Linear type systems have a long and storied history, but not a clear path forward to integrate with existing languages such as OCaml or Haskell. In this paper, we study a linear type system designed with two crucial properties in mind: backwards-compatibility and code reuse across linear and non-linear users of a library. Only then can the benefits of linear types permeate conventional functional programming. Rather than bifurcate types into linear and non-linear counterparts, we instead attach linearity to function arrows. Linear functions can receive inputs from linearly-bound values, but can also operate over unrestricted, regular values.To demonstrate the efficacy of our linear type system -both how easy it can be integrated in an existing language implementation and how streamlined it makes it to write programs with linear types -we implemented our type system in ghc, the leading Haskell compiler, and demonstrate two kinds of applications of linear types: mutable data with pure interfaces; and enforcing protocols in I/O-performing functions. This version includes an Appendix that gives an operational semantics for the core language, and proofs of the metatheoretical results stated in the paper.

This is the accepted version of the paper.This version of the publication may differ from the final published version. Abstract Reynolds' abstraction theorem shows how a typing judgement in System F can be translated into a relational statement (in second order predicate logic) about inhabitants of the type. We obtain a similar result for a single lambda calculus (a pure type system), in which terms, types and their relations are expressed. Working within a single system dispenses with the need for an interpretation layer, allowing for an unusually simple presentation. While the unification puts some constraints on the type system (which we spell out), the result applies to many interesting cases, including dependently-typed ones. Permanent

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Reynolds' abstraction theorem shows how a typing judgement in System F can be translated into a relational statement (in second order predicate logic) about inhabitants of the type. We obtain a similar result for a single lambda calculus (a pure type system), in which terms, types and their relations are expressed. Working within a single system dispenses with the need for an interpretation layer, allowing for an unusually simple presentation. While the unification puts some constraints on the type system (which we spell out), the result applies to many interesting cases, including dependently-typed ones.

We propose to unify the treatment of a broad range of modalities in typed lambda calculi. We do so by defining a generic structure of modalities, and show that this structure arises naturally from the structure of intuitionistic logic, and as such finds instances in a wide range of type systems previously described in literature. Despite this generality, this structure has a rich metatheory, which we expose.

We describe a systematic method to build a logic from any programming language described as a Pure Type System (PTS). The formulas of this logic express properties about programs. We define a parametricity theory about programs and a realizability theory for the logic. The logic is expressive enough to internalize both theories. Thanks to the PTS setting, we abstract most idiosyncrasies specific to particular type theories. This confers generality to the results, and reveals parallels between parametricity and realizability.

This is the unspecified version of the paper.This version of the publication may differ from the final published version. Permanent repository link: R O S S P A T E R S O NCity University, London, UK (e-mail:)ross@soi.city.ac.uk) AbstractReynolds' abstraction theorem (Reynolds, J. C. (1983) Types, abstraction and parametric polymorphism, Inf. Process. 83(1), 513-523) shows how a typing judgement in System F can be translated into a relational statement (in second-order predicate logic) about inhabitants of the type. We obtain a similar result for pure type systems (PTSs): for any PTS used as a programming language, there is a PTS that can be used as a logic for parametricity. Types in the source PTS are translated to relations (expressed as types) in the target. Similarly, values of a given type are translated to proofs that the values satisfy the relational interpretation. We extend the result to inductive families. We also show that the assumption that every term satisfies the parametricity condition generated by its type is consistent with the generated logic.

We investigate the influence that document context exerts on human acceptability judgements for English sentences, via two sets of experiments. The first compares ratings for sentences presented on their own with ratings for the same set of sentences given in their document contexts. The second assesses the accuracy with which two types of neural models-one that incorporates context during training and one that does not-predict these judgements. Our results indicate that: (1) context improves acceptability ratings for ill-formed sentences, but also reduces them for well-formed sentences; and (2) context helps unsupervised systems to model acceptability. 1