This document presents a compositional encoding of the asynchronous π-calculus (πa) into the join-calculus. It discusses the key differences between πa and the join-calculus in terms of their primitives for parallelism, communication, and restriction. It then reviews an existing encoding by Fournet and Gonthier, and Gorla's criteria for a good encoding in terms of compositionality, name invariance, and operational correspondence. The goal is to define a new encoding of πa into the join-calculus that satisfies Gorla's criteria for being a good encoding.

1. A Compositional Encoding of the Asynchronous
π-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, 2012
YR-CONCUR 2012

2. WraP – Writing and Publishing a Scientific Paper
Supervision by Uwe Nestmann and
Kirstin Peters
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3. Scope of Interest
Asynchronous π (πa ) Join-Calculus
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4. Scope of Interest
Asynchronous π (πa ) Join-Calculus
P |Q Parallelism P |Q
xv Send xv
x(w).P Receive def x w | o P in . . . | o
x(w)∗ .P Recursion def x w P in . . .
x(w).P | x v Communication def x w P in x v
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5. Scope of Interest
Asynchronous π (πa ) Join-Calculus
P |Q Parallelism P |Q
xv Send xv
x(w).P Receive def x w | o P in . . . | o
x(w)∗ .P Recursion def x w P in . . .
x(w).P | x v Communication def x w P in x v
Restriction
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6. Scope of Interest
Asynchronous π (πa ) Join-Calculus
P |Q Parallelism P |Q
xv Send xv
x(w).P Receive def x w | o P in . . . | o
x(w)∗ .P Recursion def x w P in . . .
x(w).P | x v Communication def x w P in x v
(νx)(x(w).P | x v ) Restriction
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7. Scope of Interest
Asynchronous π (πa ) Join-Calculus
P |Q Parallelism P |Q
xv Send xv
x(w).P Receive def x w | o P in . . . | o
x(w)∗ .P Recursion def x w P in . . .
x(w).P | x v Communication def x w P in x v
(νx)(x(w).P | x v ) Restriction def x w P in x v
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8. Scope of Interest
Asynchronous π (πa ) Join-Calculus
P |Q Parallelism P |Q
xv Send xv
x(w).P Receive def x w | o P in . . . | o
x(w)∗ .P Recursion def x w P in . . .
x(w).P | x v Communication def x w P in x v
(νx)(x(w).P | x v ) Restriction def x w P in x v
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9. Scope of Interest
π
Asynchronous π (πa ) j Join-Calculus
P |Q Parallelism P |Q
xv Send xv
x(w).P Receive def x w | o P in . . . | o
x(w)∗ .P Recursion def x w P in . . .
x(w).P | x v Communication def x w P in x v
(νx)(x(w).P | x v ) Restriction def x w P in x v
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10. An Encoding is Non-Trivial
|
no (νx)
xv
x(w)
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11. An Encoding is Non-Trivial
|
no (νx)
xv
x(w)
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12. An Encoding is Non-Trivial
| |
no (νx)
xv xv
x(w) def x
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13. An Encoding is Non-Trivial
| |
no (νx)
xv xv
x(w) def x
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14. The Encoding of Fournet and Gonthier
[Fournet and Gonthier(1996)]
First Layer
Second Layer
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15. The Encoding of Fournet and Gonthier
[Fournet and Gonthier(1996)]
First Layer
x v → xo v
P |Q → P |Q
Second Layer
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16. The Encoding of Fournet and Gonthier
[Fournet and Gonthier(1996)]
First Layer
x v → xo v
P |Q → P |Q
x(w).P → def κ P in xi
Second Layer
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17. The Encoding of Fournet and Gonthier
[Fournet and Gonthier(1996)]
First Layer
x v → xo v
P |Q → P |Q
x(w).P → def κ P in xi
(νx)P → def xi | xo κ in P
Second Layer
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18. The Encoding of Fournet and Gonthier
[Fournet and Gonthier(1996)]
First Layer
x v → xo v
P |Q → P |Q
x(w).P → def κ P in xi
(νx)P → def xi | xo κ in P
Second Layer
Restrict all free names
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19. The Encoding of Fournet and Gonthier
[Fournet and Gonthier(1996)]
First Layer
x v → xo v
P |Q → P |Q
x(w).P → def κ P in xi
(νx)P → def xi | xo κ in P
Second Layer
Restrict all free names
Is this good?
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20. Gorla’s Criteria for Good Encodings [Gorla(2010)]
.
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21. Gorla’s Criteria for Good Encodings [Gorla(2010)]
I. Compositionality
.
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22. Gorla’s Criteria for Good Encodings [Gorla(2010)]
I. Compositionality
.
II. Name Invariance
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23. Gorla’s Criteria for Good Encodings [Gorla(2010)]
I. Compositionality
Syntactic Criteria
.
II. Name Invariance
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24. Gorla’s Criteria for Good Encodings [Gorla(2010)]
I. Compositionality
Syntactic Criteria
.
II. Name Invariance
III. Operational Correspondence
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25. Gorla’s Criteria for Good Encodings [Gorla(2010)]
I. Compositionality
Syntactic Criteria
.
IV. Divergence Reflection II. Name Invariance
III. Operational Correspondence
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26. Gorla’s Criteria for Good Encodings [Gorla(2010)]
V. Success Sensitiveness I. Compositionality
Syntactic Criteria
.
IV. Divergence Reflection II. Name Invariance
III. Operational Correspondence
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27. Gorla’s Criteria for Good Encodings [Gorla(2010)]
V. Success Sensitiveness I. Compositionality
Syntactic Criteria
Semantic Criteria
.
IV. Divergence Reflection II. Name Invariance
III. Operational Correspondence
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28. Fournet and Gonthier’s Encoding Revisited
First level encoding is compositional and operationally correspondent
for closed terms
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29. Fournet and Gonthier’s Encoding Revisited
First level encoding is compositional and operationally correspondent
for closed terms
First level encoding is not operationally correspondent for open
terms
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30. Fournet and Gonthier’s Encoding Revisited
First level encoding is compositional and operationally correspondent
for closed terms
First level encoding is not operationally correspondent for open
terms
First+Second level encoding is not compositional
S. Mennicke (TU Braunschweig) Compositional πa ⇒ Join YR-CONCUR 2012 7 / 15

