model, that no kind of deductive mathematical logic could escape the limitation. This became important later when researchers attempted to extend Prolog (which had some basis in logic programming) to concurrent computation using message passing. (See the section below). What does the mathematical theory of Actors have to say about this? A closed system is defined to be one which does not communicate with the outside. Actor model theory provides the means to characterize all the possible computations of a closed Actor system using the Representation Theorem [Hewitt 2007] as follows: The mathematical denotation denoted by a closed system is found by constructing increasingly better approximations from an initial behavior called using a behavior approximating function to construct a denotation (meaning ) for as follows: In this way, the behavior of can be mathematically characterized in terms of all its possible behaviors (including those involving unbounded nondeterminism). So mathematical logic can characterize (as opposed to implement) all the possible computations of a closed Actor system. A limitation of logic due to lack of information An open Actor system is one in which the addresses of outside Actors can be passed into in the middle of computations so that can communicate with these outside Actors. These outside Actors can then in turn communicate with Actors internal to using addresses supplied to them by . Due to limitation of the inability to deduce arrival orderings, knowledge of what messages are sent from outside would not enable the response of to be deduced. When other models of concurrent systems (e.g., process calculi) are used to implement open systems, these systems also can have behavior that depends on arrival time orderings and so cannot be implemented by logical deduction. Prolog-like concurrent systems were claimed to be based on mathematical logic Keith Clark, Hervé Gallaire, Steve Gregory, Vijay Saraswat, Udi Shapiro, Kazunori Ueda, etc. developed a family of Prolog-like concurrent message passing systems using unification of shared variables and data structure streams for messages. Claims were made that these systems were based on mathematical logic. This kind of system was used as the basis of the Japanese Fifth Generation Project (ICOT). Carl Hewitt and Gul Agha [1991] argued that these Prolog-like concurrent systems were neither deductive nor logical: like the Actor model, the Prolog-like concurrent systems were based on message passing and consequently were subject to the same indeterminacy. Logical operations and system efficiency Hewitt