writing and Publishing a Scientific Paper

Supervision by Uwe Nestmann andKirstin Peters

S. Mennicke (TU Braunschweig) Compositional πa ⇒ Join YR-CONCUR 2012 2 / 15

Scope of Interest

Asynchronous π (πa) Join-Calculus


Scope of Interest

Asynchronous π (πa)

x〈v〉

x(w).P

x(w)∗.P

x(w).P |x〈v〉

P |Q

Join-Calculus

x〈v〉

def x〈w〉 | o〈〉 . P in . . . | o〈〉

def x〈w〉 . P in . . .

def x〈w〉 . P in x〈v〉

P |Q

Send

Receive

Recursion

Communication

Parallelism


Scope of Interest


x〈v〉

x(w).P

x(w)∗.P

x(w).P |x〈v〉

P |Q

Join-Calculus

x〈v〉

def x〈w〉 | o〈〉 . P in . . . | o〈〉

def x〈w〉 . P in . . .


P |Q

Send

Receive

Recursion

Communication

Restriction

Parallelism


Scope of Interest


x〈v〉

x(w).P

x(w)∗.P

x(w).P |x〈v〉

(νx)(x(w).P |x〈v〉)

P |Q

Join-Calculus

x〈v〉

def x〈w〉 | o〈〉 . P in . . . | o〈〉

def x〈w〉 . P in . . .


P |Q

Send

Receive

Recursion

Communication

Restriction

Parallelism


Scope of Interest


x〈v〉

x(w).P

x(w)∗.P

x(w).P |x〈v〉

(νx)(x(w).P |x〈v〉)

P |Q

Join-Calculus

x〈v〉

def x〈w〉 | o〈〉 . P in . . . | o〈〉

def x〈w〉 . P in . . .



P |Q

Send

Receive

Recursion

Communication

Restriction

Parallelism


Scope of Interest


x〈v〉

x(w).P

x(w)∗.P

x(w).P |x〈v〉

(νx)(x(w).P |x〈v〉)

P |Q

Join-Calculus

x〈v〉

def x〈w〉 | o〈〉 . P in . . . | o〈〉

def x〈w〉 . P in . . .



P |Q

Send

Receive

Recursion

Communication

Restriction

Parallelism


Scope of Interest


x〈v〉

x(w).P

x(w)∗.P

x(w).P |x〈v〉

(νx)(x(w).P |x〈v〉)

P |Q

Join-Calculus

x〈v〉

def x〈w〉 | o〈〉 . P in . . . | o〈〉

def x〈w〉 . P in . . .



P |Q

Send

Receive

Recursion

Communication

Restriction

Parallelism

J Kπj


An Encoding is Non-Trivial

|

x(w)

x〈v〉no (νx)



|

x(w)

x〈v〉no (νx)



|

def x

x〈v〉

|

x(w)

x〈v〉no (νx)



|

def x

x〈v〉

|

x(w)

x〈v〉no (νx)


The Encoding of Fournet and Gonthier[Fournet and Gonthier(1996)]

First Layer

x〈v〉 7→ xo〈v〉P |Q 7→ P |Qx(w).P 7→ def κ . JP K in xi(νx)P 7→ def xi |xo . κ in JP K

Second Layer

Restrict all free names

Is this good?



First Layerx〈v〉 7→ xo〈v〉P |Q 7→ P |Q

x(w).P 7→ def κ . JP K in xi(νx)P 7→ def xi |xo . κ in JP K

Second Layer


Is this good?



First Layerx〈v〉 7→ xo〈v〉P |Q 7→ P |Qx(w).P 7→ def κ . JP K in xi

(νx)P 7→ def xi |xo . κ in JP K

Second Layer


Is this good?



First Layerx〈v〉 7→ xo〈v〉P |Q 7→ P |Qx(w).P 7→ def κ . JP K in xi(νx)P 7→ def xi |xo . κ in JP K

Second Layer


Is this good?




Second LayerRestrict all free names

Is this good?




Second LayerRestrict all free names

Is this good?S. Mennicke (TU Braunschweig) Compositional πa ⇒ Join YR-CONCUR 2012 5 / 15

Gorla’s Criteria for Good Encodings [Gorla(2010)]

J.K



J.K

I. Compositionality



J.K

I. Compositionality

II. Name Invariance



Syntactic Criteria

J.K

I. Compositionality

II. Name Invariance



Syntactic Criteria

J.K

I. Compositionality

II. Name Invariance

III. Operational Correspondence



Syntactic Criteria

J.K

I. Compositionality

II. Name Invariance


IV. Divergence Reflection



Syntactic Criteria

J.K

I. Compositionality

II. Name Invariance



V. Success Sensitiveness



Semantic CriteriaSyntactic Criteria

J.K

I. Compositionality

II. Name Invariance



V. Success Sensitiveness


Fournet and Gonthier’s Encoding Revisited

First level encoding is compositional and operationally correspondentfor closed terms

