Computer Science Department
School of Computer Science, Carnegie Mellon University


Tridirectional Typechecking

Joshua Dunfield, Frank Pfenning

March 2004

This is an extended version of "Tridirectional typechecking", appearing in
ACM Symposium on Principles of Programming Languages (POPL 04), January 2004.

Keywords: Type systems, intersection types, union types, type refinements, bidirectional typechecking, datasort refinements, index refinements

In prior work we introduced a pure type assignment system that encompasses a rich set of property types, including intersections, unions, and universally and existentially quantified dependent types. This system was shown sound with respect to a call-by-value operational semantics with effects, yet is inherently undecidable. In this paper we provide a decidable formulation for this system based on bidirectional checking, combining type synthesis and analysis following logical principles. The presence of unions and existential quantification requires the additional ability to visit subterms in evaluation position before the context in which they occur, leading to a tridirectional type system. While soundness with respect to the type assignment system is immediate, completeness requires the novel concept of contextual type annotations, introducing a notion from the study of principal typings into the source program.

36 pages

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