Using Middle-Out Reasoning to Control the Synthesis of Tail-Recursive Programs

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 specification to be proved realisable in a constructive logic. Restrictions on the form of the proof ensure that the synthesised program is tail-recursive. Th

A general technique for automatically optimizing programs through the use of proof plans

The use of {\em proof plans} -- formal patterns of reasoning for theorem proving -- to control the (automatic) synthesis of efficient programs from standard definitional equations is described. A general framework for synthesizing efficient programs, using tools such as higher-order unification, has been developed and holds promise for encapsulating an otherwise diverse, and often ad hoc, range of transformation techniques. A prototype system has been implemented. We illustrate the methodology by a novel means of affecting {\em constraint-based} program optimization through the use of proof plans for mathematical induction. Proof plans are used to control the (automatic) synthesis of functional programs, specified in a standard equational form, {$\cal E$}, by using the proofs as programs principle. The goal is that the program extracted from a constructive proof of the specification is an optimization of that defined solely by {$\cal E$}. Thus the theorem proving process is a form of program optimization allowing for the construction of an efficient, {\em target}, program from the definition of an inefficient, {\em source}, program. The general technique for controlling the syntheses of efficient programs involves using {$\cal E$} to specify the target program and then introducing a new sub-goal into the proof of that specification. Different optimizations are achieved by placing different characterizing restrictions on the form of this new sub-goal and hence on the subsequent proof. Meta-variables and higher-order unification are used in a technique called {\em middle-out reasoning} to circumvent eureka steps concerning, amongst other things, the identification of recursive data-types, and unknown constraint functions. Such problems typically require user intervention