Nondeterministic polynomial-time Blum-Shub-Smale Machines over the reals give rise to a discrete complexity class between NP and PSPACE. Several problems, mostly from real algebraic geometry / polynomial systems, have been shown complete (under many-one reduction by polynomial-time Turing machines) for this class. We exhibit a new one based on questions about expressions built from cross products only. MotivationThe Millennium Question "P vs. NP" asks whether polynomial-time algorithms that may guess, and then verify, bits can be turned into deterministic ones. It arose from the Cook-Levin-Theorem asserting Boolean Satisfiability to be complete for NP; which initiated the identification of more and more other natural problems also complete [GaJo79].The Millennium Question is posed [Smal98] also for models able to guess objects more general than bits. More precisely a Blum-Shub-Smale (BSS) machine over a ring R may operate on elements from R within unit time. It induces the nondeterministic polynomial-time complexity class NP R ; for which the following problem FEAS R has been shown complete [BSS89, MAIN THEOREM]: Given a system of multivariate polynomials with 0s and ±1s as coefficients, does it admit a joint root from R ?BSS machines over R coincide with the real-RAM model from Computational Geometry [BKOS97] and underlie algorithms in Semialgebraic Geometry [Gius91, Lece00, BüSc09]. They give rise to a particularly rich structural complexity theory resembling the classical Turing Machine-based one -but often (unavoidably) with surprisingly different proofs [Bürg00,BaMe13]. It is known that NP ⊆ BP(NP 0 R ) ⊆ PSPACE holds [Grig88,Cann88,HRS90,Rene92]. FEAS R and FEAS 0 R are sometimes referred to as existential theory over the reals. However even in this highly important case R = R, and in striking contrast to NP, relatively few other natural problems have yet been identified as complete: * Supported in parts by the Marie Curie International Research Staff Exchange Scheme Fellowship 294962 within the 7th European Community Framework Programme † e.g. as lists of monomials and their coefficients or as algebraic expressions

We contribute performance functions to model the quality of service (QoS) of web services in horizontal composites. Performance functions are thereby used to formalize service level agreements (SLAs) to enable automated verification. Inter-Component rating is a model for gathering additional QoS information from web services that rate the output of their predecessor web services in composites. Furthermore, we contribute two approaches to verify if a web service adheres to its claimed QoS described with a performance function. For this verification, claimed performance is compared with reading points gathered from spot-testing the quality of the web service. The first approach we discuss is function interpolation, the second one applies methods from computational trust. The paper also discusses the advantages and drawbacks of both approaches as well as the influence of measuring errors.