Reactive Turing Machines with Infinite Alphabets

The notion of Reactive Turing machines (RTM) was proposed as an orthogonal extension of Turing machines with interaction. RTMs are used to define the notion of executable transition system in the same way as Turing machines are used to define the notion of computable function on natural numbers. RTMs inherited finiteness of all sets involved from Turing machines, and as a consequence, in a single step, an RTM can only communicate elements from a finite set of data. Some process calculi such as, e.g., the π-calculus, essentially use a form of infinite data in their definition and hence it immediately follows that transition systems specified in these calculi are not executable. On closer inspection, however, the π-calculus does not appear to use the infinite data in a non-computable manner. In this paper, we investigate several ways to relax the finiteness requirement. We start by considering a variant of RTMs in which all sets are allowed to be infinite, and then refine by adding extra restrictions. The most restricted variant of RTMs in which the sets of action and data symbols are still allowed to be infinite is the notion of RTM with atoms. We propose a notion of transition systems with atoms, and show that every effective legal transition system with atoms is executable by RTM with atoms. In such a way, we show that processes specified in the πcalculus are executable by RTM with atoms.