31. Fournet and Gonthier’s Encoding Revisited
First level encoding is compositional and operationally correspondent
for closed terms
First level encoding is not operationally correspondent for open
terms
First+Second level encoding is not compositional
Our Approach
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32. Fournet and Gonthier’s Encoding Revisited
First level encoding is compositional and operationally correspondent
for closed terms
First level encoding is not operationally correspondent for open
terms
First+Second level encoding is not compositional
Our Approach
We introduce send/receive requests carrying the channel names
S. Mennicke (TU Braunschweig) Compositional πa ⇒ Join YR-CONCUR 2012 7 / 15

33. Fournet and Gonthier’s Encoding Revisited
First level encoding is compositional and operationally correspondent
for closed terms
First level encoding is not operationally correspondent for open
terms
First+Second level encoding is not compositional
Our Approach
We introduce send/receive requests carrying the channel names
We keep the main idea of restriction
S. Mennicke (TU Braunschweig) Compositional πa ⇒ Join YR-CONCUR 2012 7 / 15

34. Fournet and Gonthier’s Encoding Revisited
First level encoding is compositional and operationally correspondent
for closed terms
First level encoding is not operationally correspondent for open
terms
First+Second level encoding is not compositional
Our Approach
We introduce send/receive requests carrying the channel names
We keep the main idea of restriction
We implement a protocol to handle communication
[Peters and Nestmann(2011)]
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35. Fournet and Gonthier’s Encoding Revisited
First level encoding is compositional and operationally correspondent
for closed terms
First level encoding is not operationally correspondent for open
terms
First+Second level encoding is not compositional
Our Approach
We introduce send/receive requests carrying the channel names
We 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
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36. The Big Picture
π
(1) x v j = def lf t, f | f alse f | f alse in
(2) def lt t, f | true t | f alse in
(3) def l t, f lt t, f | lf t, f in sr x, v, l | true
π
(4) P |Q j = 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) P π | Q π
j j
π
(16) x(v).P j = def k v, l | once
(17) def t P π in
j
(18) def f once in l t, f
(19) in rr x, k | once
π
(20) (νx)P j = def x 0 in P π j
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37. What our Encoding does
P = (νx)(y a | x(u).Q) | x b | y(z).R
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38. What our Encoding does
P = (νx)(y a | x(u).Q) | x b | y(z).R
|
νx |
| x y
y x
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39. What our Encoding does
P = (νx)(y a | x(u).Q) | x b | y(z).R
|
νx |
| x x y y
y y x x
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40. What our Encoding does
P = (νx)(y a | x(u).Q) | x b | y(z).R
|
νx |
| y x x y y
y x x
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41. What our Encoding does
P = (νx)(y a | x(u).Q) | x b | y(z).R
| y
νx |
| y x x y y
y x x
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42. What our Encoding does
P = (νx)(y a | x(u).Q) | x b | y(z).R
| y
νx |
| y x x x y y
y x
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43. What our Encoding does
P = (νx)(y a | x(u).Q) | x b | y(z).R
| y x
νx |
| y x x x y y
y x
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44. What our Encoding does
P = (νx)(y a | x(u).Q) | x b | y(z).R
| y
νx | x
| y x x y y
y x
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45. What our Encoding does
P = (νx)(y a | x(u).Q) | x b | y(z).R
| y
νx | x y
| y x x y
y x
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46. What our Encoding does
P = (νx)(y a | x(u).Q) | x b | y(z).R
| y y
νx | x y
| y x x y
y x
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47. What our Encoding does
P = (νx)(y a | x(u).Q) | x b | y(z).R
| y y
x
νx | x y
| y x x y
y x
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48. What our Encoding does
P = (νx)(y a | x(u).Q) | x b | y(z).R
| y y
x
νx | x y
| y x x y
y x
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49. What our Encoding does
P = (νx)(y a | x(u).Q) | x b | y(z).R
| y y
x
νx | x y
| y x x y
y x
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50. What our Encoding does
P = (νx)(y a | x(u).Q) | x b | y(z).R
| y y
x
νx | x y
| y x x y
y x
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51. A Receive Requests (x(v).P )
x
