Logic and Complexity

Abstract

The most important classical time and space complexity classes, such as PTIME, NP, or PSPACE, have clean definitions in terms of resource-bounded Turing machines. It is well-known (though still surprising) that most natural decision problems are complete for one of these classes; the consequence is a clear and simple complexity theoretic classification of these problems. However, if more refined complexity issues such as approximability, limited nondeterminism, or parameterizations are taken into account, the landscape of complexity classes becomes much more unwieldy. This means that the natural problems tend to fall into a large number of apparently different classes. Furthermore, these classes usually do not have clean machine characterizations, but can only be identified through their complete problems.