Lambda the Ultimate

The project Incremental λ-Calculus is just starting (compared to more mature approaches like self-adjusting computation), with a first publication last year.

A theory of changes for higher-order languages — incrementalizing λ-calculi by static differentiation
Paolo Giarusso, Yufei Cai, Tillmann Rendel, and Klaus Ostermann. 2014

If the result of an expensive computation is invalidated by a small change to the input, the old result should be updated incrementally instead of reexecuting the whole computation. We incrementalize programs through their derivative. A derivative maps changes in the program’s input directly to changes in the program’s output, without reexecuting the original program. We present a program transformation taking programs to their derivatives, which is fully static and automatic, supports first-class functions, and produces derivatives amenable to standard optimization.

We prove the program transformation correct in Agda for a family of simply-typed λ-calculi, parameterized by base types and primitives. A precise interface specifies what is required to incrementalize the chosen primitives.

We investigate performance by a case study: We implement in Scala the program transformation, a plugin and improve performance of a nontrivial program by orders of magnitude.

I like the nice dependent types: a key idea of this work is that the "diffs" possible from a value v do not live in some common type diff(T), but rather in a value-dependent type diff(v). Intuitively, the empty list and a non-empty list have fairly different types of possible changes. This makes change-merging and change-producing operations total, and allow to give them a nice operational theory. Good design, through types.

(The program transformation seems related to the program-level parametricity transformation. Parametricity abstract over equality justifications, differentiation on small differences.)