# Partial evaluation

In computing, partial evaluation is a technique for several different types of program optimization by specialization. The most straightforward application is to produce new programs that run faster than the originals while being guaranteed to behave in the same way.

A computer program prog is seen as a mapping of input data into output data:

${\displaystyle prog:I_{\text{static}}\times I_{\text{dynamic}}\to O,}$

where ${\displaystyle I_{\text{static}}}$, the static data, is the part of the input data known at compile time.

The partial evaluator transforms ${\displaystyle \langle prog,I_{\text{static}}\rangle }$ into ${\displaystyle prog^{*}:I_{\text{dynamic}}\to O}$ by precomputing all static input at compile time. ${\displaystyle prog^{*}}$ is called the "residual program" and should run more efficiently than the original program. The act of partial evaluation is said to "residualize" ${\displaystyle prog}$ to ${\displaystyle prog^{*}}$.

## Futamura projections

A particularly interesting example of the use of partial evaluation, first described in the 1970s by Yoshihiko Futamura,[1] is when prog is an interpreter for a programming language.

If Istatic is source code designed to run inside that interpreter, then partial evaluation of the interpreter with respect to this data/program produces prog*, a version of the interpreter that only runs that source code, is written in the implementation language of the interpreter, does not require the source code to be resupplied, and runs faster than the original combination of the interpreter and the source. In this case prog* is effectively a compiled version of Istatic.

This technique is known as the first Futamura projection, of which there are three:

1. Specializing an interpreter for given source code, yielding an executable.
2. Specializing the specializer for the interpreter (as applied in #1), yielding a compiler.
3. Specializing the specializer for itself (as applied in #2), yielding a tool that can convert any interpreter to an equivalent compiler.

They were described by Futamura in Japanese in 1971[2] and in English in 1983.[3]