Exponential Gaps in our Knowledge

Automata on infinite objects were the key to the solution of several fundamental decision problems in mathematics and logic. Today, automata on infinite objects are used for formal specification and verification of reactive systems. The practical importance of automata in formal methods has motivated a re-examination of the complexity of problems and constructions involving automata. For most problems and constructions, the situation is satisfying, in the sense that even if there is a gap between the upper and the lower bound, it is small. For some fundamental and highly practical cases, however, the gap between the upper and the lower bound is exponential or even larger. The article surveys several such frustrating cases, studies features that they share, and describes recent efforts (with partial success) to close the gaps.