Mathematically speaking, inferring the “formal language” which can describe a set of expressions. In the slightly looser sense used by linguists studying natural human language, discovering the syntactic rules of a given language, which is kinda the same thing but with every term sloppier, and the subject matter itself messier.
This is already a crazy complex area, and being naturally perverse, I am interested in an especially esoteric corner of it, to whit, grammars of things that aren't speech; inferring design grammars, say, could allow you to produce more things off the same “basic plan” from some examples of the thing; look at enough trees and you know how to build the rest of the forest, that kind of thing. I'm especially interested in things expressed not as a sequence of symbols from a finite alphabet  i.e. not over the free monoid, or over the free monoid but the symbol expression is hidden. This is a boutique interest. I also care about probabilistic ones, i.e. assigning measures to these things. That part is not at all boutique.
Normally design grammars deal with simple languages, such as, say “regular” languages. I'm interested in things a rung or two up the Chomsky hierarchy  Contextfree grammars, maybe even contextsensitive ones.
See also design grammars, iterated function systems and my research proposal in this area.
Things to read

Peter Norvig on Chomsky and statistical versus explanatory models of natural language syntax. Full of sick burns.
In January of 2011, television personality Bill O'Reilly weighed in on more than one culture war with his statement “tide goes in, tide goes out. Never a miscommunication. You can't explain that,” which he proposed as an argument for the existence of God. […] O'Reilly realizes that it doesn't matter what his detractors think of his astronomical ignorance, because his supporters think he has gotten exactly to the key issue: why? He doesn't care how the tides work, tell him why they work. Why is the moon at the right distance to provide a gentle tide, and exert a stabilizing effect on earth's axis of rotation, thus protecting life here? Why does gravity work the way it does? Why does anything at all exist rather than not exist? O'Reilly is correct that these questions can only be addressed by mythmaking, religion or philosophy, not by science.
Chomsky has a philosophy based on the idea that we should focus on the deep whys and that mere explanations of reality don't matter. In this, Chomsky is in complete agreement with O'Reilly. (I recognize that the previous sentence would have an extremely low probability in a probabilistic model trained on a newspaper or TV corpus.)

Cosma Shalizi's inevitable mention in 3, 2, 1… go
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