The Ehrenfeucht-Fraïssé Method and the Planted Clique Conjecture

The Ehrenfeucht-Fraı̈ssé method for first-order logic and further logics relevant in descriptive complexity has been quite successful. However, for key problems such as P 6= NP or NP 6= co-NP no progress has been achieved using it. We show that for these problems we can not get the board for the corresponding Ehrenfeucht-Fraı̈ssé game in polynomial output time, even if we allow probabilistic methods to obtain the board. In order to get this result in the probabilistic case, we need an additional hypothesis, namely that there is an algorithm, the verifier, verifying in a reasonable time that the two structures of the board satisfy the same properties expressible in a suitable fragment of the logic. The (non)existence of such a verifier is related to a logic version of the planted clique conjecture.