## Presheaf models for the π-calculus (1997)

Venue: | In Proc. CTCS’97, volume 1290 of LNCS |

### BibTeX

@INPROCEEDINGS{Stark97presheafmodels,

author = {Ian Stark},

title = {Presheaf models for the π-calculus},

booktitle = {In Proc. CTCS’97, volume 1290 of LNCS},

year = {1997},

pages = {106--126},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Abstract. The finite π-calculus has an explicit set-theoretic functor-category model that is known to be fully abstract for strong late bisimulation congruence. We characterize this as the initial free algebra for an appropriate set of operations and equations in the enriched Lawvere theories of Plotkin and Power. Thus we obtain a novel algebraic description for models of the π-calculus, and validate an existing construction as the universal such model. The algebraic operations are intuitive, covering name creation, communication of names over channels, and nondeterminism; the equations then combine these features in a modular fashion. We work in an enriched setting, over a “possible worlds ” category of sets indexed by available names. This expands significantly on the classical notion of algebraic theories, and in particular allows us to use nonstandard arities that vary as processes evolve. Based on our algebraic theory we describe a category of models for the π-calculus, and show that they all preserve bisimulation congruence. We develop a direct construction of free models in this category; and generalise previous results to prove that all free-algebra models are fully abstract. 1

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Citation Context .... Our method builds on a recent line of research by Plotkin and Power who use algebraic theories in enriched categories to capture “notions of computation”, in particular Moggi’s computational monads =-=[18, 26, 27, 28]-=-. The general idea is to describe a computational feature — I/O, state, nondeterminism — by stating a characteristic collection ⋆ Research supported by an EPSRC Advanced Research Fellowship http://www... |

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Citation Context ...r each of these features. This precision in integrating different aspects of computation is a significant benefit of the algebraic approach over existing techniques for combining computational monads =-=[13, 15, 19, 37]-=-. The structure of the paper is as follows. In §2 we review the relevant properties of algebraic theories and the functor category Set I . We then set out our proposed algebraic theory of π in §3. Fol... |

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Citation Context ...ible extensions and further applications of this work. 2 Background We outline relevant material on algebraic theories and the target category Set I . For π-calculus information, see one of the books =-=[16, 32]-=- or Parrow’s handbook chapter [23]. 2.1 Algebras and Notions of Computation We sketch very briefly the theoretical basis for our development: for more on enriched algebraic theories see Robinson’s cle... |

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Citation Context ...cations of the π-calculus with domain-specific terms, equations and processes. There are many such ad-hoc extensions, notably those brought together by Abadi and Fournet under the banner of applied π =-=[1]-=-. In ongoing work, Plotkin has given a construction for modal logics from algebraic theories. Applying this to the theory of π gives a modal logic for the π-calculus up to bisimulation congruence. Thi... |

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Citation Context ...r each of these features. This precision in integrating different aspects of computation is a significant benefit of the algebraic approach over existing techniques for combining computational monads =-=[13, 15, 19, 37]-=-. The structure of the paper is as follows. In §2 we review the relevant properties of algebraic theories and the functor category Set I . We then set out our proposed algebraic theory of π in §3. Fol... |

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Citation Context ...the choice of that name. This functor is well known, for example as dynamic allocation in [6, 7]; it also appears as the atom abstraction operator [N]X of FM-set theory identified by Gabbay and Pitts =-=[9, 24]-=-. Note that shifting the object of names gives a coproduct: δN ∼ = (N + 1). The representable objects in Set I are 1, N, (N ⊗N), (N ⊗N ⊗N), . . . The finitely presentable objects are the finite colimi... |

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Citation Context ...changes, so does this set of values. Functor categories of possible worlds like this are well established for modelling local state in programming languages [20, 22, 30] and local names in particular =-=[17, 25, 35]-=-. Similar categories of varying sets also appear in models for variable binding [5] and name binding (see, for example, [33] and citations there). Category Set I is complete and cocomplete, with limit... |

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Citation Context ...the choice of that name. This functor is well known, for example as dynamic allocation in [6, 7]; it also appears as the atom abstraction operator [N]X of FM-set theory identified by Gabbay and Pitts =-=[9, 24]-=-. Note that shifting the object of names gives a coproduct: δN ∼ = (N + 1). The representable objects in Set I are 1, N, (N ⊗N), (N ⊗N ⊗N), . . . The finitely presentable objects are the finite colimi... |

