Recursive Program Optimization Through Inductive Synthesis Proof Transformation

The research described in this paper involved developing transformation techniques which increase the efficiency of the noriginal program, the source, by transforming its synthesis proof into one, the target, which yields a computationally more efficient algorithm. We describe a working proof transformation system which, by exploiting the duality between mathematical induction and recursion, employs the novel strategy of optimizing recursive programs by transforming inductive proofs. We compare and contrast this approach with the more traditional approaches to program transformation, and highlight the benefits of proof transformation with regards to search, correctness, automatability and generality

A survey of program transformation with special reference to unfold/fold style program development

This paper consists of a survey of current, and past, work on *program transformation* for the purpose of optimization. We first discuss some of the general methodological frameworks for program modification, such as *analogy*, *explanation based learning*, *partial evaluation*, *proof theoretic optimization*, and the *unfold/fold* technique. These frameworks are not mutually exclusive, and the latter, unfold/fold, is certainly the most widely used technique, in various guises, for program transformation. Thus we shall often have occasion to: compare the relative merits of systems that employ the technique in some form, *and*; compare the unfold/fold systems with those that employ alternative techniques. We also include (and compare with unfold/fold) a brief survey of recent work concerning the use of *formal methods* for program transformation

Designing correct recursive circuits using semantics-preserving transformations of nets

This paper will present a method of formal synthesis to design correct recursive circuits by using semantics-preserving transformations of nets (SPTNs). Its theoretical base is an algebraic calculus of nets. The calculus of nets is a hardware-specific calculus, and the transformations are circuit transformations themselves. Thus, it is much better adapted to the synthesis domain. The start point of the method is a conceptually simple specification for the required function. This specification can be easily proved to be correct, thereby the perplexed problem of the specification validation can be avoided. The specification is described compactly and graphically by a small kernel of recursive equations, and the synthesis task is simplified to transform these recursive equations in in the kernel. Because only semantics-preserving transformations are allowed in synthesis procedures, the synthesis result is not only a hardware implementation, but also a proof of correctness. We will illustrate two ways to transform a basic sorter into a odd-even-merging sorter, one being based on local incremantal transformations and the other being based on global partitions. The results show that there are circuits of practical interest, which can derived formally by using this method

Generic Encodings of Constructor Rewriting Systems

Rewriting is a formalism widely used in computer science and mathematical logic. The classical formalism has been extended, in the context of functional languages, with an order over the rules and, in the context of rewrite based languages, with the negation over patterns. We propose in this paper a concise and clear algorithm computing the difference over patterns which can be used to define generic encodings of constructor term rewriting systems with negation and order into classical term rewriting systems. As a direct consequence, established methods used for term rewriting systems can be applied to analyze properties of the extended systems. The approach can also be seen as a generic compiler which targets any language providing basic pattern matching primitives. The formalism provides also a new method for deciding if a set of patterns subsumes a given pattern and thus, for checking the presence of useless patterns or the completeness of a set of patterns.Comment: Added appendix with proofs and extended example