Pure Type Systems without Explicit Contexts

2017

Type theories, from the early days of Montague Semantics (Montague 1974) to the recent work of using rich or modern type theories, have a long history of being employed as foundational languages of natural language semantics. In this introductory chapter, we will describe and discuss the development of type theories as foundational languages of mathematics, as well as their applications as foundational languages for formal semantics. In the end, a brief description of each chapter in the volume will follow.

ACM SIGPLAN Notices, 2007

We present a model of recursive and impredicatively quantified types with mutable references. We interpret in this model all of the type constructors needed for typed intermediate languages and typed assembly languages used for object-oriented and functional languages. We establish in this purely semantic fashion a soundness proof of the typing systems underlying these TILs and TALs---ensuring that every well-typed program is safe. The technique is generic, and applies to any small-step semantics including λ-calculus, labeled transition systems, and von Neumann machines. It is also simple, and reduces mainly to defining a Kripke semantics of the Gödel-Löb logic of provability. We have mechanically verified in Coq the soundness of our type system as applied to a von Neumann machine.

2012

We extend Pure Type Systems with a function turning each term M of type A into a dummy M of the same type ( ⋅ is not an identity, in that M ≠ M ). Intuitively, a dummy represents an unknown, canonical object of the given type: dummies are opaque (cannot be internally inspected), and irrelevant in the sense that dummies of a same type are convertible to each other. This latter condition makes convertibility in PTS with dummies (DPTS) stronger than usual, hence raising not trivial consistency issues. DPTS offer an alternative approach to (proof) irrelevance, tagging irrelevant information at the level of terms and not of types, and avoiding the annoying syntactical duplication of products, abstractions and applications into an explicit and an implicit version, typical of systems like ICC * .

Electronic Notes in Theoretical Computer Science, 1997

We present a notion of classical pure type system, which extends the formalism of pure type system with a double negation operator.

Journal of Logic, Language and Information, 2006

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.

This thesis presents a new algorithm for Normalisation by Evaluation for different type theories. This algorithm is later used to define a type-checking algorithm and to prove other meta-theoretical results about predicative type systems.

1997

We present a notion of classical pure type system, which extends the formalism of pure type system with a double negation operator.

Electronic Notes in Theoretical Computer Science, 1998

This article explores the use of types constrained by the de nition of functions of given types. This notion supports both overloading and a form of subtyping, and is related to Haskell type classes and System O. We study an extension of the Damas-Milner system, in which o verloaded functions can be de ned. The inference system presented uses a context-independent o verloading policy, speci ed by m e a n s of a predicate used in a single inference rule. The treatment of overloading is less restrictive t h a n in similar systems. Type annotations are not required, but can be used to simplify inferred types. The work motivates the use of constrained types as parameters of other, higher-order types.