Descriptive and Computational Complexity

Computational complexity began with the natural physical notions of time and space Given a property S an important issue is the computational complexity of checking whether or not an input satis es S For a long time the notion of complexity referred to the time or space used in the compu tation A mathematician might ask What is the complexity of expressing the property S It should not be surprising that these two questions that of checking and that of expressing are related However it is startling how closely tied they are when the second question refers to expressing the prop erty in rst order logic Many complexity classes originally de ned in terms of time or space resources have precise de nitions as classes in rst order logic In Fagin gave a characterization of nondeterministic polynomial time NP as the set of properties expressible in second order existential logic We will begin with this result and then survey some more recent work relating rst order expressibility to computational complexity Some of the results arising from this approach include characterizing polynomial time P as the set of properties expressible in rst order logic plus a least xed point operator to be de ned later and showing that the set of rst order inductive de nitions for nite structures is closed under complementation