# Lambda cube

The lambda cube. Direction of each arrow is direction of inclusion.

In mathematical logic and type theory, the λ-cube is a framework for exploring the axes of refinement in Thierry Coquand's calculus of constructions, starting from the simply typed lambda calculus (written as λ→ in the cube diagram) as the vertex of a cube placed at the origin, and the calculus of constructions (higher-order dependently typed polymorphic lambda calculus; written as λPω in the diagram) as its diametrically opposite vertex. Each axis of the cube represents a new form of abstraction:

All eight calculi include the most basic form of abstraction, values depending on values, ordinary functions as in the simply typed lambda calculus. The richest calculus in the cube, with all three abstractions, is the calculus of constructions. All eight calculi are strongly normalizing.

Subtyping however is not represented in the cube, even though systems like ${\displaystyle F_{<:}^{\omega }}$, known as higher-order bounded quantification, which combines subtyping and polymorphism are of practical interest, and can be further generalized to bounded type operators. Further extensions to ${\displaystyle F_{<:}^{\omega }}$ allow the definition of purely functional objects; these systems were generally developed after the lambda cube paper was published.[1]

The idea of the cube is due to the mathematician Henk Barendregt (1991). The framework of pure type systems generalizes the lambda cube in the sense that all corners of the cube, as well as many other systems can be represented as instances of this general framework.[2] This framework predates the lambda cube by a couple of years. In his 1991 paper, Barendregt also defines the corners of the cube in this framework.

In his Habilitation à diriger les recherches Lambda-Prolog de A à Z... ou presque [1], Olivier Ridoux gives a cut-out template of the Lambda cube (p. 70) and also a dual representation of the cube as an octahedron, where the 8 vertices are replaced by faces, as well as a dual representation as a dodecahedron, where the 12 edges are replaced by faces.

## Notes

1. ^ Pierce, 2002, chapters 31 and 32
2. ^ Pierce, 2002, p. 466

## References

• Morten Heine Sørensen, Paweł Urzyczyn, Lectures on the Curry-Howard isomorphism, Elsevier, 2006, ISBN 0-444-52077-5, chapter 14, "Pure type systems and the lambda cube"
• H.P. Barendregt, Introduction to generalized type systems, Journal of Functional Programming, 1(2):125-154, April 1991.
• Olivier Ridoux, Lambda-Prolog de A à Z... ou presque, Habilitation à Diriger les Recherches, Université de Rennes 1, 1988 (French) PostScript, PDF, html.
• Pierce, Benjamin (2002). Types and Programming Languages. MIT Press. ISBN 0-262-16209-1.