On the Cubic Bottleneck in Subtyping and Flow Analysis

A variety of program analysis methods have worst case time complexity that grows cubicly in the length of the program being analyzed. Cubic complexity typically arises in control flow analyses and the inference of recursive types (including object types). It is often said that such cubic performance can not be improved because these analyses require "dynamic transitive closure". Here we prove linear time reductions from the problem of determining membership for languages defined by 2-way nondeterministic pushdown automata (2NPDA) to problems of flow analysis and typability in the Amadio-Cardelli type system. An O(n 3 ) algorithm was given for 2NPDA acceptability in 1968 and is still the best known. The reductions are factored through the problem of "monotone closure" and we propose linear time reduction of the monotone closure as a method of establishing "monotone closure hardness" for program analysis problems. A sub-cubic procedure for a monotone closure hard problem would imply a ..