Delimited control operators prove Double-negation Shift

We propose an extension of minimal intuitionistic predicate logic, based on delimited control operators, that can derive the predicate-logic version of the Double-negation Shift schema, while preserving the disjunction and existence properties

A Direct Version of Veldman's Proof of Open Induction on Cantor Space via Delimited Control Operators

First, we reconstruct Wim Veldman's result that Open Induction on Cantor space can be derived from Double-negation Shift and Markov's Principle. In doing this, we notice that one has to use a countable choice axiom in the proof and that Markov's Principle is replaceable by slightly strengthening the Double-negation Shift schema. We show that this strengthened version of Double-negation Shift can nonetheless be derived in a constructive intermediate logic based on delimited control operators, extended with axioms for higher-type Heyting Arithmetic. We formalize the argument and thus obtain a proof term that directly derives Open Induction on Cantor space by the shift and reset delimited control operators of Danvy and Filinski

Perspectives for proof unwinding by programming languages techniques

In this chapter, we propose some future directions of work, potentially beneficial to Mathematics and its foundations, based on the recent import of methodology from the theory of programming languages into proof theory. This scientific essay, written for the audience of proof theorists as well as the working mathematician, is not a survey of the field, but rather a personal view of the author who hopes that it may inspire future and fellow researchers

An interpretation of the Sigma-2 fragment of classical Analysis in System T

We show that it is possible to define a realizability interpretation for the $\Sigma_2$-fragment of classical Analysis using G\"odel's System T only. This supplements a previous result of Schwichtenberg regarding bar recursion at types 0 and 1 by showing how to avoid using bar recursion altogether. Our result is proved via a conservative extension of System T with an operator for composable continuations from the theory of programming languages due to Danvy and Filinski. The fragment of Analysis is therefore essentially constructive, even in presence of the full Axiom of Choice schema: Weak Church's Rule holds of it in spite of the fact that it is strong enough to refute the formal arithmetical version of Church's Thesis

Continuation-passing Style Models Complete for Intuitionistic Logic

A class of models is presented, in the form of continuation monads polymorphic for first-order individuals, that is sound and complete for minimal intuitionistic predicate logic. The proofs of soundness and completeness are constructive and the computational content of their composition is, in particular, a $\beta$-normalisation-by-evaluation program for simply typed lambda calculus with sum types. Although the inspiration comes from Danvy's type-directed partial evaluator for the same lambda calculus, the there essential use of delimited control operators (i.e. computational effects) is avoided. The role of polymorphism is crucial -- dropping it allows one to obtain a notion of model complete for classical predicate logic. The connection between ours and Kripke models is made through a strengthening of the Double-negation Shift schema

From delimited CPS to polarisation

Appeared in the author's PhD thesis (Chapter III) along with more details. See (and cite) Guillaume Munch-Maccagnoni, Syntax and Models of a non-Associative Composition of Programs and Proofs, Université Paris-Diderot - Paris VII, 2013, .The understanding of continuation-passing style (CPS) translations, an historical source of denotational semantics for programming languages, benefits from notions brought by linear logic, such as focalisation, polarities and the involutive negation. Here we aim to show how linear logic helps as well when continuations are delimited, i.e. return and can be composed, in the sense of Danvy and Filinski. First we provide a polarised calculus with delimited control (first-class delimited continuations) which is, at the level of computation, a variant of Girard's polarised classical logic LC. It contains variants of delimited control calculi which spawned as answers to the question "what order of evaluation can we consider with delimited control?". Thus our polarised calculus is one answer which is unifying to some degree. Subsequently we decompose the calculus through polarised linear logic. The only difference with non-delimited continuations is the use of specific exponentials, that account for the specific semantics of the target of delimited CPS translation, as well as annotations of type-and-effect systems. As a by-product, we obtain an explanation of CPS translations through a factoring, each step of which accounts for distinct phenomena of CPS translations. Although the factoring also holds for non-delimited CPS translations, it did not appear in its entirety before

A Contextual Reconstruction of Monadic Reflection

With the help of an idea of contextual modal logic, we define a logical system lambda^{refl} that incorporates monadic reflection, and then investigate delimited continuations through the lens of monadic reflection. Technically, we firstly prove a certain universality of continuation monad, making the character of monadic reflection a little more clear. Next, moving focus to delimited continuations, we present a macro definition of shift/reset by monadic reflection. We then prove that lambda^{refl}_{2cont}, a restriction of lambda^{refl}, has exactly the same provability as lambda^{s/r}_{pure}, a system that incorporates shift/reset. Our reconstruction of monadic reflection opens up a path for investigation of delimited continuations with familiar monadic language

A Type-Theoretic Foundation of Delimited Continuations

International audienceThere is a correspondence between classical logic and programming language calculi with first-class continuations. With the addition of control delimiters, the continuations become composable and the calculi become more expressive. We present a fine-grained analysis of control delimiters and formalise that their addition corresponds to the addition of a single dynamically-scoped variable modelling the special top-level continuation. From a type perspective, the dynamically-scoped variable requires effect annotations. In the presence of control, the dynamically-scoped variable can be interpreted in a purely functional way by applying a store-passing style. At the type level, the effect annotations are mapped within standard classical logic extended with the dual of implication, namely subtraction. A continuation-passing-style transformation of lambda-calculus with control and subtraction is defined. Combining the translations provides a decomposition of standard CPS transformations for delimited continuations. Incidentally, we also give a direct normalisation proof of the simply-typed lambda-calculus with control and subtraction

On Subexponentials, Synthetic Connectives, and Multi-level Delimited Control

International audienceWe construct a partially-ordered hierarchy of delimited control operators similar to those of the CPS hierarchy of Danvy and Filinski. However, instead of relying on nested CPS translations, these operators are directly interpreted in linear logic extended with subexponentials (i.e., multiple pairs of ! and ?). We construct an independent proof theory for a fragment of this logic based on the principle of focusing. It is then shown that the new constraints placed on the permutation of cuts correspond to multiple levels of delimited control