Using Middle-out Reasoning to Control the Synthesis of Tail-recursive Programs 1

We describe a novel technique for the automatic synthesis of tail-recursive programs. The technique is to specify the required program using the standard equations and then synthesise the tail-recursive program using the proofs as programs technique. This requires the speciication to be proved realisable in a constructive logic. Restrictions on the form of the proof ensure that the synthesised program is tail-recursive. The automatic search for a synthesis proof is controlled by proof plans, which are descriptions of the high-level structure of proofs of this kind. We have extended the known proof plans for inductive proofs by adding a new form of generalisation and by making greater use of middle-out reasoning. In middle-out reasoning we postpone decisions in the early part of the proof by the use of meta-variables which are instantiated, by uniication, during later parts of the proof. Higher order uniication is required, since these meta-variables can represent higher order objects. The program synthesised is automatically veriied to ensure that it satisses its speciication. This type of veriication is contrasted with template-based transformation approaches which require proofs that the general transformations described by the templates preserve equivalence. The technique described is more general than template-based approaches, since it is not tied to program patterns which must be speciied in advance. Detailed information about proof structure enables it to use a wider repertoire of rewritings in a more goal-directed way than comparable transformational techniques.