Bounded Fixpoints for Complex Objects

We investigate a query language for complex-object databases, which is designed to (1) express only tractable queries, and (2) be as expressive over flat relations as first order logic with fixpoints. The language is obtained by extending the nested relational algebra NRA, of [9], with a “bounded fixpoint” operator. Extending the flat case [15, 21] , all PTime computable queries over ordered databases are expressible in this language. The main result consists in proving that this language is a conservative extension of the first order logic with fixpoints, or of the while-queries (depending on the interpretation of the bounded fixpoint: inflationary or partial). The proof technique uses indexes, to encode complex objects into flat relations, and is strong enough to allow for the encoding of NRA with unbounded fixpoints into flat relations. We also define a complex object logic based language with fixpoints and prove that its range restricted fragment is equivalent to NRA with bounded fixpoints.

[1]  Peter Buneman,et al.  Structural Recursion as a Query Language , 1992, DBPL.

[2]  Dirk Van Gucht,et al.  Possibilities and limitations of using flat operators in nested algebra expressions , 1988, PODS '88.

[3]  Limsoon Wong,et al.  Normal forms and conservative properties for query languages over collection types , 1993, PODS.

[4]  Saharon Shelah,et al.  Fixed-point extensions of first-order logic , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[5]  Victor Vianu,et al.  Tractable query languages for complex object databases , 1991, PODS '91.

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

[7]  Jianwen Su,et al.  On the Expressive Power of Database Queries with Intermediate Types , 1991, J. Comput. Syst. Sci..

[8]  David Harel,et al.  Structure and complexity of relational queries , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[9]  Serge Abiteboul,et al.  Expressive Power of Query Languages , 1992, Theoretical Studies in Computer Science.

[10]  Patrick C. Fischer,et al.  Nested Relational Structures , 1986, Adv. Comput. Res..

[11]  Jianwen Su,et al.  Untyped sets, invention, and computable queries , 1989, PODS '89.

[12]  Serge Abiteboul,et al.  An Extensible Rule-Based Language with Complex Objects and data-Functions , 1989, DBPL.

[13]  Dirk Van Gucht,et al.  Converting nested algebra expressions into flat algebra expressions , 1992, TODS.

[14]  Limsoon Wong,et al.  Naturally Embedded Query Languages , 1992, ICDT.

[15]  Serge Abiteboul,et al.  Object identity as a query language primitive , 1989, SIGMOD '89.

[16]  Patrick C. Fischer,et al.  Multilevel Nested Relational Structures , 1988, J. Comput. Syst. Sci..

[17]  Catriel Beeri,et al.  On the power of languages for manipulation of complex objects , 1987, VLDB 1987.

[18]  Limsoon Wong,et al.  Aggregate Functions, Conservative Extensions, and Linear Orders , 1993, DBPL.

[19]  Moshe Y. Vardi.  The complexity of relational query languages (Extended Abstract) , 1982, STOC '82.

[20]  Marc Gyssens,et al.  A Comparison between Algebraic Query Languages for Flat and Nested Databases , 1991, Theor. Comput. Sci..

[21]  Marc Gyssens,et al.  The Powerset Algebra as a Natural Tool to Handle Nested Database Relations , 1992, J. Comput. Syst. Sci..

[22]  Serge Abiteboul,et al.  Non First Normal Form Relations: An Algebra Allowing Data Restructuring , 1986, J. Comput. Syst. Sci..

[23]  Neil Immerman,et al.  Relational Queries Computable in Polynomial Time , 1986, Inf. Control..

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

[25]  S. Abiteboul,et al.  Fixpoint extensions of first-order logic and datalog-like languages , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[26]  Hans-Jörg Schek,et al.  The relational model with relation-valued attributes , 1986, Inf. Syst..