Gradual type theory

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. These dynamism judgments build on prior work in blame calculi and on the "gradual guarantee" theorem of gradual typing. Combined with the ordinary extensionality (η) 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. This construction is a semantic analogue of the interpretation of gradual typing using contracts, and use it to build some concrete domain-theoretic models of gradual typing. CC Creative Commons 7:2 M.S. New and D.R. Licata Vol. 16:1"strictly less than", and similarly for other terms such as "greater than", "more dynamic", etc. 7:4 M.S. New and D.R. Licata Vol. 16:1 7:6 M.S. New and D.R. Licata Vol. 16:1 7:8 M.S. New and D.R. Licata Vol. 16:1 2.2. PTT Signatures. While gradual type theory proves that most operational rules of gradual typing are equivalences, some must be added as axioms. Compare Moggi's monadic metalanguage [22]: since it is a general theory of monads, it is not provable that an effect is 7:12 M.S. New and D.R. Licata Vol. 16:1

Graduality from embedding-projection pairs

“…Our analysis of graduality as observational approximation and dynamism as ep pairs builds on the axiomatic and denotational semantics of graduality for a call-by-name language presented in [New and Licata 2018]. The semantics there gives axioms of type and term dynamism that imply that upcasts and downcasts are embedding-projection pairs.…”

“…Using the ep pairs, we provide a more semantic formulation and proof of graduality. First, we use our analysis of type dynamism as denoting ep pairs to define graduality as a kind of observational error approximation up to upcast/downcast, building on the axiomatic semantics of graduality in New and Licata [2018]. We then prove the graduality theorem using our logical relation for error approximation.…”

Graduality and parametricity: together again for the first time

“…In this section we prove the central metatheoretic results of the paper: that our surface language satisfies both graduality and parametricity theorems. Each of these is considered a technical challenge to prove: parametricity is typically proven by a logical relation and graduality is proven either by a simulation argument [Siek et al 2015] or a logical relation [New and Ahmed 2018;New et al 2019], so in the worst case this would require two highly technical developments. However, we show that this is not necessary: the logical relations proof for graduality is general enough that the parametricity theorem is a corollary of the reflexivity of the logical relation.…”

Towards Gradually Typed Capabilities in the Pi-Calculus

“…However, adding recursive types is part of our future work so that we would be able to describe more precisely cases of the like. Recursive types have been addressed previously in gradual typing implementations [9,25,12,13], and in semantics formulations [20,15]. We plan on building on this body of work.…”