Abstract

The best way we know of describing the semantics of parametric polymorphism is relational parametricity, whose central result is Reynolds’ Abstraction Theorem. It has many striking consequences, including “free theorems”, non-inhabitation results, and encodings of algebraic datatypes.

Relational parametricity is a principle of invariance: the behaviour of polymorphic code is invariant under changes of data representation. Invariance results also abound in mathematics and physics. The area of a triangle is invariant with respect to rotations and reflections; the determinant of a matrix is invariant under changes of basis; and Newton’s laws are the same in all inertial frames.

In this talk, I will talk about recent work inspired by this connection, in which types are indexed by attributes with algebraic structure, and polymorphism over such attributes expresses invariance results. I will describe in detail its application to computational geometry, in which polymorphic types reflect that the behaviour of a program is invariant under varieties of affine transformation. This generalises earlier work on types for units of measure. I will also describe applications to information-flow security and continuity analysis.

This is joint work with Robert Atkey and Patricia Johann of the University of Strathclyde.