On the Big-R Notation for Describing Iterative and Recursive Behaviors

Abstract

The iterative and recursive control structures are the most fundamental mechanisms of computing, because they make programming more effective and expressive. However, these constructs are perhaps the most diverse and confusable instructions in programming languages at both syntactic and semantic levels. Although a wide variety of ways have been proposed for describing iterations, there is still a lack of a unified mathematical notation that may be used to express the notion of repetitive, recursive, and predicative behaviors and architectures in computing. This paper introduces the big-R notation that provides a unifying mathematical treatment of iterations and recursions in computing. Mathematical models of iterations and recursions are developed using logical inductions. Based on the mathematical model of the big-R notation, fundamental properties of iterative and recursive behaviors of software are comparatively analyzed. The big-R notation has been adopted and implemented in real-time process algebra (RTPA) and its supporting tools. A wide range of applications of the big-R notation are identified for rigorously describing iterations and recursions in computing and software engineering. Case studies demonstrate that a convenient notation may dramatically reduce the difficulty and complexity in expressing a frequently used and highly recurring concept and notion in computing.