arXiv subjects

Steffen van Bakel

Publications and source records attributed to Steffen van Bakel.

14 recordsLinked to original sources

arXiv2026

Comparing Call-by-Name and Call-by-Value Reduction and Reduction Strategies in Calculi for Classical Logic

We define call-by-name and call-by-value reduction and reduction strategies for the calculi slmu (symmetric lmu), lmmt, and Xs (X with implicit substitution). We establish a strong relation between these notions through defining a single interpretation from slmu to lmmt that respects normal reduction, as well as the call-by-name and call-by-value reduction in slmu within the their counterpart in lmmt; for the strategies, we will show similar, but weaker results. We also define a single mapping from lmmt to Xs, and show that this also respects all three notions. We then continue with studying the natural encoding of Xs into lmmt, and show that only full reduction is respected, but that reduction steps are needed to model substitution, so the CBN and CBV strategies cannot be respected. We conclude with studying the combination of our efforts and define an interpretation of slmu into Xs, and show that CBN and CBV reduction are respected. This result underlines that Xs and lmmt are similar, but different calculi, and that the nature of slmu makes that any encoding into either can never fully respect the strategies.

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arXiv2023

Adding Negation to Lambda Mu

We present $\cal L$, an extension of Parigot's $λμ$-calculus by adding negation as a type constructor, together with syntactic constructs that represent negation introduction and elimination. We will define a notion of reduction that extends $λμ$'s reduction system with two new reduction rules, and show that the system satisfies subject reduction. Using Aczel's generalisation of Tait and Martin-Löf's notion of parallel reduction, we show that this extended reduction is confluent. Although the notion of type assignment has its limitations with respect to representation of proofs in natural deduction with implication and negation, we will show that all propositions that can be shown in there have a witness in $\cal L$. Using Girard's approach of reducibility candidates, we show that all typeable terms are strongly normalisable, and conclude the paper by showing that type assignment for $\cal L$ enjoys the principal typing property.

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arXiv2018

Intersection Types for the lambda-mu Calculus

We introduce an intersection type system for the lambda-mu calculus that is invariant under subject reduction and expansion. The system is obtained by describing Streicher and Reus's denotational model of continuations in the category of omega-algebraic lattices via Abramsky's domain-logic approach. This provides at the same time an interpretation of the type system and a proof of the completeness of the system with respect to the continuation models by means of a filter model construction. We then define a restriction of our system, such that a lambda-mu term is typeable if and only if it is strongly normalising. We also show that Parigot's typing of lambda-mu terms with classically valid propositional formulas can be translated into the restricted system, which then provides an alternative proof of strong normalisability for the typed lambda-mu calculus.

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arXiv2017

Characterisation of Approximation and (Head) Normalisation for $λμ$ using Strict Intersection Types

We study the strict type assignment for lambda-mu that is presented in [van Bakel'16]. We define a notion of approximants of lambda-mu-terms, show that it generates a semantics, and that for each typeable term there is an approximant that has the same type. We show that this leads to a characterisation via assignable types for all terms that have a head normal form, and to one for all terms that have a normal form, as well as to one for all terms that are strongly normalisable.

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arXiv2016

Proceedings Sixth International Workshop on Classical Logic and Computation

The workshop series intends to cover research that investigates the computational aspects of classical logic and mathematics. Its focus is on unwinding the computational content of logical principles and proof in mathematics based on these principles, aiming to bring together researchers from both fields and exchange ideas. Classical Logic and Computation (CL&C) 2016 was the sixth edition of this workshop series held as a satellite to FSCD 2016 on June 23, 2016 in Porto, Portugal. In this sixth edition we received 11 submissions of both short and full papers. Eight (8) of these were selected to present at the meeting in Porto, and five (5) full papers were initially accepted to appear at this EPTCS special volume of which one was subsequently withdrawn by its authors. An invited talk was given by Marc Bezem (U. of Bergen): Coherent Logic - an overview. Other topics covered by this years submissions included: computational content of proofs using nonstandard analysis, a structured grammar-based approach to the Herbrand content of proofs, semantics of the lambda-mu calculus, normalization of classical natural deduction proofs, proof mining of noneffective proofs in convex optimization and algebra by functional interpretations. I like to thank the members of the program committee for their excellent work: Steffen van Bakel (London), Stefano Berardi (Torino), Fernando Ferreira (Lisboa), Hugo de'Liguoro (Torino), Alexandre Miquel (Montevideo). Ulrich Kohlenbach (Darmstadt, PC Chair)

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arXiv2015

Orchestrated Session Compliance

We investigate the notion of orchestrated compliance for client/server interactions in the context of session contracts. Devising the notion of orchestrator in such a context makes it possible to have orchestrators with unbounded buffering capabilities and at the same time to guarantee any message from the client to be eventually delivered by the orchestrator to the server, while preventing the server from sending messages which are kept indefinitely inside the orchestrator. The compliance relation is shown to be decidable by means of 1) a procedure synthesising the orchestrators, if any, making a client compliant with a server, and 2) a procedure for deciding whether an orchestrator behaves in a proper way as mentioned before.

