Turing Oracles as Solutions for Incomputable Problems

A Turing Oracle (hereafter oracle) is a black-box function (i.e., no implementation description is given) which solves an incomputable function and yields its answer in a single step. An oracle machine is a combination of a normal computational system which also has access to an oracle. If the oracle is well-defined in its abilities, it can be used to reason about the process even if the process as a whole is incomputable. An oracle machine, then, is a regular machine (i.e., a normal computable function) which is connected to an oracle (i.e., the function has access to an operation which is incomputable). 

Alan Turing describes the oracle machine as follows  

Let us suppose that we are supplied with some unspecified means of solving number theoretic problems a kind of oracle as it were. We will not go any further into the nature of this oracle than to say that it cannot be a machine. With the help of the oracle we could form a new kind of machine (call them o-machines), having as one of its fundamental processes that of solving a given number theoretic problem. (Turing, 1939, 4) 

Even though the values of functions based on oracle machines cannot be computed (since they are by definition incomputable), it is still possible to reason about which problems are reducible to oracles and which oracles they are reducible to. Posed another way, if a programmer had an oracle for a given problem, what other problems could be solved For instance, there is an incomputable function called Rado s Sigma 

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