Recursion Removal/Introduction by Formal Transformation: An Aid to Program Development and Program Comprehension

The transformation of a recursive program to an iterative equivalent is a fundamental operation in computer science. In the reverse direction, the task of reverse engineering (analysing a given program in order to determine its specification) can be greatly ameliorated if the program can be re-expressed in a suitable recursive form. However, the existing recursion removal transformations, such as the techniques discussed by Knuth [1] and Bird [2], can only be applied in the reverse direction if the source program happens to match the structure produced by a particular recursion removal operation. In this paper we describe a much more powerful recursion removal and introduction operation which describes its source and target in the form of an action system (a collection of labels and calls to labels). A simple, mechanical restructuring operation can be applied to a great many iterative programs, which will put them in a suitable form for recursion introduction. Our transformation generates strictly more iterative versions than the standard methods, including those of Knuth and Bird [1, 2]. With the aid of this theorem we prove a (somewhat counterintuitive) result for programs that contain sequences of two or more recursive calls: under a reasonable commutativity condition, ‘depth-first’ execution is more general than ‘breadth-first’ execution. In ‘depth-first’ execution, the execution of each recursive call is completed, including all sub-calls, before execution of the next call is started. In ‘breadth-first’ execution, each recursive call in the sequence is partially executed but any sub-calls are temporarily postponed. This result means that any breadth-first program can be reimplemented as a corresponding depth-first program, but the converse does not hold. We also treat the case of ‘random-first’ execution, where the execution order is implementation dependent. For the more restricted domain of tree searching we show that a ‘breadth-first’ search is the most general form. We also give two examples of recursion introduction as an aid to formal reverse engineering.