First level encoding is not operationally correspondent for opentermsFirst+Second level encoding is not compositional

Our Approach

We introduce send/receive requests carrying the channel namesWe keep the main idea of restrictionWe implement a protocol to handle communication[Peters and Nestmann(2011)]We need matching to decide which requests may cooperate



First level encoding is compositional and operationally correspondentfor closed termsFirst level encoding is not operationally correspondent for openterms

First+Second level encoding is not compositional

Our Approach




First level encoding is compositional and operationally correspondentfor closed termsFirst level encoding is not operationally correspondent for opentermsFirst+Second level encoding is not compositional

Our Approach





Our Approach





Our ApproachWe introduce send/receive requests carrying the channel names

We keep the main idea of restrictionWe implement a protocol to handle communication[Peters and Nestmann(2011)]We need matching to decide which requests may cooperate




Our ApproachWe introduce send/receive requests carrying the channel namesWe keep the main idea of restriction

We implement a protocol to handle communication[Peters and Nestmann(2011)]We need matching to decide which requests may cooperate




Our ApproachWe introduce send/receive requests carrying the channel namesWe keep the main idea of restrictionWe implement a protocol to handle communication[Peters and Nestmann(2011)]

We need matching to decide which requests may cooperate




Our ApproachWe introduce send/receive requests carrying the channel namesWe keep the main idea of restrictionWe implement a protocol to handle communication[Peters and Nestmann(2011)]We need matching to decide which requests may cooperate


The Big Picture

(1) J x〈v〉 Kπj = def lf 〈t, f〉 | false〈〉 . f〈〉 | false〈〉 in(2) def lt〈t, f〉 | true〈〉 . t〈〉 | false〈〉 in(3) def l〈t, f〉 . lt〈t, f〉 | lf 〈t, f〉 in sr〈x, v, l〉 | true〈〉(4) J P |Q Kπj = def rrτ 〈c, k〉 | trans0〈m〉 . m〈c, k〉 | trans0〈m〉 in(5) def chain〈trans0〉 | srτ 〈c, v, l〉.(6) def mup〈c, k〉 | trans〈m〉 . m〈c, k〉 | trans〈m〉 in(7) def m〈c′, k〉.(8) [c = c′]k〈v, l〉 |mup〈c′, k〉(9) in trans0〈m〉 | chain〈trans〉

(10) in chain〈trans0〉 |(11) def sr〈c, v, l〉 . srup〈c, v, l〉 | srτ 〈c, v, l〉 in(12) def srup〈c, v, l〉 . sr〈c, v, l〉 in(13) def rr〈c, k〉 . rrup〈c, k〉 | rrτ 〈c, k〉 in(14) def rrup〈c, k〉 . rr〈c, k〉 in(15) J P Kπj | J Q Kπj(16) J x(v).P Kπj = def k〈v, l〉 | once〈〉.(17) def t〈〉 . J P Kπj in(18) def f〈〉 . once〈〉 in l〈t, f〉(19) in rr〈x, k〉 | once〈〉(20) J (νx)P Kπj = def x〈〉 . 0 in J P Kπj


What our Encoding does

P = (νx)(y〈a〉 |x(u).Q) |x〈b〉 | y(z).R



P = (νx)(y〈a〉 |x(u).Q) |x〈b〉 | y(z).R

|

νx |

|

y x

x y



P = (νx)(y〈a〉 |x(u).Q) |x〈b〉 | y(z).R

|

νx |

|

y x

x y

y x

x y



P = (νx)(y〈a〉 |x(u).Q) |x〈b〉 | y(z).R

|

νx |

|

y x

x y

x

x yy



P = (νx)(y〈a〉 |x(u).Q) |x〈b〉 | y(z).R

|

νx |

|

y x

x y

x

x yy

y



P = (νx)(y〈a〉 |x(u).Q) |x〈b〉 | y(z).R

|

νx |

|

y x

x yx yy

y

x



P = (νx)(y〈a〉 |x(u).Q) |x〈b〉 | y(z).R

|

νxνx |

|

y x

x yx yy

y

x

x



P = (νx)(y〈a〉 |x(u).Q) |x〈b〉 | y(z).R

|

νx |

|

y x

x y yy

y

x

x



P = (νx)(y〈a〉 |x(u).Q) |x〈b〉 | y(z).R

|

νx |

|

y x

x yy

y

x

x y



P = (νx)(y〈a〉 |x(u).Q) |x〈b〉 | y(z).R

|

νx |

|

y x

x yy

y

x

x y

y



P = (νx)(y〈a〉 |x(u).Q) |x〈b〉 | y(z).R

|

νx |

|

y x

x yy

y

x

x y

y

x



P = (νx)(y〈a〉 |x(u).Q) |x〈b〉 | y(z).R

|

νx |

|

y x

x yy

y

x

x y

y

x



P = (νx)(y〈a〉 |x(u).Q) |x〈b〉 | y(z).R

|

νx |

|

y x

x yy

y

x

x y

y

x



P = (νx)(y〈a〉 |x(u).Q) |x〈b〉 | y(z).R

|

νx |

|

y x

x yy

y

x

x y

y

x


A Receive Requests (x(v).P )

