TITLE: Searching the symmetric group
QUESTION [8 upvotes]: You want to design a set of yes/no questions for quickly searching the symmetric group. The questions have to be of the form "Does your permutation move $a_1$ to $b_1$ or $a_2$ to $b_2$ or ... or $a_k$ to $b_k$?" Given a random permutation, you will ask all of your questions about that permutation, and your goal is to know what the permutation is once you know the answers to your questions.
In particular, note that you aren't allowed to have later questions depend on the answers to earlier questions. You always ask the same questions of whichever random permutation you get.
Here's a slightly different way to phrase the type of question you're allowed to ask. If we represent the elements of the symmetric group by permutation matrices, then the questions you can ask involve picking a (0,1)-matrix and asking if the random permutation matrix has any 1 where your chosen matrix has a 1 (i.e., if I apply the componentwise AND function to the random matrix and your chosen matrix, do I get 0 or 1?)
Two questions:

How many such questions are needed in order to search $S_n$? How much bigger than $log_2 n!$ is this?
Are there "good" strategies for designing sets of questions that can be used for any n? Can we achieve the minimum number of questions for each n with a single strategy?

REPLY [6 votes]: It certainly isn't difficult to obtain an upper bound of $n\lceil \log_2 n \rceil = O(\log_2 n!)$.  All you need to do is to separately learn each bit of each value of the permutation.  This looks like it is only roughly optimal, but actually it is already $(1+o(1))(\log_2 n!)$.  (Thanks to t3suji for noticing this.)  However, the $o(1)$ term only decays logarithmically in this algorithm, so you could ask whether you could make it $(1+O(n^{-\alpha}))(\log_2 n!)$.
I agree with Michael that it seems similar to the asymptotic question of sorting with comparisons.  However, in the usual version of that problem, you are allowed to move elements based on an incomplete list of comparisons, which then amounts to adaptive comparisons.  You could call this new question "non-adaptive Mastermind for permutations with matrix guesses".  Michael also suggested the restricted problem of permutation guesses, which seems like a more difficult problem to me.