Languages that Capture Complexity Classes

We present in this paper a series of languages adequate for expressing exactly those properties checkable in a series of computational complexity classes For example we show that a property of graphs respectively groups binary strings etc is in polynomial time if and only if it is expressible in the rst order language of graphs respectively groups binary strings etc together with a least xed point operator As another example a property is in logspace if and only if it is expressible in rst order logic together with a deterministic transitive closure operator The roots of our approach to complexity theory go back to when Fagin showed that the NP properties are exactly those expressible in second order existential sentences It follows that second order logic expresses exactly those properties which are in the polynomial time hierarchy We show that adding suitable transitive closure operators to second order logic results in languages capturing polynomial space and exponential time respectively The existence of such natural languages for each important complexity class sheds a new light on complexity theory These languages rea rm the impor tance of the complexity classes as much more than machine dependent issues Furthermore a whole new approach is suggested Upper bounds algorithms can be produced by expressing the property of interest in one of our languages Lower bounds may be demonstrated by showing that such expression is impos sible For example from the above we know that P NP if and only if every second order property is already expressible using rst order logic plus least xed point Similarly nondeterministic logspace is di erent from P just if there is some sentence using the xed point operator which cannot be expressed with a single application of transitive closure