Languages which 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 graph property is in polynomial time if and only if it is expressible in the language of first order graph theory together with a least fixed point operator. As another example, a group theoretic property is in the logspace hierarchy if and only if it is expressible in the language of first order group theory together with a transitive closure operator. In this paper we also introduce a reduction between problems that is new to complexity theory. First order translations, as the name implies, are fixed first order sentences which translate one kind of structure into another. We present problems which are complete for logspace, nondeterministic logspace, polynomial time, etc., via first order translations.

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