Range problem

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In computability theory, a range problem is a weakened form of a search problem. It consists of two functions fl and fu (the lower and upper bounds) and a linear ordering < on the ranges of f1 and f2. A Turing machine solves a range problem if, for any x, the machine eventually halts with an output y such that f1(x) < y < f2(x).

For example, given any function f with range in R and any g : N → R, the strong range problem StrongRangeg(f) is given by lower bound

f(x)\cdot\left(1-\frac{1}{1-g(\vert x\vert)}\right)

and upper bound

f(x)\cdot\left(1-\frac{1}{1+g(\vert x\vert)}\right).

Note that g is passed the length of x, not the value, which need not even be a number.

This article incorporates material from range problem on PlanetMath, which is licensed under the GFDL.