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 i>k, such that whenever two i>k-tuples satisfy the same properties which are expressible in first order logic with up to i>k variables (FOi>k), then there is an automorphism which maps each of these i>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 i>k (RRMi>k) for every natural number i>k. This leads to the definition of classes of queries denoted as i>Qi>Ci>Qi>k. We believe these classes to be of interest in their own right. For each i>k>0, we define the subclass i>Qi>Ci>Qi>k as the total queries in the class i>Ci>Q of computable queries which preserve realization of properties expressible in FOi>k. The nature of these classes is implicit in the work of S. Abiteboul, M. Vardi and V. Vianu. We prove i>Qi>Ci>Qi>kei>total(RRMi>k) for every i>k>0. We also prove that these classes form a i>strict hierarchy within a strict subclass of i>total(i>Ci>Q). This hierarchy is orthogonal to the usual classification of computable queries in time-space-complexity classes. We prove that the computability power of RRMi>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 i>k-homogeneity, allows RRMi>k machines to achieve i>completeness.

A study of homogeneity in relational databases

In database theory, the term $\\textit{database transformation}$ was used to refer to a unifying treatment for computable queries and updates. Recently, it was shown that non-deterministic database transformations can be captured exactly by a variant of ASMs, the so-called Database Abstract State Machines (DB-ASMs). In this article we present a logic for DB-ASMs, extending the logic of Nanchen and St\\\"ark for ASMs. In particular, we develop a rigorous proof system for the logic for DB-ASMs, which is proven to be sound and complete. The most difficult challenge to be handled by the extension is a proper formalisation capturing non-determinism of database transformations and all its related features such as consistency, update sets or multisets associated with DB-ASM rules. As the database part of a state of database transformations is a finite structure and DB-ASMs are restricted by allowing quantifiers only over the database part of a state, we resolve this problem by taking update sets explicitly into the logic, i.e. by using an additional modal operator $[X]$, where $X$ is interpreted as an update set $\\Delta$ generated by a DB-ASM rule. The DB-ASM logic provides a powerful verification tool to study properties of database transformations.

A complete logic for Database Abstract State Machines1

We introduce a restriction of second order logic, SO$^F$, for finite structures. In this restriction the quantifiers range over relation closed by the equivalence relation $\equiv^{FO}$. In this equivalence relation the equivalence classes are formed by $k$-tuples whose \textit{FO type} is the same, for some integer $k\geq 1$. This logic is a proper extension of SO$^\omega$ logic defined by A. Dawar. In the SO$^F$ existential fragment, $\Sigma^{1,F}_1$, we can express rigidity, which cannot be expressed in SO$^\omega$. We define the complexity class NP$^F$ by using a variation of the relational machine of S. Abiteboul and V. Vianu and we prove that this complexity class is captured by $\Sigma^{1,F}_1$.

A Second-Order Logic in Which Variables Range over Relations with Complete First-Order Types

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 quantiflers (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 C∞ωk and L∞ωk, respectively.

Relational databases and homogeneity in logics with counting

We explore the connection between the concept of relational complexity introduced by S. Abiteboul, M. Vardi and V. Vianu and the restricted second-order logic SO i¾? introduced by A. Dawar. In relational complexity, the basis for measuring the complexity of computing queries with relational machines, is the number of different FO k-types realized by the input database. In SO i¾?, the second-order quantifiers range over relations that are closed under equality of FO k-types of k-tuples. We give a direct proof (in the style of the proof of Fagin's theorem) of the result of A. Dawar on the fact that the existential fragment of SO i¾? captures relationalNP. Then we define formally the concept of relational machine with relational oracleand show the exact correspondence between the prenex fragments of SO i¾? and the levels of the relationalpolynomial-time hierarchy.

The Relational Polynomial-Time Hierarchy and Second-Order Logic

It follows from the famous Fagin's 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.

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

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.

Jose Maria Turull Torres

Papers

A study of homogeneity in relational databases

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

Reflective Relational Machines Working on Homogeneous Databases