Jose Maria Turull Torres

A study of homogeneity in relational databases

We define four different properties of relational databases which are related tothe notion of homogeneity in classical model theory. The main question for their definition is, for any given database to determine the minimum integer k, such that whenever two k-tuples satisfy the same properties which are expressible in first order logic with up to k variables (FO k ), then there is an automorphism which maps each of these k-tuples onto each other. We study these four properties as a means to increase the computational power of subclasses of the reflective relational machines (RRMs) of bounded variable complexity. These were introduced by S. Abiteboul, C. Papadimitriou and V. Vianu and are known to be incomplete. For this sake we first give a semantic characterization of the subclasses of total RRM with variable complexity k (RRM k ) for every natural number k. This leads to the definition of classes of queries denoted as Q C Q k . We believe these classes to be of interest in their own right. For each k>0, we define the subclass Q C Q k as the total queries in the class C Q of computable queries which preserve realization of properties expressible in FO k . The nature of these classes is implicit in the work of S. Abiteboul, M. Vardi and V. Vianu. We prove Q C Q k =total(RRM k ) for every k>0. We also prove that these classes form a strict hierarchy within a strict subclass of total(C Q). This hierarchy is orthogonal to the usual classification of computable queries in time-space-complexity classes. We prove that the computability power of RRM k machines is much greater when working with classes of databases which are homogeneous, for three of the properties which we define. As to the fourth one, we prove that the computability power of RRM with sublinear variable complexity also increases when working on databases which satisfy that property. The strongest notion, pairwise k-homogeneity, allows RRM k machines to achieve completeness.

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 with the TIME-SPACE hierarchy defined with respect to 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 and where 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 increase of computation power of the introduced machine.

Reflective Relational Machines Working on Homogeneous Databases

We define four different properties of relational databases which are related to the notion of homogeneity in classical Model Theory. The main question for their definition is, for any given database, which is the minimum integer k, such that whenever two k-tuples satisfy the same properties which are expressible in First Order Logic with up to k variables (FOk), then there is an automorphism which maps each of these k-tuples onto each other? We study these four properties as a means to increase the computational power of sub-classes of Reflective Relational Machines (RRM) of bounded variable complexity. For this sake we give first a semantic characterization of the sub-classes of total RRM with variable complexity k, for every natural k, with the classes of queries which we denote as QCQk. We prove that these classes form a strict hierarchy in a strict sub-class of total(CQ). And it follows that it is orthogonal with the usual classification of computable queries in Time and Space complexity classes. We prove that the computability power of RRMk machines is much bigger when working with classes of databases which are homogeneous, for three of the properties which we define. As to the fourth one, we prove that the computability power of RRM with sub-linear variable complexity also increases when working on databases which satisfy that property. The strongest notion, pairwise k-homogeneity, allows RRMk machines to achieve completeness.

Erratum for: A Study of Homogeneity in Relational Databases [Annals of Mathematics and Artificial Intelligence 33(2) (2001) 379---414]

In definition 3.5 in [3], where we define the class QCQ, for every k 1, we need to add a second requirement for a query to be in the class QCQ, namely that the answer to every query f in the class QCQ is always the union of complete FO types. Previously, we considered that property to be implied by our definition, but it turns out that it is not so. There are total computable queries which for some k 1 preserve realization of FO types for k-tuples, but whose evaluation on some databases defines a relation which has not complete FO types for k-tuples. One example of such queries [1] is the query which defines a total order in the class of Odd Multipedes of Gurevich and Shellah [2]. We give below the correct definition, which should replace definition 3.5 in [3].

Semantic Classifications of Queries to Relational Databases

We survey different semantic classifications of queries to relational databases, as well as the different systems of partial isomorphisms developed as algebraic (hence, semantic) characterizations of equality of theories for databases, in FO and in weaker logics. We introduce an abstract notion of similarity for databases, as equality in the theories for a given logic and we define different sub-classes of queries by requiring that the queries in a given sub-class should not distinguish between databases which are similar. Then we use this general strategy to present a new semantic classification, with the class of bounded variable logics with counting (C k ), as the target logics. One important consequence of the definition of the two semantic classifications of queries which we present here is their orthogonality with the TIME-SPACE hierarchy defined in Turing machine complexity, allowing the definition of finer complexity classes by intersecting orthogonal hierarchies.

Expressing properties in second- and third-order logic: hypercube graphs and SATQBF

It follows from the famous Fagins theorem that all problems in NP are expressible in existential second-order logic (ESO), and vice versa. Indeed, there are well-known ESO characterizations of NP-complete problems such as 3-colorability, Hamiltonicity and clique. Furthermore, the ESO sentences that characterize those problems are simple and elegant. However, there are also NP problems that do not seem to possess equally simple and elegant ESO characterizations. In this work, we are mainly interested in this latter class of problems. In particular, we characterize in second-order logic the class of hypercube graphs and the classes SATQBF_k of satisfiable quantified Boolean formulae with k alternations of quantifiers. We also provide detailed descriptions of the strategies followed to obtain the corresponding nontrivial second-order sentences. Finally, we sketch a third-order logic sentence that defines the class SATQBF = \bigcup_{k \geq 1} SATQBF_k. The sub-formulae used in the construction of these complex second- and third-order logic sentences, are good candidates to form part of a library of formulae. Same as libraries of frequently used functions simplify the writing of complex computer programs, a library of formulae could potentially simplify the writing of complex second- and third-order queries, minimizing the probability of error.

Arity and alternation: a proper hierarchy in higher order logics

We study the effect of simultaneously bounding the maximal-arity of the higher-order variables and the alternation of quantifiers in higher-order logics, as to their expressive power on finite structures (or relational databases). Let

Games on Trees and Syntactical Complexity of Formulas

\mathit{AA}^i(r,m)

Redundant Relations in Relational Databases: A Model Theoretic Perspective.

be the class of (i + 1)-th order logic formulae where all quantifiers are grouped together at the beginning of the formulae, forming m alternating blocks of consecutive existential and universal quantifiers, and such that the maximal-arity (a generalization of the concept of arity, not just the maximal of the arities of the quantified variables) of the higher-order variables is bounded by r. Note that, the order of the quantifiers in the prefix may be mixed. We show that, for every i ≥ 1, the resulting

Towards an ASM Thesis for Reflective Sequential Algorithms

\mathit{AA}^i(r,m)