Discrete Time Process Algebra with Relative Timing

Process algebra in the form of ACP describes the main features of imperative concurrent programming without explicit mention of time. Implicitly, time is present in the interpretation of sequential composition: in p · q the process p must be executed before q. A quantative view on the relation between process execution and progress of time is absent in ACP, however. Process algebras can be developed that provide standardised features to incorporate a quantative view on time. An option is to represent time by means of non-negative reals, and to have time stamps on actions. Another option is to divide time in slices indexed by natural numbers, to have an implicit or explicit time stamping mechanism that provides each action with the time slice in which it occurs and to have a time order within each time slice only. We use the phrase discrete time process algebra if an enumeration of time slices is used. The objective of this note is to extend ACP to a discrete time process algebra. We consider discrete time process algebras with relative timing, where timing refers to the execution of the previous action. We present the so-called two-phase version, where the passage of time and the execution of actions is separated. Another version of discrete time process algebra uses absolute timing, where all timing refers to an absolute clock. Finally, we have discrete process algebra with parametric timing, where absolute and relative timing are integrated. There are many practical uses conceivable for timed process algebras. In particular, we mention the ToolBus. This ToolBus contains a program notation called T which is syntactically sugared discrete time process algebra. Programs in T are called T-scripts. The runtime system is also described in terms of discrete time process algebra. By using randomised symbolic execution the ToolBus implementation enacts that the axioms of process algebra can be viewed as correctness preserving transformations of T-scripts. A comparable part of discrete time process algebra that is used to describe T-scripts has also been used for the description of φ-SDL, flat SDL, a subset of SDL that leaves out modularisation and concentrates on timing aspects. We design our algebras (or rather their specifications) in a modular, incremental way.