## Linear Logical Relations for Session-Based Concurrency

Citations: | 5 - 5 self |

### BibTeX

@MISC{Pérez_linearlogical,

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

title = {Linear Logical Relations for Session-Based Concurrency},

year = {}

}

### OpenURL

### Abstract

Abstract. In prior work we proposed an interpretation of intuitionistic linear logic propositions as session types for concurrent processes. The type system obtained from the interpretation ensures fundamental properties of session-based typed disciplines—most notably, type preservation, session fidelity, and global progress. In this paper, we complement and strengthen these results by developing a theory of logical relations. Our development is based on, and is remarkably similar to, that for functional languages, extended to an (intuitionistic) linear type structure. A main result is that well-typed processes always terminate (strong normalization). We also introduce a notion of observational equivalence for sessiontyped processes. As applications, we prove that all proof conversions induced by the logic interpretation actually express observational equivalences, and explain how type isomorphisms resulting from linear logic equivalences are realized by coercions between interface types of session-based concurrent systems. 1

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Citation Context ...s always fresh, our typed calculus conforms to an internal mobility discipline [3], without loss of expressiveness. The composition rules (Tcut/Tcut ! ) follow the “composition plus hiding” principle =-=[1]-=-, extended to a name passing setting. Other linear typing rules for parallel composition (as in, e.g., [13]) are derivable—see [4]. As we consider π-calculus terms up to structural congruence, typabil... |

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Citation Context ...communication correctness: dialogues between participants are structured into sessions, the basic units of communication; descriptions of the interaction patterns are then abstracted as session types =-=[11]-=-, which are statically checked against specifications. These specifications are usually given in the π-calculus [16], so we obtain processes communicating through so-called session channels connecting... |

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Citation Context ...siveness. The composition rules (Tcut/Tcut ! ) follow the “composition plus hiding” principle [1], extended to a name passing setting. Other linear typing rules for parallel composition (as in, e.g., =-=[13]-=-) are derivable—see [4]. As we consider π-calculus terms up to structural congruence, typability is closed under ≡ by definition. πDILL enjoys the usual properties of equivariance, weakening, and cont... |

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Citation Context ...d (2) P → Q if and only if P τ −→≡ Q. 3 Session Types as Dual Intutionistic Linear Logic Propositions As anticipated in the introduction, the type structure coincides with intuitionistic linear logic =-=[10, 2]-=-, omitting atomic formulas and the additive constants ⊤ and 0. Definition 3.1 (Types). Types (A, B, C) are given by A, B ::= 1 | !A | A ⊗ B | A⊸B | A � B | A ⊕ B Types are assigned to (channel) names,... |

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Citation Context ...tion; descriptions of the interaction patterns are then abstracted as session types [11], which are statically checked against specifications. These specifications are usually given in the π-calculus =-=[16]-=-, so we obtain processes communicating through so-called session channels connecting exactly two subsystems. The discipline of session types ensures session protocols in which actions always occur in ... |

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Citation Context ... correspond directly to intuitionistic linear logic connectives. By erasing processes, typing judgments in πDILL correspond to DILL, a sequent formulation of Barber’s dual intuitionistic linear logic =-=[2, 6]-=-. Below we only provide some intuitions of this correspondence; see [4] for details. DILL is equipped with a faithful proof term assignment, so sequents have the form Γ ; ∆ ⊢ D : C, where Γ is the unr... |

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Citation Context ...rather an instantiation of the method on our canonical linear type structure. Another interpretation of session types as linear logic propositions is proposed in [7]. It is based on soft linear logic =-=[15]-=-, and so the exponential “!” is treated following a non canonical discipline that uses two different typing environments. Hence, typing rules and judgements in [7] are rather different from ours. A bo... |

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Citation Context ...d (2) P → Q if and only if P τ −→≡ Q. 3 Session Types as Dual Intutionistic Linear Logic Propositions As anticipated in the introduction, the type structure coincides with intuitionistic linear logic =-=[10, 2]-=-, omitting atomic formulas and the additive constants ⊤ and 0. Definition 3.1 (Types). Types (A, B, C) are given by A, B ::= 1 | !A | A ⊗ B | A⊸B | A � B | A ⊕ B Types are assigned to (channel) names,... |

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Citation Context ...t) input. The forwarding construct [x ↔ y] equates names x and y; it is a primitive representation of a copycat process, akin to the link processes used in internal mobility encodings of name passing =-=[3]-=-. Also, this construct allows for a simple identity axiom in the type system [25]. The remaining three operators define a minimal labeled choice mechanism, comparable to the n-ary branching constructs... |

30 | Session-based distributed programming in Java
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Citation Context ...th termination guarantees should be highly beneficial for the increasingly growing number of implementations (libraries, programming language extensions) based on session types foundations—see, e.g., =-=[12, 17, 20]-=-. In the second part of the paper, we present two applications of the basic theory, which bear witness to its complementarity with the other properties derived from the interpretation. The application... |