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52. A Receive Requests (x(v).P )
def k
rr x, k
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53. A Receive Requests (x(v).P )
rr x, k
def k
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54. A Receive Requests (x(v).P )
def k
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55. A Receive Requests (x(v).P )
k v, l
def k
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56. A Receive Requests (x(v).P )
k v, l
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57. A Receive Requests (x(v).P )
if l then P
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58. B Send Requests (x v )
x
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59. B Send Requests (x v )
def l
sr x, v, l
true
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60. B Send Requests (x v )
sr x, v, l
def l
true
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61. B Send Requests (x v )
def l
true
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62. B Send Requests (x v )
l t, f
def l
true
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63. B Send Requests (x v )
t
def l
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64. B Send Requests (x v )
l t, f
def l
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65. B Send Requests (x v )
f
def l
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66. C Restriction (νx)
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67. C Restriction (νx)
¬x
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68. D Parallel Composition ( | )
Remember this?
π
P |Q j = 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 in
def 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 in
def srup c, v, l sr c, v, l in
def rr c, k rrup c, k | rrτ c, k in
def rrup c, k rr c, k in
P π | Q π
j j
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69. D Parallel Composition ( | )
|
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70. D Parallel Composition ( | )
|
sr x1 , v1 , l1
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71. D Parallel Composition ( | )
|
rr y1 , k1
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72. D Parallel Composition ( | )
sr x1 , v1 , l1 , . . . , sr xn , vn , ln
rr y1 , k1 , . . . , rr yn , kn
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73. D Parallel Composition ( | )
sr/rr
sr x1 , v1 , l1 , . . . , sr xn , vn , ln
rr y1 , k1 , . . . , rr yn , kn
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74. D Parallel Composition ( | )
sr x1 , v1 , l1 , . . . , sr xn , vn , ln
rr y1 , k1 , . . . , rr yn , kn
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75. D Parallel Composition ( | )
sr x1 , v1 , l1 , . . . , sr xn , vn , ln
rr y1 , k1 , . . . , rr yn , kn
sr x1 , v1 , l1
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76. D Parallel Composition ( | )
sr x1 , v1 , l1 , . . . , sr xn , vn , ln
rr y1 , k1 , . . . , rr yn , kn
sr x1 , v1 , l1
sr x2 , v2 , l2
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77. D Parallel Composition ( | )
rr y1 , k1 , . . . , rr yn , kn
sr x1 , v1 , l1
sr x2 , v2 , l2
...
sr xn , vn , ln
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78. D Parallel Composition ( | )
rr y1 , k1 , . . . , rr yn , kn
sr x1 , v1 , l1 rr yi , ki if x1 = yi then ki v1 , l1
sr x2 , v2 , l2
...
sr xn , vn , ln
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79. D Parallel Composition ( | )
rr y1 , k1 , . . . , rr yn , kn
sr x1 , v1 , l1
sr x2 , v2 , l2 rr yi , ki if x2 = yi then ki v2 , l2
...
sr xn , vn , ln
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80. D Parallel Composition ( | )
k2 v1 , l1 , k9 v42 , l42
sr x1 , v1 , l1
sr x2 , v2 , l2
...
sr xn , vn , ln
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81. D Parallel Composition ( | )
k9 v42 , l42
sr x1 , v1 , l1 k2 v1 , l1
sr x2 , v2 , l2
...
sr xn , vn , ln
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82. Take-Home-Points
1 There is a good encoding from πa to Join: ours!
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83. Take-Home-Points
1 There is a good encoding from πa to Join: ours!
2 It is compositional
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84. 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
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85. 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!
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86. 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 Results
for 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.
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