Turing machines, transition systems, and interaction

We present Persistent Turing Machines (PTMs), a new way of interpreting Turing-machine computation, one that is both interactive and persistent. We show that the class of PTMs is isomorphic to a very general class of effective transition systems. One may therefore conclude that the extensions to the Turing-machine model embodied in PTMs are sufficient to make Turing machines expressively equivalent to transition systems. We also define the persistent stream language (PSL) of a PTM and a corresponding notion of PSL-equivalence, and consider the infinite hierarchy of successively finer equivalences for PTMs over finite interaction-stream prefixes. We show that the limit of this hierarchy is strictly coarser than PSL-equivalence, a "gap" whose presence can be attributed to the fact that the transition systems corresponding to PTM computations naturally exhibit unbounded nondeterminism. We also consider amnesic PTMs and a corresponding notion of equivalence based on amnesic stream languages (ASLs). It can be argued that amnesic stream languages are representative of the classical view of Turing-machine computation. We show that the class of ASLs is strictly contained in the class of PSLs. Furthermore, the hierarchy of PTM equivalence relations collapses for the subclass of amnesic PTMs. These results indicate that, in a stream-based setting, the extension of the Turing-machine model with persistence is a nontrivial one, and provide a formal foundation for reasoning about programming concepts such as objects with static attributes.