TITLE: Structure theorems for Turing-decidable languages?
QUESTION [16 upvotes]: Languages decidable by weak models of computation often have certain necessary characteristics, e.g. the pumping lemma for regular languages or the pumping lemma for context-free languages.  Such characteristics are useful for determining when a language cannot be decided by that weak model of computation.

Do Turing-decidable languages have any such necessary characteristics?  If not, why not?

Such a characteristic should avoid mention of Turing machines.
Motivation:  The only way I know to show that a language is not Turing-decidable is diagonalization, or reducing to a language where diagonalization has already been applied.  It would be nice to have another method.
I suspect that there aren't any nice characteristics of this type known, or they'd probably be very well-known; but an argument for why they can't exist could be illuminating.
I'd also be interested in the same question with the phrase "Turing-decidable" replaced with any complexity class, e.g. NP.  This question seems more tractable (as there are fewer languages under examination).
Edit:  Given the comments, I'll be a bit more precise about what I mean by "characteristics."  I would be thrilled with either of the following two possibilities:

1) A property that allowed one to show a language was not in the relevant class (Turing-decidable, NP, whatever) without using diagonalization.

Or

2)  A property that can be stated in a language too weak to define Turing machines.

REPLY [5 votes]: It should be noted that the idea of finding a "property" that can be used to show non-membership in a certain class of languages was a driving force in computability theory in the 1950's or so.  Specifically, Emil Post posed the question: are there computably enumerable (aka "c.e.", also known as Turing-recognizable (but not necessarily decidable)) sets that are undecidable and strictly Turing-below the Halting problem (note that any c.e. set is Turing-below the halting problem).  That is, we want a c.e. set A such that $\emptyset <_T A <_T H$, where $H$ is the halting problem.  This became known as Post's Problem.  Further, he proposed Post's Program, which was to find a property of c.e. sets that implied the required conditions, and then to find an A with this property.  Examples of such candidate properties are being "simple" (its a technical term).  
Post's problem was solved in the positive by Friedberg and Muchnik, but by using a clever sort of delayed diagonalization, a sort of "injury argument".  However, this did not show that Post's program could be solved in the positive, but indeed Harrington and Soare showed in 1991 that there is such a property satisfying Post's conditions (and a corresponding set A satisfying this property).
See the wikipedia article: http://en.wikipedia.org/wiki/Turing_degree#Post.27s_problem as well as the Harrington-Soare article explaining some of the background (its open-access): http://www.ncbi.nlm.nih.gov/pmc/articles/PMC52904/