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(Show Context)
Citation Context ...r a survey). In the first part of this paper, we study termination in session-based concurrency. The starting point is our interpretation of (intutionistic) linear logic propositions as session types =-=[4]-=-, which has provided the first purely logical account of session types. In the interpretation, types are assigned to names (denoting communication channels) and describe their session protocol. This w... |

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(Show Context)
Citation Context ... to be a typed context bisimulation. ⊓⊔ α18 Jorge A. Pérez, Luís Caires, Frank Pfenning, and Bernardo Toninho 7 Related Work Termination in the π-calculus using logical relations has been studied in =-=[26, 21]-=-. Neither of these works considers session types; hence, the technical details of the logical relations are very different, with semantic interpretations of types relying on constraints on the syntax ... |

17 | Strong Normalisation in the Pi-Calculus
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(Show Context)
Citation Context ... to be a typed context bisimulation. ⊓⊔ α18 Jorge A. Pérez, Luís Caires, Frank Pfenning, and Bernardo Toninho 7 Related Work Termination in the π-calculus using logical relations has been studied in =-=[26, 21]-=-. Neither of these works considers session types; hence, the technical details of the logical relations are very different, with semantic interpretations of types relying on constraints on the syntax ... |

12 | Dependent session types via intuitionistic linear type theory
- Toninho, Caires, et al.
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(Show Context)
Citation Context ...ive representation of a copycat process, akin to the link processes used in internal mobility encodings of name passing [3]. Also, this construct allows for a simple identity axiom in the type system =-=[25]-=-. The remaining three operators define a minimal labeled choice mechanism, comparable to the n-ary branching constructs found in standard session π-calculi (see, e.g., [11]). Without loss of generalit... |

8 | Towards concurrent type theory - Caires, Pfenning, et al. - 2012 |

8 | Haskell session types with (almost) no class
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(Show Context)
Citation Context ...th termination guarantees should be highly beneficial for the increasingly growing number of implementations (libraries, programming language extensions) based on session types foundations—see, e.g., =-=[12, 17, 20]-=-. In the second part of the paper, we present two applications of the basic theory, which bear witness to its complementarity with the other properties derived from the interpretation. The application... |

6 | B.: Proof-carrying code in a session-typed process calculus
- Pfenning, Caires, et al.
(Show Context)
Citation Context ...t appeal to extraneous technical devices, and preserves a standard, intuitive treatment of “!”. This is particularly desirable for extensions/generalizations of our framework, such as the proposed in =-=[25, 19]-=-. Loosely related to typed context bisimilarity is [27], where a form of linear bisimilarity is proposed; following a linear type structure, it treats some visible actions as internal actions, thus le... |

4 | Mobile processes and termination
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(Show Context)
Citation Context ...ation in concurrent calculi, however, is known to be hard: in (variants of) the π-calculus, proofs require heavy constraints on the language and/or its types, often relying on ad-hoc machineries (see =-=[8]-=- for a survey). In the first part of this paper, we study termination in session-based concurrency. The starting point is our interpretation of (intutionistic) linear logic propositions as session typ... |

4 |
A short survey of isomorphisms of types
- Cosmo
(Show Context)
Citation Context ...⊓⊔ Type Isomorphisms. In type theory, types A and B are called isomorphic if there are morphisms (proofs in our case) πA of B ⊢ A and πB of A ⊢ B which compose to the identity in both ways—see, e.g., =-=[9]-=-. We adapt this notion to our setting, using typed context bisimilarity to account for isomorphisms in linear logic. (Below, we write P 〈˜x〉 for a process parametric on a sequence of names x1, . . . ,... |

4 | Safe parallel programming with session java
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(Show Context)
Citation Context ...th termination guarantees should be highly beneficial for the increasingly growing number of implementations (libraries, programming language extensions) based on session types foundations—see, e.g., =-=[12, 17, 20]-=-. In the second part of the paper, we present two applications of the basic theory, which bear witness to its complementarity with the other properties derived from the interpretation. The application... |

3 |
Giamberardino. Soft session types
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- 2011
(Show Context)
Citation Context ... not an adaptation of the method, but rather an instantiation of the method on our canonical linear type structure. Another interpretation of session types as linear logic propositions is proposed in =-=[7]-=-. It is based on soft linear logic [15], and so the exponential “!” is treated following a non canonical discipline that uses two different typing environments. Hence, typing rules and judgements in [... |

2 | On asynchronous session semantics
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(Show Context)
Citation Context ... as internal actions, thus leading to an equivalence larger than standard bisimilarity which is a congruence. The only work on behavioral equivalences for session-based concurrency we are aware of is =-=[14]-=-. It studies the behavioral theory of a π-calculus with asynchronous session communication and an event inspection primitive for buffered messages. The aim is to capture the distinction between order-... |