Call-by-name Gradual Type Theory

TL;DR

This paper introduces gradual type theory for call-by-name languages, defining core concepts via universal properties and providing semantic models that justify standard cast behaviors and the gradual guarantee.

Contribution

It develops a novel logical and type-theoretic framework for call-by-name gradual typing, with universal property-based definitions and semantic models.

Findings

Most operational cast behaviors are uniquely determined by the gradual guarantee.

Semantic models validate standard cast definitions and highlight violations by non-standard ones.

The theory connects gradual typing with categorical semantics via equipments.

Abstract

We present gradual type theory, a logic and type theory for call-by-name gradual typing. We define the central constructions of gradual typing (the dynamic type, type casts and type error) in a novel way, by universal properties relative to new judgments for gradual type and term dynamism, which were developed in blame calculi and to state the "gradual guarantee" theorem of gradual typing. Combined with the ordinary extensionality ($\eta$) principles that type theory provides, we show that most of the standard operational behavior of casts is uniquely determined by the gradual guarantee. This provides a semantic justification for the definitions of casts, and shows that non-standard definitions of casts must violate these principles. Our type theory is the internal language of a certain class of preorder categories called equipments. We give a general construction of an equipment interpreting gradual type theory from a 2-category representing non-gradual types and programs, which is a semantic analogue of Findler and Felleisen's definitions of contracts, and use it to build some concrete domain-theoretic models of gradual typing.