On the Hardness of Computable Sets

What makes a problem hard to compute, and how does one categorize problems that accurately captures this notion of hardness? Computer science has traditionally approached this issue with two classification schemas: resource-bounded complexity classes (deterministic or nondeterministic time and space bound computations) and bounded reducibilities (≤m and ≤ T reductions). We show that neither of these approaches accurately capture the phenomena of being “hard to compute” The development of complexity classes has blurred the concept of simplicity and complexity. That a function (or the sets characterized by them) cannot be computed in time t does not imply it always requires more than time t, but only that it sometimes does. So nonmembership in a complexity class only implies infinitely often (i.o.) complexity. From this view, classification of problems by complexity classes is a classification of problems that are almost-everywhere easy. Here we study problems that require more than time t for all but a finite number of inputs, that is, problems that are almost-everywhere (a.e.) complex. That we can show separation of complexity classes by a.e. complex sets underscores the fact that simpilicity and complexity are not complementary notions. The common existence of a e. complex sets motivates us to re-examine our notions of what we mean when we say a problem is “hard,” or more precisely, what we are trying to say, when we say that a problem is C-hard (or C-complete) for some complexity class C. We present results that prove the existence of sets (problems) that in a very mathematically meanigful way should be considered hard for a given complexity class, but this hardness is not captured by our existing mechanisms for classification of a problems complexity. We show that polynomial bounded reducibilities do not accurately capture the notion of “hard to compute.” Specifically, we construct sets that hard to compute in the truest sense (a.e. complex) that are incomparable (under polynomial bounded reducibilities), but still lie within the same complexity class. These results bring in question thr commonly held interpretation that bounded reducibilities capture the notion of sets “being within a polynomial of the complexity” of each other.