We have previously introduced the static dependency pair method that proves termination by analyzing the static recursive structure of various extensions of term rewriting systems for handling higher-order functions. The key is to succeed with the formalization of recursive structures based on the notion of strong computability, which is introduced for the termination of typed λ-calculi. To bring the static dependency pair method close to existing functional programs, we also extend the method to term rewriting models in which functional abstractions with patterns are permitted. Since the static dependency pair method is not sound in general, we formulate a class; namely, accessibility, in which the method works well. The static dependency pair method is a very natural reasoning; therefore, our extension differs only slightly from previous results. On the other hand, a soundness proof is dramatically difficult.

For simply-typed term rewriting systems (STRSs) and higher-order rewrite systems (HRSs) a la Nipkow, we proposed a method for proving termination, namely the static dependency pair method. The method combines the dependency pair method introduced for first-order rewrite systems with the notion of strong computability introduced for typed λ-calculi. This method analyzes a static recursive structure based on definition dependency. By solving suitable constraints generated by the analysis, we can prove termination. In this paper, we extend the method to rewriting systems for functional programs (RFPs) with product, algebraic data, and ML-polymorphic types. Although the type system in STRSs contains only product and simple types and the type system in HRSs contains only simple types, our RFPs allow product types, type constructors (algebraic data types), and type variables (ML-polymorphic types). Hence, our RFPs are more representative of existing functional programs than STRSs and HRSs. Therefore, our result makes a large contribution to applying theoretical rewriting techniques to actual problems, that is, to proving the termination of existing functional programs.

Deforestation is a well-known program transformation technique which eliminates intermediate data structures that are passed between functions. One of its weaknesses is the inability to deforest programs using accumulating parameters. We show how certain kinds of intermediate lists produced by accumulating parameters can be deforested. In this paper we introduce an accumulative variant of foldr, called rdlof, and show the composition of functions defined by foldr and rdlof. As a simplified instance of foldr and rdlof, we then examine dmap, an accumulative extension of map, and give the corresponding fusion rules. While the associated composition rules cannot capture all deforestation problems, they can handle accumulator fusion of fold- and map-style functions in a simple manner. The rules for accumulator fusion presented here can also be viewed as a restricted composition scheme for attribute grammars, which in turn may help us to bridge the gap between the attribute and functional worlds.

In logic programming or functional programming languages, data objects, such as terms and lists, are immutable. In a basic implementation of such language, updating one element of an aggregate (contiguous data structure, such as an array) involves making a new copy of the whole aggregate. However, such copying can be expensive, and can be avoided by using a destructive update. We introduce the concept of a wrapper which enables destructive operation on an immutable object. Based on this concept, we designed the reversible functor as a solution to the aggregate update problem. We implemented the reversible functor in the existing SB-Prolog system and carried out several benchmarks. These benchmark results show its effectiveness. When using a large functor and updating it many times, the performance is improved dramatically by implementing the reversible functor. It incurs some overhead at runtime, but the amount is small and acceptable.