## Process Realizability (2000)

Venue: | In Foundations of Secure Computation |

Citations: | 9 - 1 self |

### BibTeX

@INPROCEEDINGS{Abramsky00processrealizability,

author = {Samson Abramsky},

title = {Process Realizability},

booktitle = {In Foundations of Secure Computation},

year = {2000},

pages = {167--180},

publisher = {IOS Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper aims to give a readable and reasonably accessible account of some ideas linking the currently still largely separate worlds of concurrency theory and process algebra, on the one hand, and type theory, categorical models and realizability on the other. Background in process algebra may be found in standard texts such as [Hoa85, Hen88, Mil89, Ros97]; while background in realizability, categorical models etc. is provided by texts such as [GLT89, AL91, Cro93, AC98, BW99]. A modest background in either or both of these fields should suffice to understand the main ideas. Most of the detailed verification of properties of the formal definitions we will present is left as a series of exercises. The diligent reader who attempts a number of these should get some feeling for the interplay between concrete process-theoretic notions, and more abstract logical and categorical ideas, which is characteristic of this topic. It is this interplay which makes the topic a fascinating one for the author; I hope this brief introduction, to a field which is still wide open for further development, succeeds in conveying something of this fascination to the reader.

### Citations

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(Show Context)
Citation Context ... this fascination to the reader. 2 2 CCS with simultaneous actions Our universe of realizers will be a minor extension of one of the most standard and widely-used process calculi, namely Milner's CCS =-=[Mil89]. The extension is t-=-o allow "compound" actions, consisting of the simultaneous performance of several "atomic" actions. This idea of compound actions is present in the synchronous process calculus SCC... |

723 |
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(Show Context)
Citation Context ...ds to a left (or forwards) application: hQjP \Delta = ((Q[l] j P )nL l )[r \Gamma1 ] which we can visualize as follows: Q P aes\Omega \Psi . . . . . . . . . We use a Dirac-style bra-ket notation (cf. =-=[Dir67]-=-) to denote the left application of P to Q. The idea is that Q is relabelled into the left part of P 's name space, and we then restrict on the left name space so that P and Q are forced to interact t... |

582 | The Theory and Practice of Concurrency - Roscoe - 1998 |

404 | Testing equivalences for processes
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(Show Context)
Citation Context ...ongruence on guarded terms. 4 Discussion Our reason for working with failures equivalence ([BHR84, BR83]) is that it, or one of its variants such as the testing equivalences of Hennessy and De Nicola =-=[DNH83]-=-, or the Failures-Divergences model of Brookes and Roscoe [BR84], seem to be the finest equivalences which will suffice for our purposes. In particular, the realizability for the additive connectives ... |

393 | Algebraic Theory of Processes - Hennessy - 1988 |

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213 | Intensional Interpretations of Functionals of Finite Type - Tait - 1967 |

210 |
Data types as lattices
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(Show Context)
Citation Context ... 2.5 Some basic combinators Since processes are untyped, we will build a type-free universe of realizers. As usual, this will require a little coding (cf. the Kleene algebra K 0 [Kle45], graph models =-=[Sco76]-=- etc.), but in our setting this will take a very simple form. We simply split the set of names into two infinite disjoint sets N = N ls[ N r and fix bijections l : N = \Gamma! N l r : N = \Gamma! N r ... |

122 | Interaction categories and the foundations of typed concurrent programming - Abramsky, Gay, et al. - 1996 |

81 |
On the interpretation of intuitionistic number theory
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(Show Context)
Citation Context ...ng denotational model. 2.5 Some basic combinators Since processes are untyped, we will build a type-free universe of realizers. As usual, this will require a little coding (cf. the Kleene algebra K 0 =-=[Kle45]-=-, graph models [Sco76] etc.), but in our setting this will take a very simple form. We simply split the set of names into two infinite disjoint sets N = N ls[ N r and fix bijections l : N = \Gamma! N ... |

80 | Categories for Types - Crole - 1993 |

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(Show Context)
Citation Context ...ith failures equivalence ([BHR84, BR83]) is that it, or one of its variants such as the testing equivalences of Hennessy and De Nicola [DNH83], or the Failures-Divergences model of Brookes and Roscoe =-=[BR84]-=-, seem to be the finest equivalences which will suffice for our purposes. In particular, the realizability for the additive connectives of Linear Logic will not work in a fully satisfactory way if we ... |

60 | Realizability Toposes and Language Semantics - Longley - 1994 |

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41 |
Proofs as processes
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(Show Context)
Citation Context ...sting of the work on interaction categories into a realizability framework. Indeed, the essential ideas on the process interpretation of proofs go back to a 1991 lecture on "Proofs as Processes&q=-=uot; (see [Abr94b]-=-). 2.1 Names, co-names and actions As usual with CCS, we introduce two disjoint, countable sets N of names, and N of co-names, with a bijection ( ) : N = \Gamma! N , which we extend to an involution (... |

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3 | R.: A specification structure for deadlockfreedom of synchronous processes. Theoretical Computer Science 222
- Abramsky, Gay, et al.
- 1999
(Show Context)
Citation Context ...lled Cut-elimination by process interaction. We can also prove that Cut-elimination terminates in our model. We outline the argument. For a detailed account in the interaction categories setting, see =-=[AGN99]-=-. Firstly, we say that a process P diverges, written P*, if there is a sequence (P n j n 2 !) with P = P 0 and P n \Gamma! P n+1 for all n 2 !. We say that P is convergent, written P+, if it does not ... |

1 | Behavioural Equivalences induced by Programming Logics - Brookes, Rounds - 1983 |