## Relational Parametricity for Polymorphic Session Types

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Citations: | 4 - 3 self |

### BibTeX

@MISC{Caires_relationalparametricity,

author = {Luís Caires and Jorge A. Pérez and Frank Pfenning and Bernardo Toninho},

title = {Relational Parametricity for Polymorphic Session Types},

year = {}

}

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### Abstract

We introduce a theory of polymorphic concurrent processes, which arises from an interpretation of second-order intuitionistic linear logic propositions as polymorphic session types, in the style of the Girard-Reynolds polymorphic λ-calculus. The interpretation naturally generalizes recent discoveries on the correspondence between linear logic propositions and session types. In our proposed theory, polymorphism accounts for the exchange of abstract communication protocols, and dynamic instantiation of heterogeneous interfaces. Well-typed processes enjoy a strong form of subject reduction (type preservation) and global progress, but also termination (strong normalization) and relational parametricity (representation independence). The latter two properties are obtained by adapting proof techniques in the functional setting to linear session types.

### Citations

456 |
The Pi-calculus: a Theory of Mobile Processes
- Sangiorgi, Walker
- 2001
(Show Context)
Citation Context ... remarks. For space reasons, most proofs are omitted; technical details of the above results are available in the online appendix [6]. 2 Polymorphic Session Types We consider a synchronous π-calculus =-=[24]-=- extended with binary guarded choice, channel links, and type input and output prefixes. The syntax of processes/types is as follows: Definition 2.1 (Processes, Session Types). Given an infinite set Λ... |

373 |
Types, abstraction, and parametric polymorphism
- Reynolds
- 1983
(Show Context)
Citation Context ...g, and Bernardo Toninho of T ω η [Γ ; ∆ ⊢ T ] to the binary setting. Using ≈L we prove a parametricity result for well-typed processes (Theorem 4.12), along the lines of Reynold’s abstraction theorem =-=[23]-=-, which allows us to ultimately show that ≈L and ∼ = coincide (Theorem 4.13). Below, we follow closely the presentation by Harper [14] for System F. Barbed Congruence. We define barbed congruence as t... |

369 |
Towards a theory of type structure
- Reynolds
- 1974
(Show Context)
Citation Context ... stepping stone of the correspondence: obtaining, by purely logical means, a natural extension of session types with parametric polymorphism in the sense of the Girard-Reynolds polymorphic λ-calculus =-=[22,12]-=-. In the extended interpretation, these correspond to an input and an output of a session type, respectively. This is rather remarkable, since session types denote communication protocol behavior, par... |

286 | Computational interpretations of linear logic
- Abramsky
- 1993
(Show Context)
Citation Context ... this question. By doing so, we further strengthen the logical foundations for structured communications, and contribute to the study of the connections between logic and models of concurrency (e.g., =-=[1,3,16,2]-=-). We summarize the technical achievements of this paper. We develop a system of polymorphic session-typed processes, and establish its fundamental properties: subject reduction (type preservation) an... |

228 |
Une extension de l’interpretation de Gödel a l’analyse, et son application a l’elimination des coupure dans l’analyse et la theorie des types
- Girard
- 1971
(Show Context)
Citation Context ... stepping stone of the correspondence: obtaining, by purely logical means, a natural extension of session types with parametric polymorphism in the sense of the Girard-Reynolds polymorphic λ-calculus =-=[22,12]-=-. In the extended interpretation, these correspond to an input and an output of a session type, respectively. This is rather remarkable, since session types denote communication protocol behavior, par... |

212 | Language primitives and type discipline for structured communication-based programming
- Honda, Vasconcelos, et al.
- 1998
(Show Context)
Citation Context ...ngoing research program [7,25,20,8,19,9], we have been investigating logical foundations for session-based concurrency, based on a fresh interpretation of linear logical propositions as session types =-=[15,17]-=-, sequent proofs as π-calculus processes, and cut elimination as process communication. The result is a tight correspondence, in the Curry-Howard style; well-typed processes enjoy strong forms of type... |

98 | The Polymorphic Pi-Calculus: Theory and Implementation
- Turner
- 1995
(Show Context)
Citation Context ...our work seems to be the first one considering impredicative polymorphism for session types, by relying solely on linear logic principles. Polymorphism in the π-calculus was first addressed by Turner =-=[27]-=-, who studied a simply-typed π-calculus (roughly, the discipline in which types of names describe the objects they can carry), with a form of polymorphism based on existential quantification. In proce... |

