Call-by-name Gradual Type Theory

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 (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, 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. This work is licensed under a Creative Commons “Attribution 4.0 International” license. ∗This material is based upon work supported by the National Science Foundation under grant CCF-1453796. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. †This research was partially supported by the United States Air Force Research Laboratory under agreement numbers FA-95501210370 and FA-95501510053. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the United States Air Force Research Laboratory, the U.S. Government or Carnegie Mellon University. 1 ar X iv :1 80 2. 00 06 1v 2 [ cs .P L ] 2 3 M ar 2 01 8