Some Considerations on Universality

Without the investigation of the concept of universality it is quite possible that our modern computers wouldn’t exist. Although they are finite automata, at least theoretically, practically they can be treated as Turing machines processing any special program which can be seen as a special Turing machine. This paper presents some general considerations on that concept, presenting in some more detail universality for general Turing machines, deterministic as well as non-deterministic ones. Universal devices not only have been considered for Turing machines (TM’s), but also for cyclic Post machines (CPM’s), cellular automata (CA’s), random access machines (RAM’s), and others. Very small universal devices mainly have been constructed for DTM’s, CPM’s, and RAM’s. Until recently, the encoding of special deterministic Turing machines (DTM’s) for small universal (deterministic) Turing machines (U(D)TM’s) was achieved via universal tag systems, needing exponential space and time for the simulation. A polynomial amount of space and time was shown in [7, 8]. Universality has mainly been investigated intensively for the class of Turing computability whereas there exist only few publications on universality for restricted systems, having the same restriction. Such a restriction is e.g. space or time complexity, or working conditions. Furthermore, the concept of universality also should be investigated for other underlying structures than free monoids, e.g. commutative monoids or more general structures. Actually the ideas presented here arose from the question if there exists a universal (higher order) Petri nets capable of simulating all special Petri nets but not having power of Turing computability. The concept of universality discussed here is somehow restricted, since there should not just be a black box with input special machine (e.g. a DTM) together with its input, and output of the somehow simulated result, without considering also e.g. space and time amount of the simulation [4, 1]. It should also somehow simulate the (local) behaviour (e.g. in space and time). To clarify the problem universality for Turing machines will be considered in more detail.