In this paper we deal with the logical description of complexity classes arising in the real number model of computation introduced by Blum, Shub, and Smale [4]. We adapt the approach of descriptive complexity theory for this model developped in [14] and extend it to capture some further complexity classes over the reals by logical means. Among the latter we find NC R , PAR R , EXP R and some others more.

We use sets of assignments, a.k.a. teams, and measures on them to define probabilities of first-order formulas in given data. We then axiomatise first-order properties of such probabilities and prove a completeness theorem for our axiomatisation. We use the Hardy–Weinberg Principle of biology and the Bell’s Inequalities of quantum physics as examples.

We describe a logic which is the same as first-order logic except that it allows control over the information that passes down from formulas to subformulas. For example the logic is adequate to express branching quantifiers. We describe a compositional semantics for this logic; in particular this gives a compositional meaning to formulas of the 'information-friendly' language of Hintikka and Sandu. For first-order formulas the semantics reduces to Tarski's semantics for first-order logic. We prove that two formulas have the same (...) interpretation in all structures if and only if replacing an occurrence of one by an occurrence of the other in a sentence never alters the truth-value of the sentence in any structure. (shrink)

We introduce and study certain classes of optimization problems over the real numbers. The classes are defined by logical means, relying on metafinite model theory for so called R-structures . More precisely, based on a real analogue of Fagin's theorem [12] we deal with two classes MAX-NPR and MIN-NPR of maximization and minimization problems, respectively, and figure out their intrinsic logical structure. It is proven that MAX-NPR decomposes into four natural subclasses, whereas MIN-NPR decomposes into two. This gives a real (...) number analogue of a result by Kolaitis and Thakur [13] in the Turing model. Our proofs mainly use techniques from [17]. Finally, approximation issues are briefly discussed. (shrink)