90 |
Dual intuitionistic linear logic
- Barber
- 1996
(Show Context)
Citation Context ...ssion types 2.1 Second-order Intuitionistic Linear Logic This presentation of second-order linear logic consists of a natural extension of a sequent calculus formulation of linear logic known as DILL =-=[2,10]-=-. To handle the second-order quantifiers, we add an additional context Ω to our sequent judgment that keeps track of predicate variables. Thus, our main judgment is of the form Ω; Γ ; ∆ ⊢ D : A, meani... |

64 |
Subtyping for session types in the pi calculus
- Gay, Hole
- 2005
(Show Context)
Citation Context ...plines have only considered forms of bounded polymorphism, in which quantification over types is limited using subtyping. Gay [11] was the first to study this idea for the session types discipline in =-=[10]-=-. There, bounded polymorphism is associated exclusively to branch and choice types: each branch is quantified by a type variable with upper and lower bounds; the scope of a variable is a whole branch ... |

56 | Behavioral equivalence in the polymorphic pi-calculus
- Pierce, Sangiorgi
(Show Context)
Citation Context ...ressed as explicit type parameters in input/output prefixes: messages are composed of a type and values of that type, and so type variables are relevant at the level of messages. Sangiorgi and Pierce =-=[21]-=- proposed a behavioral theory for Turner’s framework. None of these works addresses termination of well-typed processes nor studies relational parametricity. Building up on [21],14 Luís Caires, Jorge... |

50 | A Judgmental Analysis of Linear Logic - Chang, Chaudhuri, et al. - 2003 |

41 |
On the π-calculus and linear logic
- Bellin, Scott
- 1994
(Show Context)
Citation Context ... this question. By doing so, we further strengthen the logical foundations for structured communications, and contribute to the study of the connections between logic and models of concurrency (e.g., =-=[1,3,16,2]-=-). We summarize the technical achievements of this paper. We develop a system of polymorphic session-typed processes, and establish its fundamental properties: subject reduction (type preservation) an... |

33 | Session types as intuitionistic linear propositions
- Caires, Pfenning
- 2010
(Show Context)
Citation Context ... session types. Our theory enables us to exploit behavioral genericity which, to the best of our knowledge, is novel in the realm of session-typed concurrent processes. In an ongoing research program =-=[7,25,20,8,19,9]-=-, we have been investigating logical foundations for session-based concurrency, based on a fresh interpretation of linear logical propositions as session types [15,17], sequent proofs as π-calculus pr... |

24 | Genericity and the pi-calculus
- Berger, Honda, et al.
- 2005
(Show Context)
Citation Context ...o Jeffrey and Rathke [18] show that weak bisimulation is fully abstract for observational equivalence for an asynchronous polymorphic π-calculus with name equality testing. Berger, Honda, and Yoshida =-=[4,5]-=- proposed a π-calculus with polymorphism (universal/existential quantification), with a typing structure based on linearity, so-called action types, and duality principles. As in our case, they prove ... |

24 | Bounded polymorphism in session types
- Gay
- 2008
(Show Context)
Citation Context ...ore detailed comparisons. To the best of our knowledge, session type disciplines have only considered forms of bounded polymorphism, in which quantification over types is limited using subtyping. Gay =-=[11]-=- was the first to study this idea for the session types discipline in [10]. There, bounded polymorphism is associated exclusively to branch and choice types: each branch is quantified by a type variab... |

16 | A concurrent model for linear logic
- Beffara
- 2006
(Show Context)
Citation Context ... this question. By doing so, we further strengthen the logical foundations for structured communications, and contribute to the study of the connections between logic and models of concurrency (e.g., =-=[1,3,16,2]-=-). We summarize the technical achievements of this paper. We develop a system of polymorphic session-typed processes, and establish its fundamental properties: subject reduction (type preservation) an... |

13 | F.: Dependent session types via intuitionistic linear type theory. In: PPDP’11
- Toninho, Caires, et al.
- 2011
(Show Context)
Citation Context ... session types. Our theory enables us to exploit behavioral genericity which, to the best of our knowledge, is novel in the realm of session-typed concurrent processes. In an ongoing research program =-=[7,25,20,8,19,9]-=-, we have been investigating logical foundations for session-based concurrency, based on a fresh interpretation of linear logical propositions as session types [15,17], sequent proofs as π-calculus pr... |

