Algorithm for inverting a bijective function. Does there exist a generalized algorithm for finding the inverse function of an arbitrary bijective function?


In order for this algorithm to be
useful, it must eventually halt once
the correct answer is found.
Beyond the requirement that it must find the solution eventually, there is no time constraints on the time it takes to find or run the inverse function (with that in mind, something better than bruteforce guess-and-check, would be more interesting). 


For example, if such a generalized algorithm existed it could solve for a decompression algorithm for a lossless compression algorithm.

EDIT:

I really liked Evgenij Thorstensen assumptions as they summed up my question fairly well.

Assumptions


Computable bijective functions over a fixed alphabet (say {0, 1})
Represented by a deterministic Turing machine (DTM) that computes it
The proposed algorithm would be able to solve for a DTM that would invert the original bijective function output.


Another shot at explaining it:

Given: Bijective function F that maps X from domain A onto Y from domain B.

Proposed algorithm should be able to solve for a Bijective function G that maps Y from domain B onto X from domain A, such that G(F(X))=X and F * G = I where I is the identity function.
There have been papers on automatically converting algorithms for bijective functions into algorithms for the inverse function; my own first conference paper was one such. But the class of algorithms that can be inverted in this way is severely limited, as the other answers already suggest.