x



def krr〈x, k〉



def k

rr〈x, k〉



def k



def k

k〈v, l〉



k〈v, l〉



if l then JP K


B Send Requests (x〈v〉)

x

def lsr〈x, v, l〉true〈〉

def ltrue〈〉

sr〈x, v, l〉l〈t, f〉

def l

t〈〉l〈t, f〉f〈〉



x


def ltrue〈〉


def l

t〈〉l〈t, f〉f〈〉



x


def ltrue〈〉

sr〈x, v, l〉

l〈t, f〉

def l

t〈〉l〈t, f〉f〈〉



x


def ltrue〈〉


def l

t〈〉l〈t, f〉f〈〉



x


def ltrue〈〉

sr〈x, v, l〉

l〈t, f〉

def l

t〈〉l〈t, f〉f〈〉



x


def ltrue〈〉


def l

t〈〉

l〈t, f〉f〈〉



x


def ltrue〈〉


def l

t〈〉

l〈t, f〉

f〈〉



x


def ltrue〈〉


def l

t〈〉l〈t, f〉

f〈〉


C Restriction (νx)


C Restriction (νx)

¬x


D Parallel Composition ( | )

Remember this?J P |Q Kπj = def rrτ 〈c, k〉 | trans0〈m〉 . m〈c, k〉 | trans0〈m〉 in

def chain〈trans0〉 | srτ 〈c, v, l〉.def mup〈c, k〉 | trans〈m〉 . m〈c, k〉 | trans〈m〉 indef m〈c′, k〉.

[c = c′]k〈v, l〉 |mup〈c′, k〉in trans0〈m〉 | chain〈trans〉

in chain〈trans0〉 |def sr〈c, v, l〉 . srup〈c, v, l〉 | srτ 〈c, v, l〉 indef srup〈c, v, l〉 . sr〈c, v, l〉 indef rr〈c, k〉 . rrup〈c, k〉 | rrτ 〈c, k〉 indef rrup〈c, k〉 . rr〈c, k〉 inJ P Kπj | J Q Kπj



|



|

sr〈x1, v1, l1〉



|

rr〈y1, k1〉



sr〈x1, v1, l1〉, . . . , sr〈xn, vn, ln〉rr〈y1, k1〉, . . . , rr〈yn, kn〉




sr/rr







sr〈x1, v1, l1〉




sr〈x1, v1, l1〉

sr〈x2, v2, l2〉



rr〈y1, k1〉, . . . , rr〈yn, kn〉

sr〈x1, v1, l1〉

sr〈x2, v2, l2〉

. . .

sr〈xn, vn, ln〉



rr〈y1, k1〉, . . . , rr〈yn, kn〉

sr〈x1, v1, l1〉

sr〈x2, v2, l2〉

. . .

sr〈xn, vn, ln〉

rr〈yi, ki〉 if x1 = yi then ki〈v1, l1〉



rr〈y1, k1〉, . . . , rr〈yn, kn〉

sr〈x1, v1, l1〉

sr〈x2, v2, l2〉

. . .

sr〈xn, vn, ln〉

rr〈yi, ki〉 if x2 = yi then ki〈v2, l2〉



sr〈x1, v1, l1〉

sr〈x2, v2, l2〉

. . .

sr〈xn, vn, ln〉

k2〈v1, l1〉, k9〈v42, l42〉



sr〈x1, v1, l1〉

sr〈x2, v2, l2〉

. . .

sr〈xn, vn, ln〉

k9〈v42, l42〉

k2〈v1, l1〉


Take-Home-Points

1 There is a good encoding from πa to Join: ours!

2 It is compositional3 Strong Conjecture: It is operationally correspondent

Thank you!


Take-Home-Points

1 There is a good encoding from πa to Join: ours!2 It is compositional

3 Strong Conjecture: It is operationally correspondent

Thank you!


Take-Home-Points

1 There is a good encoding from πa to Join: ours!2 It is compositional3 Strong Conjecture: It is operationally correspondent

Thank you!


Take-Home-Points

1 There is a good encoding from πa to Join: ours!2 It is compositional3 Strong Conjecture: It is operationally correspondent

Thank you!


Bibliography

Cédric Fournet and Georges Gonthier.The reflexive chemical abstract machine and the join-calculus.pages 372–385, 1996.

D. Gorla.Towards a Unified Approach to Encodability and Separation Resultsfor Process Calculi.Information and Computation, 208(9) 1031–1053, 2010.

K. Peters and U. Nestmann.Breaking Symmetries.Submitted to Mathematical Structures in Computer Science, 2011.


Technology

A Compositional Encoding for the Asynchronous Pi-Calculus into the Join-Calculus