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

Constraint Databases and Geographic Information Systems

Constraint databases and geographic information systems share many applications. However, constraint databases can go beyond geographic information systems in efficient spatial and spatiotemporal data handling methods and in advanced applications. This survey mainly describes ways that constraint databases go beyond geographic information systems. However, the survey points out that in some areas constraint databases can learn also from geographic information systems

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Monadic Decomposability of Regular Relations

Monadic decomposibility - the ability to determine whether a formula in a given logical theory can be decomposed into a boolean combination of monadic formulas - is a powerful tool for devising a decision procedure for a given logical theory. In this paper, we revisit a classical decision problem in automata theory: given a regular (a.k.a. synchronized rational) relation, determine whether it is recognizable, i.e., it has a monadic decomposition (that is, a representation as a boolean combination of cartesian products of regular languages). Regular relations are expressive formalisms which, using an appropriate string encoding, can capture relations definable in Presburger Arithmetic. In fact, their expressive power coincide with relations definable in a universal automatic structure; equivalently, those definable by finite set interpretations in WS1S (Weak Second Order Theory of One Successor). Determining whether a regular relation admits a recognizable relation was known to be decidable (and in exponential time for binary relations), but its precise complexity still hitherto remains open. Our main contribution is to fully settle the complexity of this decision problem by developing new techniques employing infinite Ramsey theory. The complexity for DFA (resp. NFA) representations of regular relations is shown to be NLOGSPACE-complete (resp. PSPACE-complete)