A Process Algebra for Hybrid Systems

The work that is presented here was initiated by question of a colleague from the mechanical engineering department. He was analysing several industrial manufacturing processes (a tire production line, a cookie baking line, etc.). These processes were modeled in a procedural kind of programming language (called “χ” [AvdMR94]) extended with hybrid features such as continuous variables defined by differential equations. Having specified the process by such a program, analysis then proceeded by means of simulation. Typically, they would design a production line, and then test it by building it on a small scale to see whether it worked. So a problem would first be found on the actual production line (“valve 4C gets clogged with coagulated cookie mix”). They would try to recreate that problematic situation in their simulation tool. Then an ad-hoc solution would be found in the simulator (“if we heat the cookie mix a bit more it will not coagulate”), and finally the physical production line would be modified to reflect the “improvement”. This cycle was repeated over and over again, until confidence in the production line design had grown enough to justify substantial investments in a large scale production line. This method (of course?) proved unsatisfactory for systems larger than some critical size: the “debugging cycle” for a production line needed to be repeated ad infinitum, with each “improvement” prompting more problems than it solved. So although they were very precise and formal in specifying their processes, these specifications were then not used in any kind of formal analysis. This seemed very wasteful; so the idea arose to find a translation from the language χ to an (ACP-style) process algebra [BW90]. In that way the tedious simulation driven analysis could be replaced by, or at least augmented with, a hopefully less tedious formal analysis by process algebraic means. With the purpose described above in mind, we created an ACP-style process algebra named ACPhs (“ACP for hybrid systems”) that incorporates hybrid features. As our final goal is easy translation from the language χ to ACPhs, we focused on the ease of specifying in an intuitive way, and power to specify in a way close to χ. Sometimes, this approach forced us to make compromises on the mathematical elegance of ACPhs, and sometimes it resulted in some axioms and proofs being longer and messier than would have been necessary had we focused on theoretical beauty only. On the whole however, we believe we have created an process algebra that is both easy enough to specify in, and not too complicated to reason with.