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Citation Context ...e π-Calculus As the set of names available changes, so does this set of values. Functor categories of possible worlds like this are well established for modelling local state in programming languages =-=[20, 22, 30]-=- and local names in particular [17, 25, 35]. Similar categories of varying sets also appear in models for variable binding [5] and name binding (see, for example, [33] and citations there). Category S... |

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Citation Context ...l established for modelling local state in programming languages [20, 22, 30] and local names in particular [17, 25, 35]. Similar categories of varying sets also appear in models for variable binding =-=[5]-=- and name binding (see, for example, [33] and citations there). Category Set I is complete and cocomplete, with limits and colimits taken pointwise. It is cartesian closed, with a convenient way to ca... |

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Citation Context ...changes, so does this set of values. Functor categories of possible worlds like this are well established for modelling local state in programming languages [20, 22, 30] and local names in particular =-=[17, 25, 35]-=-. Similar categories of varying sets also appear in models for variable binding [5] and name binding (see, for example, [33] and citations there). Category Set I is complete and cocomplete, with limit... |

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Citation Context ...e π-Calculus As the set of names available changes, so does this set of values. Functor categories of possible worlds like this are well established for modelling local state in programming languages =-=[20, 22, 30]-=- and local names in particular [17, 25, 35]. Similar categories of varying sets also appear in models for variable binding [5] and name binding (see, for example, [33] and citations there). Category S... |

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Citation Context ... in [23, §9], and we now need to explore the algebraic theories they generate. Pitts and others have championed nominal sets and Fraenkel-Mostowski set theory as a foundation for reasoning with names =-=[9, 24, 34]-=-. If we move from Set I to its full subcategory of pullback-preserving functors then we have the Schanuel topos, which models FM set theory. As noted earlier, all of our constructions lie within this,... |

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Citation Context ...r each of these features. This precision in integrating different aspects of computation is a significant benefit of the algebraic approach over existing techniques for combining computational monads =-=[13, 15, 19, 37]-=-. The structure of the paper is as follows. In §2 we review the relevant properties of algebraic theories and the functor category Set I . We then set out our proposed algebraic theory of π in §3. Fol... |

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Citation Context .... The situation here is quite general, with a precise correspondence between singlesorted algebraic theories and finitary monads on Set (i.e., monads that preserve filtered colimits). Kelly and Power =-=[14, 29]-=- extend this to an enriched setting: carriers for the algebras may be from categories other than Set; the arities of operations can be not just natural numbers, but certain objects in a category; and ... |

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Citation Context .... Our method builds on a recent line of research by Plotkin and Power who use algebraic theories in enriched categories to capture “notions of computation”, in particular Moggi’s computational monads =-=[18, 26, 27, 28]-=-. The general idea is to describe a computational feature — I/O, state, nondeterminism — by stating a characteristic collection ⋆ Research supported by an EPSRC Advanced Research Fellowship http://www... |

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Citation Context ...and generalise previous results to prove that all free-algebra models are fully abstract. 1 Introduction There are by now a handful of models known to give a denotational semantics for the π-calculus =-=[2, 3, 6, 7, 8, 10, 36]-=-. All are fully abstract for appropriate operational equivalences, and all use functor categories to handle the central issue of names and name creation. In this paper we present a method for generati... |

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Citation Context ...changes, so does this set of values. Functor categories of possible worlds like this are well established for modelling local state in programming languages [20, 22, 30] and local names in particular =-=[17, 25, 35]-=-. Similar categories of varying sets also appear in models for variable binding [5] and name binding (see, for example, [33] and citations there). Category Set I is complete and cocomplete, with limit... |

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Citation Context .... Our method builds on a recent line of research by Plotkin and Power who use algebraic theories in enriched categories to capture “notions of computation”, in particular Moggi’s computational monads =-=[18, 26, 27, 28]-=-. The general idea is to describe a computational feature — I/O, state, nondeterminism — by stating a characteristic collection ⋆ Research supported by an EPSRC Advanced Research Fellowship http://www... |

