On the computational complexity of Data Flow Analysis

TL;DR

This paper analyzes the computational complexity of Data Flow Analysis, showing that computing the Maximum Fixed Point is P-complete even for small finite lattices, while computing the Meet Over all Paths is NL-complete and efficiently parallelizable.

Contribution

It establishes the complexity classes of MFP and MOP problems over finite lattices, highlighting differences in parallelizability and decidability.

Findings

MFP is P-complete for four-element lattices.

MOP is NL-complete for finite lattices.

MOP becomes undecidable for infinite lattices.

Abstract

We consider the problem of Data Flow Analysis over monotone data flow frameworks with a finite lattice. The problem of computing the Maximum Fixed Point (MFP) solution is shown to be P-complete even when the lattice has just four elements. This shows that the problem is unlikely to be efficiently parallelizable. It is also shown that the problem of computing the Meet Over all Paths (MOP) solution is NL-complete (and hence efficiently parallelizable) when the lattice is finite even for non-monotone data flow frameworks. These results appear in contrast with the fact that when the lattice is not finite, solving the MOP problem is undecidable and hence significantly harder than the MFP problem which is polynomial time computable for lattices of finite height.