TITLE: What is the relationship between "translation" and time complexity?
QUESTION [8 upvotes]: Consider the problem of deciding a language $L$; for concreteness, say that this is the graph isomorphism problem.  That is, $L$ consists of pairs of graphs $(G, H)$ such that $G\simeq H$.  Now the time complexity of deciding this problem as stated depends on how the graphs are encoded.  For example, if one were to have a "canonical" encoding of graphs (such that encoding strings are in bijective correspondence with isomorphism classes of graphs) the problem would be $O(n)$, as we could decide whether $G\simeq H$ simply by comparing the string representing $G$ to the string representing $H$.
On the other hand, if we represent a graph via its adjacency matrix, the best known algorithm (according to Wikipedia) gives only a subfactorial bound.  Now consider the time complexity of converting from one language to another.  If we let $T_1, T_2$ be the time complexity of deciding languages $L_1$ and $L_2$ respectively, and $T_{ij}$ be the time it takes a Turing machine to take a string $S$ and output another string $S'$ which is in language $j$ if and only if $S$ is in language $i$.  We have
$$T_1\leq T_2+T_{12}$$
$$T_2\leq T_1+ T_{21}$$
as given a string that we want to test for its belonging to $L_i$, we may run it through the translation $L_i \to L_j$ and then decide language $j$.  Indeed, this is a special case of a trivial "triangle inequality" for translation; the time it takes to translate from $L_1$ to $L_2$ plus the time it takes to translate from $L_2$ to $L_3$ is greater than or equal to the time it takes to translate from $L_1$ to $L_3$.   (I say it is a special case because a decision problem is the same as converting a language $L$ to the language $\{ 1 \}$.)
What I want to know is:

Can  we better quantify the relationship between the time complexity of a decision problem and the nature of the encoding?

So that this question is not prohibitively vague, let us say that I am looking for (1) related references, and (2) a measure of the complexity of an encoding which more tightly relates to time complexity of the "underlying" decision problem.

Added (7/19/2010):  The answers below, particularly Ryan Williams' excellent survey of the dependence of the time complexity of various problems on their encoding, get at the motivation to my question but not at my question itself.  In particular, it's clear that every problem may be re-encoded to allow (say) $O(\log n)$ time complexity, by padding.  My question is whether there's a reasonable way to measure this dependence.
For example, say the decision problem for $L_1$ is reducible to the decision problem for $L_2$, and vice versa, so that $L_1$ and $L_2$ in some sense represent the same problem.  Is there a way to formalize this last statement (about "representing the same problem")?  I am imagining, for example, a measure $C_i$ of the complexity of a language so that if $T_i$ is the time complexity of the language, and $L_1$ and $L_2$ are, say, easily reducible to one another, then $T_1/C_1\sim T_2/C_2$.  (Of course $C_i=T_i$ works, but ideally $C_i$ would be somehow a property of the language, rather than the decision problem.)  This is unfortunately becoming quite speculative, so again, related references would be a great answer.

REPLY [3 votes]: The general abstract setting for the issue driving your
question is the notion of reduction of equivalence
relations. The idea of this is that one equivalence
relation $E$ reduces to another $F$ with respect to some
complexity concept if there is a function $f$ in this class
such that

$x\, E\, y$ if and only if $f(x)\, F\, f(y)$

You can imagine that $E$ is the equivalence relation
arising from one way of representing mathematical objects
(graphs, algebraic structures, whatever) and $F$ is the
relation corresponding to an alternative method. The
reduction is saying that equivalence with respect to the
$E$ way of representing the objects is no more difficult
than equivalence with respect to the $F$ way of
representing them. 
I claim that understanding this reducibility relation amounts to
understanding exactly what your question is aimed at, the
question of how one manner of representing the same objects
can be simpler than another. More generally, this reducibility relation provides a very precise way to understand what it means to say that one classification problem is strictly harder than another, even when the objects in the two cases seem totally unrelated at first.
In the case you seem most interested, you could regard $E$
and $F$ as NP equivalence relations and insist that $f$ is polynomial time
computable. This is a case that has been recently
investigated by Sy Friedman, and this MO
question
arose out of a talk he gave on this topic here in New York,
and discusses as motivation some of the relevant general theory.
This appears to be a completely new research area, ripe for
progress. I would encourage anyone to enter into it.
Much of that theory is inspired by the enormous successes
of the much more developed instance of this concept,
occurring when $E$ and $F$ are Borel relations on the reals
and $f$ is a Borel function. This case is the
emerging-but-possibly-now-mature field of Borel equivalence
relation theory (see Greg Hjorth's survey
article
and Simon Thomas'
notes).
The theory of Borel equivalence relation theory has to deal
explicitly with the Borel analogues of the precise issues
you mention in your question, and has made huge
illuminating progress in understanding the structure of
Borel equivalence relations under Borel reducibility. I
mention some of the basic results in this MO
answer.
In general, for each notion of complexity, the goal is to
study the whole hierarchy of equivalence relations, to
discover its features and general structural results. The
Borel case is quite well developed by now, exhibiting many
fascinating features, but the NP case is much less well
developed. I do know personally, however, that other
researchers are working on several other natural contexts
of this idea.