Relational databases and homogeneity in logics with counting

We define a new hierarchy in the class of computable queries to relational databases, in terms of the preservation of equality of theories in fragments of first order logic with bounded number of variables with the addition of counting quantifiers (Ck). We prove that the hierarchy is strict, and it turns out that it is orthogonal to the TIME-SPACE hierarchy defined with respect to the Turing machine complexity. We introduce a model of computation of queries to characterize the different layers of our hierarchy which is based on the reflective relational machine of S. Abiteboul, C. Papadimitriou, and V. Vianu. In our model the databases are represented by their Ck theories. Then we define and study several properties of databases related to homogeneity in Ck getting various results on the change in the computation power of the introduced machine, when working on classes of databases with such properties. We study the relation between our hierarchy and a similar one which we defined in a previous work, in terms of the preservation of equality of theories in fragments of first order logic with bounded number of variables, but without counting quantifiers (FOk). Finally, we give a characterization of the layers of the two hierarchies in terms of the infinitary logics CK∞ω and LK∞ω respectively

Capturing relational NEXPTIME with a fragment of existential third order logic

We prove that the existential fragment Σ12,ω of the third order logic TOω captures the relational complexity class non deterministic exponential time. As a Corollary we have that relational machines that work in NEXPTIMEr can simulate third order relational machines that work in NEXPTIME3,r.Facultad de Informátic

Capturing relational NEXPTIME with a fragment of existential third order logic

We prove that the existential fragment Σ12,ω of the third order logic TOω captures the relational complexity class non deterministic exponential time. As a Corollary we have that relational machines that work in NEXPTIMEr can simulate third order relational machines that work in NEXPTIME3,r.Facultad de Informátic

Capturing relational NEXPTIME with a fragment of existential third order logic

We prove that the existential fragment Σ²1,ω of the third or- der logic TOω captures the relational complexity class non deterministic exponential time. As a Corollary we have that relational machines can simulate third order relational machines.XII Workshop Bases de Datos y Minería de Datos (WBDDM)Red de Universidades con Carreras en Informática (RedUNCI

Capturing relational NEXPTIME with a fragment of existential third order logic

We prove that the existential fragment Σ²1,ω of the third or- der logic TOω captures the relational complexity class non deterministic exponential time. As a Corollary we have that relational machines can simulate third order relational machines.XII Workshop Bases de Datos y Minería de Datos (WBDDM)Red de Universidades con Carreras en Informática (RedUNCI

Relative Entailment Among Probabilistic Implications

We study a natural variant of the implicational fragment of propositional logic. Its formulas are pairs of conjunctions of positive literals, related together by an implicational-like connective; the semantics of this sort of implication is defined in terms of a threshold on a conditional probability of the consequent, given the antecedent: we are dealing with what the data analysis community calls confidence of partial implications or association rules. Existing studies of redundancy among these partial implications have characterized so far only entailment from one premise and entailment from two premises, both in the stand-alone case and in the case of presence of additional classical implications (this is what we call "relative entailment"). By exploiting a previously noted alternative view of the entailment in terms of linear programming duality, we characterize exactly the cases of entailment from arbitrary numbers of premises, again both in the stand-alone case and in the case of presence of additional classical implications. As a result, we obtain decision algorithms of better complexity; additionally, for each potential case of entailment, we identify a critical confidence threshold and show that it is, actually, intrinsic to each set of premises and antecedent of the conclusion

The Existential Fragment of Third Order Logic and Third Order Relational Machines

We introduce a new sub logic of third order logic (TO), the logic TOϖ, as a semantic restriction of TO. We focus on the existential fragment of TOϖ which we denote Σ²ϖ , and we study its relational complexity by introducing a variation of the non deterministic relational machine, which we denote 3-NRM, where we allow third order relations in the relational store of the machine.We then prove that Σ²ϖ characterizes exactly NEXPTIME3,rV Workshop Aspectos teóricos de las Ciencias de la ComputaciónRed de Universidades con Carreras de Informática (RedUNCI

Characterizing downwards closed, strongly first order, relativizable dependencies

In Team Semantics, a dependency notion is strongly first order if every sentence of the logic obtained by adding the corresponding atoms to First Order Logic is equivalent to some first order sentence. In this work it is shown that all nontrivial dependency atoms that are strongly first order, downwards closed, and relativizable (in the sense that the relativizations of the corresponding atoms with respect to some unary predicate are expressible in terms of them) are definable in terms of constancy atoms. Additionally, it is shown that any strongly first order dependency is safe for any family of downwards closed dependencies, in the sense that every sentence of the logic obtained by adding to First Order Logic both the strongly first order dependency and the downwards closed dependencies is equivalent to some sentence of the logic obtained by adding only the downwards closed dependencies

Semi-automatic support for evolving functional dependencies

During the life of a database, systematic and frequent violations of a given constraint may suggest that the represented reality is changing and thus the constraint should evolve with it. In this paper we propose a method and a tool to (i) find the functional dependencies that are violated by the current data, and (ii) support their evolution when it is necessary to update them. The method relies on the use of confidence, as a measure that is associated with each dependency and allows us to understand \u201dhow far\u201d the dependency is from correctly describing the current data; and of goodness, as a measure of balance between the data satisfying the antecedent of the dependency and those satisfying its consequent. Our method compares favorably with literature that approaches the same problem in a different way, and performs effectively and efficiently as shown by our tests on both real and synthetic databases