Partial Solutions to Incomputable Functions Using Additional Axioms

Function (affectionately known as the busy beaver function). This function says, given n, what is the longest non-inhnite output of any program of size n This is an incomputable function, but it can be shown to be computable given an oracle for 

5 Partial Solutions to Incomputable Functions Using Additional Axioms 

Incomputable functions are unpredictably sensitive to initial conditions. In other words, there is no way to computably predict ahead of time the difference in behavior of the function from the differences in changes to the initial conditions. If this were possible, they would by definition not be incomputable However, partial solutions to these functions can be made by incorporating additional axioms. 

Once an axiom is known, however, then the computation of halters and nonhalters for which sufficient axioms are known becomes an algorithmic problem. Therefore, the discovery of new axioms converts subsets of problems from non-algorithmic 

Using Turing Oracles in Cognitive Models of Problem-Solving 

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