## Specification Structures and Propositions-as-Types for Concurrency (1995)

Venue: | Logics for Concurrency: Structure vs. Automata---Proceedings of the VIIIth Banff Higher Order Workshop, volume 1043 of Lecture Notes in Computer Science |

Citations: | 21 - 5 self |

### BibTeX

@INPROCEEDINGS{Abramsky95specificationstructures,

author = {Samson Abramsky and Simon Gay and Rajagopal Nagarajan},

title = {Specification Structures and Propositions-as-Types for Concurrency},

booktitle = {Logics for Concurrency: Structure vs. Automata---Proceedings of the VIIIth Banff Higher Order Workshop, volume 1043 of Lecture Notes in Computer Science},

year = {1995},

pages = {5--40},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

Many different notions of "property of interest" and methods of verifying such properties arise naturally in programming. A general framework of "Specification Structures" is presented for combining different notions and methods in a coherent fashion. This is then applied to concurrency in the setting of Interaction Categories.

### Citations

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Citation Context ...re in calculations, a different object will have to be used instead. We will not discuss this issue in the present paper. To solve the first problem we can adapt Hoare's solution of a similar problem =-=[22]. He consi-=-ders processes with one input and one output, which can be connected together in sequence. This is actually quite close to the categorical view in some ways: these processes have the "shape"... |

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Citation Context ...turn to the specific applications of this framework which in fact originally suggested it, in the setting of the first author's interaction categories. 3 Interaction Categories Interaction Categories =-=[1, 3, 4, 6]-=- are a new paradigm for the semantics of sequential and concurrent computation. This term encompasses certain known categories (the category of concrete data structures and sequential algorithms [12],... |

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Citation Context ...egory (using the autonomous structure) so it properly generalizes the standard interpretation of Cut. For some related notions which have arisen in work on coherence in compact closed categories, see =-=[13, 24]-=-. 3.1.3 SProc as a Linear Category SProc also has structure corresponding to the linear logic exponentials ! and ? . We will not need this structure in the present paper; details can be found elsewher... |

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Citation Context ...notions which is highly suggestive, particularly from a Computer Science point of view. Similar notions have been studied, for a variety of purposes, by Burstall and McKinna [28], O'Hearn and Tennent =-=[32], and Pitt-=-s [33]. Definition 1 Let C be a category. A specification structure S over C is defined by the following data: ffl a set PA of "properties over A", for each object A of C . ffl a relation R ... |

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Citation Context ...a ' (' ( /)\Omega 'feval A;B g/ '\Omega /ffg` =) 'f(f )g/ ( ': Going one step further, suppose that C is a -autonomous category, i.e. a model for the multiplicative fragment of classical linear logic =-=[11]-=-, with linear negation (\Gamma) ? , where for simplicity we assume that A ?? = A. Then we require an action (\Gamma) ? A : PA ! PA ? satisfying ' ?? = ' ' ( / = ('\Omega / ? ) ? : Under these circumst... |

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Citation Context ...rrent computation. This term encompasses certain known categories (the category of concrete data structures and sequential algorithms [12], categories of games [7], geometry of interaction categories =-=[8]-=-) as well as several new categories for concurrency. The fundamental examples of concurrent interaction categories are SProc, the category of synchronous processes, and ASProc, the category of asynchr... |

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Citation Context ... on C is illustrative. Exactly similar definitions can be given for a range of structures, including: ffl models of Classical (or Intuitionistic) Linear Logic including the additives and exponentials =-=[10]-=- ffl cartesian closed categories [15] ffl models of polymorphism [15]. 4 2.1 Examples of Specification Structures In each case we specify the category C , the assignment of properties P to objects and... |

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Citation Context ...ry in Logical Form [2], the other part arising from the local lattice-theoretic structure of the sets PD and its interaction with the global type structure. 6. C = games and partial strategies, as in =-=[9]-=-, PA = all sets of infinite plays, UfoegV iff oe is winning with respect to U; V in the sense of [7]. Then C S is the category of games and winning strategies of [7]. These examples show the scope and... |

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Citation Context ...te a different rule in the context of Linear Logic.) The usual Cut Rule ` \Gamma; A ` \Delta; A ? ` \Gamma; \Delta allows us to plug two modules together by an interface consisting of a single "p=-=ort" [5]-=-: A A ? : : : : : : This allows us to connect processes in a tree structure `j 'i `j 'i `j 'i \Gamma @ \Gamma @ \Gamma \Gamma @ @ \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta ... |

