Does anyone know of a set of problems that vary uniformly and span one of the interesting hierarchies of complexity and computability? By interesting, I mean, for example, the Polynomial Hierarchy, the Arithmetic Hierarchy, or the Analytic Hierarchy. Or maybe (N)P, (N)EXP, 2(N)EXP, $\ldots$More concretely: You can give a uniform set of problems that characterize the Arithmetic Hierarchy: $0, 0', \overline{0'}, 0'', \overline{0''},\ldots$. But these aren't always the most useful for reducing to actual problems. On the other hand, the book by Harel, Kozen and Tiuryn has a set of varying tiling problems that are NP, $\Pi^0_1$, $\Sigma^0_2$ and $\Sigma^1_1$ complete. The problems are useful for showing reductions, but it isn't entirely clear if they generalize uniformly to cover the other levels of the hierarchies they sit in.Does anyone know of such a set of concrete, uniform problems that span a hierarchy?EDIT: Just for clarification, I know that the 3 hierarchies I give above all have standard definitions in terms of alternating quantifier strength. That's not what I'm looking for. I'm looking for something different, like a game on a graph or a puzzle played with tilings.
Uniform hierarchy of problems that span complexity and computational hierarchies
cc.complexity theory;computability