On the Computation of Approximations of Database Queries

Reflective Relational Machines were introduced by S. Abiteboul, C. Papadimitriou and V. Vianu in 1994, as variations of Turing machines which are suitable for the computation of queries to relational databases. The machines are equipped with a relational store which can be accessed by means of dynamically built first order logic queries. We initiate a study on approximations of computable queries, defining, for every natural k, the k-approximation of an arbitrary computable query q. This, in turn, motivates us to define a new variation of Reflective Relational Machines by considering two different logics to express dynamic queries: one for queries and a possibly different one for updates to the relational store. We prove several results relating k-approximations of queries with the new machines, and also with classes of queries defined in terms of preservation of equality of FOk theories. Finally, we summarize a few open problems related to our work.

[1]  David Harel,et al.  Computable Queries for Relational Data Bases , 1980, J. Comput. Syst. Sci..

[2]  Serge Abiteboul,et al.  Computing with Infinitary Logic , 1995, Theor. Comput. Sci..

[3]  A. Dawar,et al.  Infinitary Logic and Inductive Definability over Finite Structures , 1995, Inf. Comput..

[4]  Heikki Mannila,et al.  Beyond Independence: Probabilistic Models for Query Approximation on Binary Transaction Data , 2003, IEEE Trans. Knowl. Data Eng..

[5]  Jeffrey D. Ullman,et al.  Principles of Database and Knowledge-Base Systems, Volume II , 1988, Principles of computer science series.

[6]  Y. Gurevich.  On Finite Model Theory , 1990 .

[7]  Yannis E. Ioannidis,et al.  Histogram-Based Approximation of Set-Valued Query-Answers , 1999, VLDB.

[8]  Alfred V. Aho,et al.  Universality of data retrieval languages , 1979, POPL.

[9]  Martin Otto,et al.  Bounded Variable Logics and Counting , 1997 .

[10]  Serge Abiteboul,et al.  Fixpoint logics, relational machines, and computational complexity , 1997, JACM.

[11]  Lauri Hella,et al.  Almost Everywhere Equivalence of Logics in Finite Model Theory , 1996, Bulletin of Symbolic Logic.

[12]  Christos H. Papadimitriou,et al.  Reflective Relational Machines , 1998, Inf. Comput..

[13]  Jeffrey D. Uuman.  Principles of database and knowledge- base systems , 1989 .

[14]  Serge Abiteboul,et al.  The power of reflective relational machines , 1994, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science.

[15]  Neil Immerman,et al.  Descriptive Complexity , 1999, Graduate Texts in Computer Science.

[16]  Jeffrey D. Ullman,et al.  Principles Of Database And Knowledge-Base Systems , 1979 .

[17]  A. Dawar.  Feasible computation through model theory , 1993 .

[18]  José Maria Turull Torres.  Relational Databases and Homogeneity in Logics with Counting , 2002, Acta Cybern..

[19]  Serge Abiteboul,et al.  Foundations of Databases , 1994 .

[20]  Serge Abiteboul,et al.  Generic Computation and its complexity , 1991, STOC '91.

[21]  Martin Otto,et al.  The expressive power of fixed-point logic with counting , 1996, Journal of Symbolic Logic.

[22]  Serge Abiteboul,et al.  Computing with First-Order Logic , 1995, J. Comput. Syst. Sci..

[23]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[24]  José Maria Turull Torres.  A study of homogeneity in relational databases , 2004, Annals of Mathematics and Artificial Intelligence.