Feasible Functions ∗

It is common to identify the class of computable functions with the class of functions computable by the Turing machine. The identification is done on the ground of the famous Church-Turing thesis. In fact, the thesis justifies only one implication: computable function are Turing computable. The other implication is quite obviously false if computability means practical computability. I am not trying to mock the Turing machine or recursion theory. They brought about a major advance in our understanding of computations. In particular, assuming the Church-Turing thesis, logicians were able to prove that some important decision problems, like Hilbert’s Tenth Problem [M], are not solvable by any algorithm. But the fact is that some “computable in principle” functions are not computable in principle in any practical sense. It is common to identify feasible functions (that is, feasibly computable functions) with those computable in polynomial time. Since a great many useful decision problems are NP [GJ], this identification makes the famous question P=?NP so central in complexity theory. On the first glance, the claim PTime → feasible seems silly. The time complexity of computing a PTime function may have a terrible lower bound, like n. No technological progress will allow us to compute such a function. Fortunately, PTime functions of practical interest tend to have lowpolynomial time complexity. It seems reasonable that, given an n-bit input, we should be able, at least in principle, spend time 7n, n logn or n to work on it. One famous proponent of the thesis feasible ↔ PTime is Steve Cook, of the University of Toronto. He and I debated the issue during the 1991 ∗London Mathematical Society Newsletter, No. 206, June 1993, 6–7.