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 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

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

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

L-rigid Databases and the Expressibility of Incomplete Relational Query Languages

The class of Computable Queries (CQ) was de ned by Chandra and Harel in 1980, as functions on structures rather than functions on numbers (as recursive functions). With this formulation of the notion of a query to a relational database (db), the field of Finite Model Theory became a suitable theoretical framework for relational databases. In this framework the notion of local expressibility of a given logic is the class of queries which can be expressed in that logic. On the other hand, in 1991, Abiteboul and Vianu de fined the Generic Machine, denoted as GMloose, which they proved to be strictly included in CQ. They also proved that this class of machines behaves as complete w.r.t. the whole class CQ when working on classes of ordered db. If we consider the structures as db instances, and if we are using a GMloose machine, it means that we will not be able to compute queries such as \give me the names of the salesmen who sell to an even number of clients unless our db is ordered. In the present Thesis, we study some properties of relational db which can increase the expressive power of relational languages or formalisms which are incomplete in the general case, when working on classes of db which satisfy that properties.Sociedad Argentina de Informática e Investigación Operativ

Preservación de consultas expresables en fragmentos existenciales en bases de datos relacionales

En el contexto de bases de datos, el concepto de preservacióon de teorí as en una cierta l ógica y entre dos estructuras relacionales A y B dadas, signi fica que el conjunto de consultas Booleanas que son expresables en dicha l ógica y que son verdaderas en la base de datos A tambi én son verdaderas en la base de datos B. As , por ejemplo ante el agregado de nuevos elementos a una base de datos es posible determinar si el conjunto de consultas Booleanas expresables en FOk(9) (y en la extensi ón in nitaria Lk 1;!(9)) se preserva. Existen juegos de fichas que caracterizan la preservacióon de teorí as para estas dos l ógicas, y es sabido que dicha preservaci ón puede determinarse en tiempo polinomial. En el presente trabajo mostramos una caracterización alternativa de la preservación de teorí as en FOk(9) (y en la extensi ón infi nitaria Lk 1;!(9)) mediante la realización de tipos de k-tuplas. Luego presentamos un algoritmo polinomial que permite determinar dicha preservación y en el que utilizamos nuestra ara teorización.Eje: I - Workshop de Ingeniería de Software y Base de DatosRed de Universidades con Carreras en Informática (RedUNCI

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

Expressibility of the logic SOF on classes of structures of bounded FO types

We introduce a new property for classes of structures (or relational database instances), that we call bounded FO types. Then we prove that on such classes the expressive power of SOF collapses to rst order logic FO. As a consequence of this we prove that SOF is strictly included in SO.Eje: Workshop Teóricos de ciencia de la computación (WATCC)Red de Universidades con Carreras en Informática (RedUNCI