Computation Beyond the Turing Limit

Even Alan Turing himself thought about a machine that could break the barrier that had been set by the Church-Turing Thesis resting upon his concept of an universal Turing machine. His Omachine enhances a conventional Turing machine by adding an oracle in a black box. In addition to the normal operations of a Turing machine, this machine can ask the oracle questions (depending on the current state and letter under the head) and the oracle answers after one time step either 0 or 1. The Turing machine can use this advice for its computation. Turing showed that the O-machine is a super-Turing machine, i.e., it can compute functions that are uncomputable on a conventional Turing machine. However, he did not elaborate on the realization of such a black box. Hence, he did not disprove the Church-Turing Thesis because the proof that an O-machine is realizable is missing. [CP00] A Turing machine with advice is similar to Turing’s O-machine. The input of such a Turing machine is supplemented with an advice sequence wn, which depends only on the length of the input n. The Turing machine can use this advice for its computation, like the O-machine used the answers of the oracle. This machine is more powerful than a conventional Turing machine. For example, the unary halting problem can be solved using an advice sequence whose lengths is polynomial in n. The solution is trivial as there is only one input of length n so that the advice wn can consist of one bit to decide if the program run on this input halts or not. If we loose the constraint that the advice has a polynomial length, we can compute all functions f : {0, 1}∗ → {0, 1} because an advice sequence of length 2 can encode the results for all possible inputs of length n. Of course, again, the problem is to realize such a machine as it is quite “difficult” to give the correct advice. A common approach to break the Turing barrier bases on dealing with real numbers. Such (not yet realized) machines are called analog computers or, in order to avoid confusion with already existing analog machines (such as differential analyzers), real computers. While a conventional computer deals only with discrete numbers (e.g., the natural numbers), a real computer can perform operations on real numbers in a single step. Obviously, this is an enhancement of computability as the set of real numbers is uncountable (in contrast to the set of natural numbers). In this context, one concrete possibility of breaking the Turing barrier is the measurement of a real-valued physical quantity up to an arbitrary precision [Cop00]. For instance, let us assume there is a physical quantity whose value corresponds to the halting probability Ω (for a given Turing machine and a given encoding scheme). In order to decide if a given program of size n halts, we just have to measure this quantity up to a precision of n bits, as the first n bits of Ω are sufficient to make this decision. [Cha90] Another example for a computation with real numbers is the classical analog recurrent neuronal network (ARNN). However, the main focus in Siegelmann’s paper is the model of analog shift maps.