Pure type systems with corecursion on streams

Severi, Paula; Vries, F. J. de

doi:10.1145/2398856.2364550

Cited by 8 publications

(1 citation statement)

References 37 publications

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“…Severi and de Vries [21] study Pure Type Systems, which include dependently typed calculi such as the Calculus of Constructions, extended with essentially a guarded recursive type of streams. Instead of restricting the unfolding of fixed points to obtain strong normalisation as we do for CloTT, their calculi allow arbitrary unfolding of fixed points.…”

Section: B Related Workmentioning

confidence: 99%

The clocks are ticking: No more delays!

Bahr

Grathwohl

Møgelberg

2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Guarded recursion in the sense of Nakano allows general recursive types and terms to be added to type theory without breaking consistency. Recent work has demonstrated applications of guarded recursion such as programming with codata, reasoning about coinductive types, as well as constructing and reasoning about denotational models of general recursive types. Guarded recursion can also be used as an abstract form of step-indexing for reasoning about programming languages with advanced features.Ultimately, we would like to have an implementation of a type theory with guarded recursion in which all these applications can be carried out and machine-checked. In this paper, we take a step towards this goal by devising a suitable reduction semantics.We present Clocked Type Theory, a new type theory for guarded recursion that is more suitable for reduction semantics than the existing ones. We prove confluence, strong normalisation and canonicity for its reduction semantics, constructing the theoretical basis for a future implementation. We show how coinductive types as exemplified by streams can be encoded, and derive that the type system ensures productivity of stream definitions.

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Section: B Related Workmentioning

confidence: 99%

The clocks are ticking: No more delays!

Bahr

Grathwohl

Møgelberg

2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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On Sessions and Infinite Data

Severi

Padovani

Tuosto

et al. 2016

Lecture Notes in Computer Science

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We define a novel calculus that combines a call-by-name functional core with session-based communication primitives. We develop a typing discipline that guarantees both normalisation of expressions and progress of processes and that uncovers an unexpected interplay between evaluation and communication. CC Creative Commons2 SEVERI ET AL.Proof. By case analysis on the derivation.Proof. By induction on the structure of processes. We only discuss the case of rule [thread].The interesting observation is that we need to use the hypothesis dom(Γ 2 ) ∩ dom(∆) = ∅ to ensure that the name of the thread does not belong to its own body. We also use Lemma 4.4 to type the body of the thread itself. Lemma 5.13 (Evaluation Contexts for Processes). If Γ C[e] : • n (IO s), then Γ = Γ 1 + Γ 2 and Γ 1 , x : • n (IO t) C[x] : • n (IO s) and Γ 2 e : • n (IO t).Proof. By induction on the structure of evaluation contexts for processes. ON SESSIONS AND INFINITE DATA 21A useful consequence of the previous lemma is the following property of contexts filled by communication expressions.Lemma 5.14. If Γ , a p :Proof. We only consider the case Γ , a p : • n T C[send a p e] : • m (IO t). By Lemma 5.13 Γ = Γ 1 + Γ 2 and Γ 1 , x : • m (IO s) C[x] : • m (IO t) and Γ 2 , a p : • n T send a p e : • m (IO s). Since rule [→E] requires at least n bullets in front of the type of send we get n ≤ m.

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Completeness of Conversion between Reactive Programs for Ultrametric Models

Severi

Vries

2013

Lecture Notes in Computer Science

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Abstract. In 1970 Friedman proved completeness of beta eta conversion in the simply-typed lambda calculus for the set-theoretical model. Recently Krishnaswami and Benton have captured the essence of Hudak's reactive programs in an extension of simply typed lambda calculus with causal streams and a temporal modality and provided this typed lambda calculus for reactive programs with a sound ultrametric semantics. We show that beta eta conversion in the typed lambda calculus of reactive programs is complete for the ultrametric model.

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Pure type systems with corecursion on streams

Cited by 8 publications

References 37 publications

The clocks are ticking: No more delays!

The clocks are ticking: No more delays!

On Sessions and Infinite Data

Completeness of Conversion between Reactive Programs for Ultrametric Models

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