29 | Combining effects: sum and tensor
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Citation Context ...effectful actions to program with; and a modal logic for specification and reasoning. This approach also gives a flexible way to express interactions between features, by combining sets of operations =-=[11, 12]-=-. For the π-calculus, we apply and expand their technique. The enriched setting supports not only models that are objects in Set I , but also arities from Set I ; so that we have operations whose arit... |

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Citation Context ...er [23]. 2.1 Algebras and Notions of Computation We sketch very briefly the theoretical basis for our development: for more on enriched algebraic theories see Robinson’s clear and detailed exposition =-=[31]-=-; the link to computations and generic effects is described in [27, 28]. There is a well-established connection between algebraic theories and monads on the category Set. For example, consider the fol... |

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Citation Context ...and generalise previous results to prove that all free-algebra models are fully abstract. 1 Introduction There are by now a handful of models known to give a denotational semantics for the π-calculus =-=[2, 3, 6, 7, 8, 10, 36]-=-. All are fully abstract for appropriate operational equivalences, and all use functor categories to handle the central issue of names and name creation. In this paper we present a method for generati... |

24 |
Comparing operational models of name-passing process calculi
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Citation Context ...and generalise previous results to prove that all free-algebra models are fully abstract. 1 Introduction There are by now a handful of models known to give a denotational semantics for the π-calculus =-=[2, 3, 6, 7, 8, 10, 36]-=-. All are fully abstract for appropriate operational equivalences, and all use functor categories to handle the central issue of names and name creation. In this paper we present a method for generati... |

22 |
A fully abstract domain model for the π-calculus
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19 | Combining computational effects: commutativity and sum
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Citation Context ...effectful actions to program with; and a modal logic for specification and reasoning. This approach also gives a flexible way to express interactions between features, by combining sets of operations =-=[11, 12]-=-. For the π-calculus, we apply and expand their technique. The enriched setting supports not only models that are objects in Set I , but also arities from Set I ; so that we have operations whose arit... |

19 |
An Introduction to the π-Calculus
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Citation Context ...ns of this work. 2 Background We outline relevant material on algebraic theories and the target category Set I . For π-calculus information, see one of the books [16, 32] or Parrow’s handbook chapter =-=[23]-=-. 2.1 Algebras and Notions of Computation We sketch very briefly the theoretical basis for our development: for more on enriched algebraic theories see Robinson’s clear and detailed exposition [31]; t... |

17 |
A fully-abstract model for the π-calculus
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15 | Enriched Lawvere Theories
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(Show Context)
Citation Context .... The situation here is quite general, with a precise correspondence between singlesorted algebraic theories and finitary monads on Set (i.e., monads that preserve filtered colimits). Kelly and Power =-=[14, 29]-=- extend this to an enriched setting: carriers for the algebras may be from categories other than Set; the arities of operations can be not just natural numbers, but certain objects in a category; and ... |

15 | A Dependent Type Theory with Names and Binding
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Citation Context ...in programming languages [20, 22, 30] and local names in particular [17, 25, 35]. Similar categories of varying sets also appear in models for variable binding [5] and name binding (see, for example, =-=[33]-=- and citations there). Category Set I is complete and cocomplete, with limits and colimits taken pointwise. It is cartesian closed, with a convenient way to calculate function spaces using natural tra... |

11 |
A fully abstract denotational semantics for the π-calculus
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4 |
Communicating and Mobile Systems: the Pi-Calculus. CUP
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Citation Context ...ible extensions and further applications of this work. 2 Background We outline relevant material on algebraic theories and the target category Set I . For π-calculus information, see one of the books =-=[16, 32]-=- or Parrow’s handbook chapter [23]. 2.1 Algebras and Notions of Computation We sketch very briefly the theoretical basis for our development: for more on enriched algebraic theories see Robinson’s cle... |

2 |
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1 |
Functor categories and store shapes. Chapter 11 of [21
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Citation Context ...e π-Calculus As the set of names available changes, so does this set of values. Functor categories of possible worlds like this are well established for modelling local state in programming languages =-=[20, 22, 30]-=- and local names in particular [17, 25, 35]. Similar categories of varying sets also appear in models for variable binding [5] and name binding (see, for example, [33] and citations there). Category S... |