A Post’s Program For Complexity Theory Harry Buhrman ∗ Leen Torenvliet † Abstract In 1944, E. Post proposed a program that would lead to the identification of separate degrees of recursively enumerable sets. Post proposed to identify structural properties that sets of different degrees would not share. Thus proving such a property for sets in one degree would imply that these sets are not in the other. We propose a similar program for the separation of complexity classes and identify three properties that are potential separators: auto-reducibility, robustness, and mitoticity. Some partial results that do separate complexity classes have already been established. Also, answering the question whether complete sets in certain classes do or do not have these properties either way gives an answer to separation problems of central interest. 1 Introduction The major quest for the complexity theory community is finding methods that may separate complexity classes. For the standard sequential hierarchy of complexity classes, P, NP, PSPACE, EXP, NEXP, EXPSPACE,... we know that e.g., P = EXP by the ancient hierarchy theorems. However in the game of separating complexity classes, the hierarchy theorems have a big drawback. Their proofs relativize, as do many other techniques currently known in complexity theory. As there are oracle worlds in which e.g., P = PSPACE and others where P = PSPACE, such proofs can never be used to determine the relations we are currently interested in. Many approaches have been tried in recent and more distant history to prove com- plexity classes different. We name a few. Lower bounds. The most direct approach to proving that complexity classes A and B are not the same is of course to take a problem in one class and prove a lower bound on it’s computational complexity that shows that it is not in the other. However, most complexity classes of interest have rather large flexibility. Most time bounds that define time complexity classes are closed under taking polynomials and hence a lower bound would imply quite a gap between A and B. As an example, better than linear lower bounds have yet to be proven for satisfiability. Diagonalization. Another way of proving that A and B are different is by con- structing a problem in B that is not in A. Therefore we take an enumeration of all * Centrum voor Wiskunde en Informatica; Kruislaan 413; 1098 SJ Amsterdam; buhrman@cwi.nl † University of Amsterdam; ILLC; Plantage Muidergracht 24; 1018 TV Amsterdam; leen@science.uva.nl 1