7/5/09 Harry Mairson (Boston)

Linear Logic and the Complexity of Control Flow Analysis

Static program analysis is about predicting the future: it's what compilers do at compile-time to predict and optimize what happens at run-time. What is the tradeoff between the efficiency and the precision of the computed prediction? Control flow analysis (CFA) is a canonical form of program analysis that answers questions like "can call site X ever call procedure P?" or "can procedure P ever be called with argument A?" Such questions can be answered exactly by Girard's geometry of interaction (GoI), but in the interest of efficiency and time-bounded computation, control flow analysis computes a crude approximation to GoI, possibly including false positives.

Different versions of CFA are parameterized by their sensitivity to calling contexts, as represented by a contour—a sequence of k labels representing these contexts, analogous to Lévy's labelled lambda calculus. CFA with larger contours is designed to make the approximation more precise. A naive calculation shows that 0CFA (i.e., with k=0) can be solved in polynomial time, and kCFA (k>0, a constant) can be solved in exponential time. We show that these bounds are exact: the decision problem for 0CFA is PTIME-hard, and for kCFA is EXPTIME-hard. Each proof depends on fundamental insights about the linearity of programs. In both the linear and nonlinear case, contrasting simulations of the linear logic exponential are used in the analysis. The key idea is to take the approximation mechanism inherent in CFA, and exploit its crudeness to do real computation.

This is joint work with David Van Horn (Brandeis University), presented at the 2008 ACM International Conference on Functional Programming.