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arXiv2014

A fully-abstract semantics of lambda-mu in the pi-calculus

We study the lambda-mu-calculus, extended with explicit substitution, and define a compositional output-based interpretation into a variant of the pi-calculus with pairing that preserves single-step explicit head reduction with respect to weak bisimilarity. We define four notions of weak equivalence for lambda-mu -- one based on weak reduction, two modelling weak head-reduction and weak explicit head reduction (all considering terms without weak head-normal form equivalent as well), and one based on weak approximation -- and show they all coincide. We will then show full abstraction results for our interpretation for the weak equivalences with respect to weak bisimilarity on processes.

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arXiv2013

Characterisation of Strongly Normalising lambda-mu-Terms

We provide a characterisation of strongly normalising terms of the lambda-mu-calculus by means of a type system that uses intersection and product types. The presence of the latter and a restricted use of the type omega enable us to represent the particular notion of continuation used in the literature for the definition of semantics for the lambda-mu-calculus. This makes it possible to lift the well-known characterisation property for strongly-normalising lambda-terms - that uses intersection types - to the lambda-mu-calculus. From this result an alternative proof of strong normalisation for terms typeable in Parigot's propositional logical system follows, by means of an interpretation of that system into ours.

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arXiv2011

From X to Pi; Representing the Classical Sequent Calculus in the Pi-calculus

We study the Pi-calculus, enriched with pairing and non-blocking input, and define a notion of type assignment that uses the type constructor "arrow". We encode the circuits of the calculus X into this variant of Pi, and show that all reduction (cut-elimination) and assignable types are preserved. Since X enjoys the Curry-Howard isomorphism for Gentzen's calculus LK, this implies that all proofs in LK have a representation in Pi.

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arXiv2011

Note on a simple type system for non-interference

We consider CCS with value passing and elaborate a notion of noninterference for the process calculi, which matches closely that of the programming language. The idea is to view channels as information carriers rather than as "events", so that emitting a secret on output channel can be considered safe, while inputting a secret may lead to some kind of leakage. This is in contrast with the standard notion of noninterference for the process calculi where any causal dependency of low-level action from any high-level action is forbidden.

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arXiv2011

Reduction in X does not agree with Intersection and Union Types (Extended abstract)

This paper defines intersection and union type assignment for the calculus X, a substitution free language that enjoys the Curry-Howard correspondence with respect to Gentzen's sequent calculus for classical logic. We show that this notion is closed for subject-expansion, and show that it needs to be restricted to satisfy subject-reduction as well, making it unsuitable to define a semantics.

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arXiv2011

Semantic Predicate Types and Approximation for Class-based Object Oriented Programming

We apply the principles of the intersection type discipline to the study of class-based object oriented programs and; our work follows from a similar approach (in the context of Abadi and Cardelli's Varsigma-object calculus) taken by van Bakel and de'Liguoro. We define an extension of Featherweight Java, FJc and present a predicate system which we show to be sound and expressive. We also show that our system provides a semantic underpinning for the object oriented paradigm by generalising the concept of approximant from the Lambda Calculus and demonstrating an approximation result: all expressions to which we can assign a predicate have an approximant that satisfies the same predicate. Crucial to this result is the notion of predicate language, which associates a family of predicates with a class.

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arXiv2011

Proceedings Third International Workshop on Classical Logic and Computation

The fact that classical mathematical proofs of simply existential statements can be read as programs was established by Goedel and Kreisel half a century ago. But the possibility of extracting useful computational content from classical proofs was taken seriously only from the 1990s on when it was discovered that proof interpretations based on Goedel's and Kreisel's ideas can provide new nontrivial algorithms and numerical results, and the Curry-Howard correspondence can be extended to classical logic via programming concepts such as continuations and control operators. The workshop series "Classical Logic and Computation" aims to support a fruitful exchange of ideas between the various lines of research on computational aspects of classical logic. This volume contains the abstracts of the invited lectures and the accepted contributed papers of the third CL&C workshop which was held jointly with the workshop "Program Extraction and Constructive Mathematics" at the University of Brno in August 21-22, 2010, as a satellite of CSL and MFCS. The workshops were held in honour of Helmut Schwichtenberg who became "professor emeritus" in September 2010. The topics of the papers include the foundations, optimizations and applications of proof interpretations such as Hilbert's epsilon substitution method, Goedel's functional interpretation, learning based realizability and negative translations as well as special calculi and theories capturing computational and complexity-theoretic aspects of classical logic such as the lambda-mu-calculus, applicative theories, sequent-calculi, resolution and cut-elimination

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arXiv2011

Sound and Complete Typing for lambda-mu

In this paper we define intersection and union type assignment for Parigot's calculus lambda-mu. We show that this notion is complete (i.e. closed under subject-expansion), and show also that it is sound (i.e. closed under subject-reduction). This implies that this notion of intersection-union type assignment is suitable to define a semantics.

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