Exotic quantifiers, complexity classes, and complete problems

We introduce some operators defining new complexity classes from existing ones in the Blum-Shub-Smale theory of computation over the reals. Each one of these operators is defined with the help of a quantifier differing from the usual ones, ∀ and ∃, and yet having a precise geometric meaning. Our agenda in doing so is twofold. On the one hand, we show that a number of problems whose precise complexity was previously unknown are complete in some of the newly defined classes. This substancially expands the catalog of complete problems in the BSS theory over the reals thus adding evidence to its appropriateness as a tool for understanding numeric computations. On the other hand, we show that some of our newly defined quantifiers have a natural meaning in complexity theoretical terms. An additional profit of our development is given by the relationship of the new complexity classes with some complexity classes in the Turing model of computation. This relationship naturally leads to a new notion in complexity over the reals (we call it “gap narrowness”) and to a series of completeness results in the discrete, classical setting