Abstract

There are properties of finite structures that are expressible with the use of Hilbert's $\epsilon$-operator in a manner that does not depend on the actual interpretation for $\epsilon$-terms, but not expressible in plain first-order. This observation strengthens a corresponding result of Gurevich, concerning the invariant use of an auxiliary ordering in first-order logic over finite structures. The present result also implies that certain non-deterministic choice constructs, which have been considered in database theory, properly enhance the expressive power of first-order logic even as far as deterministic queries are concerned, thereby answering a question raised by Abiteboul and Vianu.