Information Measure as Time Complexity

The concept of Shannon entropy of random variables was generalized to measurable functions in general, and to simple functions with finite values in particular. It is shown that the information measure of a function is related to the time complexity of search problems concerning the function in question. Formally, given a Turing reduction from a search problem y x f  ) ( to another search problem z x g  ) ( , the average amount of information about y x f  ) ( provided by querying the solution(s) of z x g  ) ( is exactly equal to the average mutual information ) ; ( g f  . As a result, the average number of queries is