Writing Constructive Proofs Yielding Efficient Extracted Programs

Abstract

The NuPRL system [3] was designed for interactive writing of machine–checked constructive proofs and for extracting algorithms from the proofs. The extracted algorithms are guaranteed to be correct11Provided that the trusted parts of the NuPRL system work correctly. which makes it possible to use NuPRL as a programming language with built-in verification[1,5,7,8,9,10]. However it turned out that proofs written without algorithmic efficiency in mind often produce very inefficient algorithms — exponential and double-exponential ones for problems that can be solved in polynomial time.In this paper we present some general principles of efficient programming in constructive type theory as well as describe a case study that shows how these principles apply to particular problems. We consider the proof of the Myhill–Nerode automata minimization theorem from the NuPRL automata library [4] which leaded to a double–exponential (in time) extracted program. Systematic use of the presented principles allowed us to build a new complexity cautious proof leading to polynomial-time algorithm extracted by the same NuPRL extractor.We believe that the principles presented in this paper in combination with other methods may lead to an efficient technique of programming-by-proofs.