The real-time process algebra (RTPA) is a set of new mathematical notations for formally describing system architectures, and static and dynamic behaviors. It is recognized that the specification of software behaviors is a three-dimensional problem known as: (i) mathematical operations, (ii) event/process timing, and (iii) memory manipulations. Conventional formal methods in software engineering were designed to describe the 1-D (type (i)) or 2-D (types (i) and (iii)) static behaviors of software systems via logic, set and type theories. However, they are inadequate to address the 3-D problems in real-time systems. A new notation system that is capable to describe and specify the 3-D real-time behaviors, the real-time process algebra (RTPA), is developed in this paper to meet the fundamental requirements in software engineering. RTPA is designed as a coherent software engineering notation system and a formal engineering method for addressing the 3-D problems in software system specification, refinement, and implementation, particularly for real-time and embedded systems. In this paper, the RTPA meta-processes, algebraic relations, system architectural notations, and a set of fundamental primary and abstract data types are described. On the basis of the RTPA notations, a system specification method and a refinement scheme of RTPA are developed. Then, a case study on a telephone switching system is provided, which demonstrates the expressive power of RTPA on formal specification of both software system architectures and behaviors. RTPA elicits and models 32 algebraic notations, which are the common core of existing formal methods and modern programming languages. The extremely small set of formal notations has been proven sufficient for modeling and specifying real-time systems, their architecture, and static/dynamic behaviors in real-world software engineering environment.
