On the computational complexity of Data Flow Analysis over finite bounded meet semilattices

Abstract

The problem of computing the meet over all paths (MOP) solution in a monotone data flow framework over an infinite meet semilattice is generally undecidable [1]. Hence, the maximum fixed point (MFP) solution, which is polynomial time computable on semi-lattices of finite height, is generally used in practice for program analysis questions in monotone data flow frameworks. However, we show that if the semi-lattice is finite, computing MOP solution is NL-complete with respect to log space reductions, which implies parallelizability and polynomial time computability. It is also shown that the problem of computing the maximum fixed point (MFP) solution is P-complete with respect to log space reductions, and hence not efficiently parallelizable, even when the flow graph is directed acyclic and the semilattice has just four elements. These results appear in contrast with the fact that when the semilattice is not finite, solving the MOP problem is significantly harder than MFP.