Definitions

# Partial evaluation

In computing, partial evaluation is a technique for program optimization by specialization.

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

$prog : I_\left\{static\right\} times I_\left\{dynamic\right\} to O$

$I_\left\{static\right\}$, the static data, is the part of the input data known at compile time.

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

## Futamura projections

A particularly interesting example of this, first described in the 1970s by Yoshihiko Futamura, is when prog is an interpreter for a programming language.

If Istatic is source code designed to run inside said 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. Compiling by specializing an interpreter
2. Compiler generation by self-application
3. Compiler generator generation by double self-application

## References

• Yoshihiko Futamura (1971). "Partial Evaluation of Computation Process – An Approach to a Compiler-Compiler". Systems, Computers, Controls 2 (5): 45–50. Reprinted in Higher-Order and Symbolic Computation 12 (4): 381–391, 1999, with a foreword.
• Charles Consel and Olivier Danvy (1993). "Tutorial Notes on Partial Evaluation". Proceedings of the Twentieth Annual ACM Symposium on Principles of Programming Languages 493–501.