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

Command injection attacks, continuations, and the Lambek calculus

This paper shows connections between command injection attacks, continuations, and the Lambek calculus: certain command injections, such as the tautology attack on SQL, are shown to be a form of control effect that can be typed using the Lambek calculus, generalizing the double-negation typing of continuations. Lambek's syntactic calculus is a logic with two implicational connectives taking their arguments from the left and right, respectively. These connectives describe how strings interact with their left and right contexts when building up syntactic structures. The calculus is a form of propositional logic without structural rules, and so a forerunner of substructural logics like Linear Logic and Separation Logic.Comment: In Proceedings WoC 2015, arXiv:1606.0583

Continuation Passing Style for Effect Handlers

We present Continuation Passing Style (CPS) translations for Plotkin and Pretnar's effect handlers with Hillerström and Lindley's row-typed fine-grain call-by-value calculus of effect handlers as the source language. CPS translations of handlers are interesting theoretically, to explain the semantics of handlers, and also offer a practical implementation technique that does not require special support in the target language's runtime. We begin with a first-order CPS translation into untyped lambda calculus which manages a stack of continuations and handlers as a curried sequence of arguments. We then refine the initial CPS translation first by uncurrying it to yield a properly tail-recursive translation and second by making it higher-order in order to contract administrative redexes at translation time. We prove that the higher-order CPS translation simulates effect handler reduction. We have implemented the higher-order CPS translation as a JavaScript backend for the Links programming language

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

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

Parameterised notions of computation

Moggi’s Computational Monads and Power et al’s equivalent notion of Freyd category have captured a large range of computational effects present in programming languages such as exceptions, side-effects, input/output and continuations. We present generalisations of both constructs, which we call parameterised monads and parameterised Freyd categories, that also capture computational effects with parameters. Examples of such are composable continuations, side-effects where the type of the state varies and input/output where the range of inputs and outputs varies. By also considering monoidal parameterisation, we extend the range of effects to cover separated side-effects and multiple independent streams of I/O. We also present two typed λ-calculi that soundly and completely model our categorical definitions — with and without monoidal parameterisation — and act as prototypical languages with parameterised effects

On the Expressive Power of User-Defined Effects: Effect Handlers, Monadic Reflection, Delimited Control

We compare the expressive power of three programming abstractions for user-defined computational effects: Bauer and Pretnar's effect handlers, Filinski's monadic reflection, and delimited control without answer-type-modification. This comparison allows a precise discussion about the relative expressiveness of each programming abstraction. It also demonstrates the sensitivity of the relative expressiveness of user-defined effects to seemingly orthogonal language features. We present three calculi, one per abstraction, extending Levy's call-by-push-value. For each calculus, we present syntax, operational semantics, a natural type-and-effect system, and, for effect handlers and monadic reflection, a set-theoretic denotational semantics. We establish their basic meta-theoretic properties: safety, termination, and, where applicable, soundness and adequacy. Using Felleisen's notion of a macro translation, we show that these abstractions can macro-express each other, and show which translations preserve typeability. We use the adequate finitary set-theoretic denotational semantics for the monadic calculus to show that effect handlers cannot be macro-expressed while preserving typeability either by monadic reflection or by delimited control. We supplement our development with a mechanised Abella formalisation