In mathematical logic and type theory, the λ-cube is a framework for exploring the axes of refinement in Coquand's calculus of constructions, starting from the simply typed lambda calculus as the vertex of a cube placed at the origin, and the calculus of constructions (higher order dependently-typed polymorphic lambda calculus) as its diametric opposite vertex. Each axis of the cube represents a new form of abstraction:
- Terms depending on types, or polymorphism (as in System F),
- Types depending on types, or type operators, and
- Types depending on terms, or dependent types (as in LF).
All eight calculi include the most basic form of abstraction, terms depending on terms, ordinary functions as in the simply-typed lambda calculus.
All eight calculi are strongly normalizing.
The idea of the cube is due to the mathematician Henk Barendregt (1991).
See also
References
- H.P. Barendregt, Introduction to generalized type systems, Journal of Functional Programming, 1(2):125-154, April 1991.
- Simon Peyton Jones and Erik Meijer, 1997. Henk: A Typed Intermediate Language
- Barendregt's Lambda Cube
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