Automatically Inferring Properties of Computer Programs Automatically Inferring Properties of Computer Programs

This thesis presents two independent pieces of research. First, we consider the problem of automatically inferring properties of programs. Our approach is to explore the application of familiar type inference principles to a \type system" suuciently expressive that the typing problem is eeectively the checking of program speciications. We present such a type system, and use familiar syntax-directed type inference rules to give a polynomial-time procedure for inferring type theorems in this type system. We discuss examples of simple functional programs and the speciication information this procedure automatically infers. The enriched notion of type allows the deenition of any recursively enumerable set as a type, and includes argument-dependent output types for functions. The inference procedure is capable for example of automatically inferring that an insertion sort program always returns a sorted permutation of its input. We present both rst-order and higher-order versions of our sample programming language and inference algorithms for both languages. We believe most of the interesting inferential challenges are already present in the rst-order case. The second piece of research we present addresses the satissability of sets of formulas in a particular set constraints language. We consider set expressions built up from set constants by union, set complement, and taking the image of a set expression under a function or relation. Previous work in this area has neglected the \Tarskian" case where the functions and relations are allowed to take on arbitrary meanings, rather than only a standard Herbrand meaning. We prove that the satissability of a nite set of subset formulas between these \Tarskian" set expressions is in nondeter-ministic doubly exponential time. Our proof is by reduction to a new Diophantine inequation solvability problem, which we show to be in nondeterministic exponential time, but conjecture to be in NP.