Explicit Substitutions for Contextual Type Theory

2015

Constructive t ype theories, such as that of Martin-L of, allow program construction and veri cation to take place within a single system: proofs may be read as programs and propositions as types. However, parts of proofs may be seen to be irrelevant from a computational viewpoint. We show h o w a form of abstract interpretation may be used to detect computational redundancy in a functional language based upon Martin-L of's type theory. T h us, without making any alteration to the system of type theory itself, we present a n automatic way of discovering and removing such redundancy. We also note that the strong normalisation property o f t ype theory means that proofs of correctness of the abstract interpretation are simpler, being based upon a set-theoretic rather than a domain-theoretic semantics.

Journal of Functional Programming, 2008

Linear typing schemes can be used to guarantee non-interference and so the soundness of in-place update with respect to a functional semantics. But linear schemes are restrictive in practice, and more restrictive than necessary to guarantee soundness of in-place update. This limitation has prompted research into static analysis and more sophisticated typing disciplines to determine when in-place update may be safely used, or to combine linear and non-linear schemes. Here we contribute to this direction by defining a new typing scheme that better approximates the semantic property of soundness of in-place update for a functional semantics. We begin from the observation that some data is used only in a "read-only" context, after which it may be safely re-used before being destroyed. Formalising the in-place update interpretation in a machine model semantics allows us to refine this observation, motivating three usage aspects apparent from the semantics that are used to annotate function argument types. The aspects are (1) used destructively, (2) used read-only but shared with result, and (3) used read-only and not shared with the result. The main novelty is aspect (2) that allows a linear value to be safely read and even aliased with a result of a function without being consumed. This novelty makes our type system more expressive than previous systems for functional languages in the literature. The system remains simple and intuitive, but it enjoys a strong soundness property whose proof is non-trivial. Moreover, our analysis features principal types and feasible type reconstruction, as shown in (Konecny, 2003b).

Theoretical Computer Science, 2011

This paper shows (1) the undecidability of the type checking and the typability problems in the domain-free lambda calculus with negation, product, and existential types, (2) the undecidability of the typability problem in the domain-free polymorphic lambda calculus, and (3) the undecidability of the type checking and the typability problems in the domain-free lambda calculus with function and existential types. The first and the third results are proved by the second result and CPS translations that reduce those problems in the domain-free polymorphic lambda calculus to those in the the domain-free lambda calculi with existential types. The key idea is the conservativity of the domain-free lambda calculi with existential types over the images of the translations.

Lecture Notes in Computer Science, 2006

Herbrand constraint solving or unification has long been understood as an efficient mechanism for type checking and inference for programs using Hindley/Milner types. If we step back from the particular solving mechanisms used for Hindley/Milner types, and understand type operations in terms of constraints we not only give a basis for handling Hindley/Milner extensions, but also gain insight into type reasoning even on pure Hindley/Milner types, particularly for type errors. In this paper we consider typing problems as constraint problems and show which constraint algorithms are required to support various typing questions. We use a light weight constraint reasoning formalism, Constraint Handling Rules, to generate suitable algorithms for many popular extensions to Hindley/Milner types. The algorithms we discuss are all implemented as part of the freely available Chameleon system.

Lecture Notes in Computer Science, 2009

Rewriting logic semantics (RLS) was proposed as a programing language definitional framework that unifies operational and algebraic denotational semantics; see and the references there. Once a language is defined as an RLS theory, many generic tools are immediately available for use with no additional cost to the designer. These include a formal inductive theorem proving environment, an efficient interpreter, a state space explorer, and even a model checker. RLS has already been used to define a series of didactic and real languages .

Essays in Honour of Gilles Kahn, 2009

This paper presents a formal description of a small functional language with dependent types. The language contains data types, mutual recursive/inductive definitions and a universe of small types. The syntax, semantics and type system is specified in such a way that the implementation of a parser, interpreter and type checker is straightforward. The main difficulty is to design the conversion algorithm in such a way that it works for open expressions. The paper ends with a complete implementation in Haskell (around 400 lines of code).

Journal of Applied Logic, 2013

The modal logic S4 can be used via a Curry-Howard style correspondence to obtain a λcalculus. Modal (boxed) types are intuitively interpreted as 'closed syntax of the calculus'. This λ-calculus is called modal type theory -this is the basic case of a more general contextual modal type theory, or CMTT.

Meseguer and Rosu [MR04,MR07] proposed rewriting logic semantics (RLS) as a programing language definitional framework that unifies operational and algebraic denotational semantics. Once a language is defined as an RLS theory, many generic tools are immediately available for use with no additional cost to the designer. These include a formal inductive theorem proving environment, an efficient interpreter, a state space explorer, and even a model checker. RLS has already been used to define a series of didactic and real languages [MR04, MR07], but its benefits in connection with defining and reasoning about type systems have not been fully investigated yet. This paper shows how the same RLS style employed for giving formal definitions of languages can be used to define type systems. The same term-rewriting mechanism used to execute RLS language definitions can now be used to execute type systems, giving type checkers or type inferencers. Since both the language and its type system ar...

2008

We show how programming language semantics and definitions of their corresponding type systems can both be written in a single framework amenable to proofs of soundness. The framework is based on full rewriting logic (not to be confused with context reduction or term rewriting), where rules can match anywhere in a term (or configuration).

ACM SIGPLAN Notices, 2003

We develop an explicit two level system that allows programmers to reason about the behavior of effectful programs. The first level is an ordinary ML-style type system, which confers standard properties on program behavior. The second level is a conservative extension of the first that uses a logic of type refinements to check more precise properties of program behavior. Our logic is a fragment of intuitionistic linear logic, which gives programmers the ability to reason locally about changes of program state. We provide a generic resource semantics for our logic as well as a sound, decidable, syntactic refinement-checking system. We also prove that refinements give rise to an optimization principle for programs. Finally, we illustrate the power of our system through a number of examples.