%T Extending CSP %A Adrian E. Lawrence %E Peter H. Welch, Andr\[`e] W. P. Bakkers %B Proceedings of WoTUG\-21: Architectures, Languages and Patterns for Parallel and Distributed Applications %X CSP, timed or untimed, has not included a general treatment of priority, although the PRI ALT constructor is an essential part of occam. This paper introduces CSPP which includes a generalization of PRI ALT in the form of a prioritized external choice P <pribox> Q. PRI PAR is also included. A new denotational semantics is introduced, although only the simplest model is outlined. The work is intended to provide a solid rogorous foundation for hardware\-software codesign. And a companion paper describes untimed HCSP which is a further extension of CSP built upon these foundations. It was first presented informally at the Twente WoTUG\-20 technical meeting.