13 | Propositions as sessions
- Wadler
- 2012
(Show Context)
Citation Context ...clusively to branch and choice types: each branch is quantified by a type variable with upper and lower bounds; the scope of a variable is a whole branch of communication. In very recent work, Wadler =-=[28]-=- proposes an interpretation of classical linear logic as session types (along the lines of [8]) which includes polymorphism; however, the theory of polymorphic sessions and parametricity is not explor... |

12 | 2010): An exact correspondence between a typed pi-calculus and polarised proof-nets
- Honda, Laurent
(Show Context)
Citation Context |

9 | Towards concurrent type theory
- Caires, Pfenning, et al.
- 2012
(Show Context)
Citation Context ... session types. Our theory enables us to exploit behavioral genericity which, to the best of our knowledge, is novel in the realm of session-typed concurrent processes. In an ongoing research program =-=[7,25,20,8,19,9]-=-, we have been investigating logical foundations for session-based concurrency, based on a fresh interpretation of linear logical propositions as session types [15,17], sequent proofs as π-calculus pr... |

9 | Proof-carrying code in a sessiontyped process calculus
- Pfenning, Caires, et al.
- 2011
(Show Context)
Citation Context ...notes inputing a type and then using the session as A, agnostic to what the actual received type can be. Note that in the presence of polymorphism the identity rule (not present in [7,8], but used in =-=[25,20,19]-=-) is necessary, since it is the only way of typing a session with a type variable. As usual, in the presence of type annotations in binders, type-checking is decidable in our system (we omit these for... |

7 | Linear logical relations for session-based concurrency
- Pérez, Caires, et al.
- 2012
(Show Context)
Citation Context |

6 |
Practical Foundations for Programming Languages. Working Draft
- Harper
- 2012
(Show Context)
Citation Context ... (Theorem 4.12), along the lines of Reynold’s abstraction theorem [23], which allows us to ultimately show that ≈L and ∼ = coincide (Theorem 4.13). Below, we follow closely the presentation by Harper =-=[14]-=- for System F. Barbed Congruence. We define barbed congruence as the largest equivalence relation on typed processes that is (i) closed under internal actions; (ii) preserves barbs— arguably the most ... |

5 |
Linear logic propositions as session types
- Caires, Pfenning, et al.
(Show Context)
Citation Context ...ial session ∃X.A denotes inputing a type and then using the session as A, agnostic to what the actual received type can be. Note that in the presence of polymorphism the identity rule (not present in =-=[7,8]-=-, but used in [25,20,19]) is necessary, since it is the only way of typing a session with a type variable. As usual, in the presence of type annotations in binders, type-checking is decidable in our s... |

5 | Functions as session-typed processes
- Toninho, Caires, et al.
- 2012
(Show Context)
Citation Context ... types (along the lines of [8]) which includes polymorphism; however, the theory of polymorphic sessions and parametricity is not explored in [28]. Further Developments. Following the developments of =-=[26]-=-, where a correct encoding of the simply typed λ-calculus as session typed π-calculus is given, we can provide a correct encoding of System F into our polymorphic session calculus presented here. The ... |

4 |
Types for dynamic interaction
- Honda
- 1993
(Show Context)
Citation Context ...ngoing research program [7,25,20,8,19,9], we have been investigating logical foundations for session-based concurrency, based on a fresh interpretation of linear logical propositions as session types =-=[15,17]-=-, sequent proofs as π-calculus processes, and cut elimination as process communication. The result is a tight correspondence, in the Curry-Howard style; well-typed processes enjoy strong forms of type... |

4 | Full abstraction for polymorphic pi-calculus
- Jeffrey, Rathke
- 2008
(Show Context)
Citation Context ... works addresses termination of well-typed processes nor studies relational parametricity. Building up on [21],14 Luís Caires, Jorge A. Pérez, Frank Pfenning, and Bernardo Toninho Jeffrey and Rathke =-=[18]-=- show that weak bisimulation is fully abstract for observational equivalence for an asynchronous polymorphic π-calculus with name equality testing. Berger, Honda, and Yoshida [4,5] proposed a π-calcul... |

3 | Polymorphic endpoint types for copyless message passing - Bono, Padovani - 2011 |

1 |
A Theory of Polymorphic Session Types
- Caires, Pérez, et al.
(Show Context)
Citation Context ...e example. Section 5 discusses related work and offers some concluding remarks. For space reasons, most proofs are omitted; technical details of the above results are available in the online appendix =-=[6]-=-. 2 Polymorphic Session Types We consider a synchronous π-calculus [24] extended with binary guarded choice, channel links, and type input and output prefixes. The syntax of processes/types is as foll... |