Concurrent Combinators for Mobile Processes

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Concurrent Combinators for Mobile Processes

• Raymond Hu
• Imperial College, 2006

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Outline

1. Introduction
2. The p-calculus
3. The cc
4. Encoding the p-calculus in the cc
5. Properties of the cc
6. Conclusion
7. Related Work

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Combinators
Introduction

• Atoms of computation
• The ?-calculus and CL (SKI) syntax and op. sem.
• ?x abstraction (elimination)
• CL-term and ß-redex behaviour (?x.M)N ?w
  N/xM
• HS86

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Outline (2)

1. Introduction
2. The p-calculus
3. The cc
4. Encoding the p-calculus in the cc
5. Properties of the cc
6. Conclusion
7. Related Work

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The p-calculus
The p-calculus

• Concurrent computation
• Process calculus syntax, op. sem., equiv.
• Synchronisation, name passing
• Communication interaction observation
• Mobility (dynamic communication topology)
• The asynchronous p-calculus
• Mil99, San01, HT92, Bou92, Pal97

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The p-calculus (2)
The p-calculus

• Semantics structural congruence () and
  reduction (?), or transition relations (?)
• Semantics-based equivalences ( , )
• San01

l


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Outline (3)

1. Introduction
2. The p-calculus
3. The cc
4. Encoding the p-calculus in the cc
5. Properties of the cc
6. Conclusion
7. Related Work

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Concurrent Combinators

• Atoms have ports connected to names
• Port polarities determine computation
• Fixed dyadic interaction (reduction)

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The cc
The cc

• 7 atoms
• Constant units of name passing behaviour
• Pcc c(u) PQ (?z)P !P 0
• Constructs have same meaning as in the p-calculus
• defined by same rules as p-calculus
• HY94, Yos98

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Duplicator
The cc

• d(abc) m(ae) ? m(be) m(ce)
• cz(bz cz) ae ? be ce

-
-
-
-
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Forwarder
The cc

• fw(ab) m(ae) ? m(be)
• az.bz ae ? be

-
-
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Killer
The cc

• k(a) m(ae) ? 0
• az.0 ae ? 0

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Left-binder
The cc

• bl(ab) m(ae) ? fw(eb)
• az.zw.bw ae ? ew.bw

-
-
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Right-binder
The cc

• br(ab) m(ae) ? fw(be)
• az.bw.zw ae ? bw.ew

-
-
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Synchroniser
The cc

• s(abc) m(ae) ? fw(bc)
• az.bw.cw. ae ? bw.cw

-
-
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cc-terms
The cc

• e.g. Switcher sw(ab) (vc)(br(ac) m(cb))

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Outline (4)

1. Introduction
2. The p-calculus
3. The cc
4. Encoding the p-calculus in the cc
5. Properties of the cc
6. Conclusion
7. Related Work

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From p-calculus to the cc
Encoding the p-calculus in the cc

• Homomorphic mapping with emulation of input
  prefix
• synchronisation
• binding
• xz.P abstraction
• HY94, Yos98

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From p-calculus to the cc (2)
Encoding the p-calculus in the cc
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Analysis of Input Prefix
Encoding the p-calculus in the cc
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Analysis of Input Prefix (2)
Encoding the p-calculus in the cc
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Analysis of Input Prefix (3)
Encoding the p-calculus in the cc

• (i) xz.(PQ) (vc1,c2)(d(xc1c2) c1z.P
  c2z.Q)

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From p-calculus to the cc (3)
-
-

• xz.(bzcz) (vc1,c2)(d(xc1c2) fw(c1b)
  fw(c2b))

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Analysis of Input Prefix (4)
Encoding the p-calculus in the cc

• (vi) xz.c(v- u) (vc)(s(xvc) c(c- u))

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From p-calculus to the cc (4)
Encoding the p-calculus in the cc
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Outline (5)

1. Introduction
2. The p-calculus
3. The cc
4. Encoding the p-calculus in the cc
5. Properties of the cc
6. Conclusion
7. Related Work

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Theorems on the cc
Properties of the cc

• xz.P xz.P
• xz.P m(xy) ? Py/z
• Proof rule induction over pref. map. rules
• P Pcc
• Proof structural induction
• HY94, Yos98

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Replication
Properties of the cc

• xz.!P (vc)(fw(xc) !cz.(P m(cz)))
• Want to show xz.!P xz.!P

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Replication (2)
Properties of the cc

• Let A !P where P P0y/z,
• B (vc)(m(cy) !cz.(P0 m(cz))),
• S (QA, QB) U , L QA, R QB

1a)
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Replication (3)
Properties of the cc
1b)
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Replication (4)
Properties of the cc

• Let A !P where P P0y/z,
• B (vc)(m(cy) !cz.(P0 m(cz))),
• S (QA, QB) U , L QA, R QB

2)
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Outline (6)

1. Introduction
2. The p-calculus
3. The cc
4. Encoding the p-calculus in the cc
5. Properties of the cc
6. Conclusion
7. Related Work

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Conclusion
Conclusion

• p-calculus
• Behaviour reduction, transition
• Equivalences
• The cc
• Concurrent combinator system
• Encoding of input prefix
• Proofs

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Outline (6)

1. Introduction
2. The p-calculus
3. The cc
4. Encoding the p-calculus in the cc
5. Properties of the cc
6. Conclusion
7. Related Work

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Minimality and Separation Yos98
Related Work

• Generation
• Basis
• Essential elements
• Minimality
• 5/7 atoms essential for the asynch.-p
• Tool for comparing expressiveness
• Separation

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More Related Work
Related Work

• Replication in Concurrent Combinators HY94
• additional atoms, emulation of replication
• Solos in Concert LV99
• two encodings of continuations in the fusion
  calculus

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Solos in Concert LV99
Related Work