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Citation Context ...turn to the specific applications of this framework which in fact originally suggested it, in the setting of the first author's interaction categories. 3 Interaction Categories Interaction Categories =-=[1, 3, 4, 6]-=- are a new paradigm for the semantics of sequential and concurrent computation. This term encompasses certain known categories (the category of concrete data structures and sequential algorithms [12],... |

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Citation Context ...4, 6] are a new paradigm for the semantics of sequential and concurrent computation. This term encompasses certain known categories (the category of concrete data structures and sequential algorithms =-=[12]-=-, categories of games [7], geometry of interaction categories [8]) as well as several new categories for concurrency. The fundamental examples of concurrent interaction categories are SProc, the categ... |

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Citation Context ...egory (using the autonomous structure) so it properly generalizes the standard interpretation of Cut. For some related notions which have arisen in work on coherence in compact closed categories, see =-=[13, 24]-=-. 3.1.3 SProc as a Linear Category SProc also has structure corresponding to the linear logic exponentials ! and ? . We will not need this structure in the present paper; details can be found elsewher... |

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Citation Context ...tually a sorting function. Martin-Lof type theory, with dependent types and equality types, can express complete total correctness specifications. In the richer theories, type checking is undecidable =-=[35]-=-. One might try to make a methodological distinction: post-hoc verification vs. constructions with intrinsic properties. However, this is more a distinction between ways in which Type Inference/Verifi... |

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Citation Context ...ve view of these standard notions which is highly suggestive, particularly from a Computer Science point of view. Similar notions have been studied, for a variety of purposes, by Burstall and McKinna =-=[28], O'Hearn -=-and Tennent [32], and Pitts [33]. Definition 1 Let C be a category. A specification structure S over C is defined by the following data: ffl a set PA of "properties over A", for each object ... |

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Citation Context ...turn to the specific applications of this framework which in fact originally suggested it, in the setting of the first author's interaction categories. 3 Interaction Categories Interaction Categories =-=[1, 3, 4, 6]-=- are a new paradigm for the semantics of sequential and concurrent computation. This term encompasses certain known categories (the category of concrete data structures and sequential algorithms [12],... |

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Citation Context ...tial correctness and termination, and to use different methods for these two aspects [16]. ffl In the field of static analysis, and particularly in the systematic framework of abstract interpretation =-=[23]-=-, a basic ingredient of the methodology is to use a range of nonstandard interpretations to gain information about different properties of interest. ffl In concurrency, it is standard to separate out ... |

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Citation Context ... highly suggestive, particularly from a Computer Science point of view. Similar notions have been studied, for a variety of purposes, by Burstall and McKinna [28], O'Hearn and Tennent [32], and Pitts =-=[33]. Definiti-=-on 1 Let C be a category. A specification structure S over C is defined by the following data: ffl a set PA of "properties over A", for each object A of C . ffl a relation R A;B ` PA \Theta ... |

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Citation Context ...nctional composition typically found in categories of mathematical structures. There is not yet a definitive axiomatisation of interaction categories, although some possibilities have been considered =-=[18]-=-. The common features of the existing examples are that they have -autonomous structure, which corresponds to the multiplicative fragment of classical linear logic [20]; products and coproducts, corre... |

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Citation Context ...fl Even in the most basic form of sequential programming, it has proved fruitful to separate out the aspects of partial correctness and termination, and to use different methods for these two aspects =-=[16]-=-. ffl In the field of static analysis, and particularly in the systematic framework of abstract interpretation [23], a basic ingredient of the methodology is to use a range of nonstandard interpretati... |

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Citation Context ...is symmetric so the roles of p and q, tester and testee, can be interchanged freely. Now we lift this symmetric relation to a self-adjoint Galois connection on sets of processes in a standard fashion =-=[14]-=-: p ? U j 8q 2 U: p ? q U ? j fp j p ? Ug: Since (\Gamma) ? is a self-adjoint Galois connection, it satisfies U ??? j U ? : We are now ready to define the specification structure D on the subcategory ... |

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Citation Context |

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Citation Context ...e for composition in SProc D will be a compositional proof rule for plugging together deadlock-free processes while preserving deadlock-freedom. Rules of this kind are known to be difficult to obtain =-=[17]-=-. ffl The concepts and techniques used in defining this specification structure and verifying that it has the required properties represent a striking transfer of techniques from Proof Theory (Tait-Gi... |