The Weakness of Self-Complementation

Model checking is a method for the verification of systems with respect to their specifications. Symbolic model-checking, which enables the verification of large systems, proceeds by evaluating fixed-point expressions over the system’s set of states. Such evaluation is particularly simple and efficient when the expressions do not contain alternation between least and greatest fixed-point operators; namely, when they belong to the alternation-free 5 -calculus (AFMC). Not all specifications, however, can be translated to AFMC, which is exactly as expressive as weak monadic second-order logic (WS2S). Rabin showed that a set 6 of trees can be expressed in WS2S if and only if both 6 and its complement can be recognized by nondeterministic Büchi tree automata. For the “only if” direction, Rabin constructed, given two nondeterministic Büchi tree automata