Some fragments of second-order logic over the reals for which satisfiability and equivalence are (un)decidable

We consider the Σ1 0-fragment of second-order logic over the vocabulary h+, ×, 0, 1, <, S1, ..., Ski, interpreted over the reals, where the predicate symbols Si are interpreted as semi-algebraic sets. We show that, in this context, satisfiability of formulas is decidable for the first-order ∃ ∗ - quantifier fragment and undecidable for the ∃ ∗∀- and ∀ ∗ -fragments. We also show that for these three fragments the same (un)decidability results hold for containment and equivalence of formulas.Fil: Grimson, Rafael. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Kuijpers, Bart. Hasselt University; Bélgic

Software Engineering and Complexity in Effective Algebraic Geometry

We introduce the notion of a robust parameterized arithmetic circuit for the evaluation of algebraic families of multivariate polynomials. Based on this notion, we present a computation model, adapted to Scientific Computing, which captures all known branching parsimonious symbolic algorithms in effective Algebraic Geometry. We justify this model by arguments from Software Engineering. Finally we exhibit a class of simple elimination problems of effective Algebraic Geometry which require exponential time to be solved by branching parsimonious algorithms of our computation model.Comment: 70 pages. arXiv admin note: substantial text overlap with arXiv:1201.434

Degrees of Monotonicity of Spatial Transformations

. We consider spatial databases that can be defined in terms of polynomial inequalities, and we are interested in monotonic transformations of spatial databases. We investigate a hierarchy of monotonicity classes of spatial transformations that is determined by the number of degrees of freedom of the transformations. The result of a monotonic transformation with k degrees of freedom on a spatial database is completely determined by its result on subsets of cardinality at most k of the spatial database. The result of a transformation in the largest class of the hierarchy on a spatial database is determined by its result on arbitrary large subsets of the database. The latter is the class of all the monotonic spatial transformations. We give a sound and complete language for the monotonic spatial transformations that can be expressed in the relational calculus augmented with polynomial inequalities and that belong to a class with a finite number of degrees of freedom. In particular, we s..