Abstract
The expressive power of first-order query languages with several classes of equality and inequality constraints is studied in this paper. We settle the conjecture that recursive queries such as parity test and transitive closure cannot be expressed in the relational calculus augmented with polynomial inequality constraints over the reals. Furthermore, noting that relational queries exhibit several forms of genericity, we establish a number of collapse results of the following form: The class of generic Boolean queries expressible in the relational calculus augmented with a given class of constraints coincides with the class of queries expressible in the relational calculus (with or without an order relation). We prove such results for both the natural and active-domain semantics. As a consequence, the relational calculus augmented with polynomial inequalities expresses the same classes of generic Boolean queries under both the natural and active-domain semantics.
In the course of proving these results for the active-domin semantics, we establish Ramsey-type theorems saying that any query involving certain kinds of constraints coincides with a constraint-free query on databases whose elements come from a certain infinite subset of the domain. To prove the collapse results for the natural semantics, we make use of techniques from nonstandard analysis and from the model theory of ordered structures.
- ABITEBOUL, S., HULL, R., AND VIANU, V. 1995. Foundations of Databases. Addison-Wesley, Reading, Mass.]] Google Scholar
- AFRATI, F., ANDRONIKOS, T., AND KAVALIEROS, T. 1995. On the expressiveness of first-order constraint languages. In Proceedings of the 1st Workshop on Constraints Databases and Their Applications. Springer-Verlag, New York, pp. 22-39.]] Google Scholar
- AFRATI, F., COSMADAKIS, S., GRUMBACH, S., AND KUPER, G. 1994. Linear vs. polynomial constraints in database query languages. In Proceedings of Conference on Principles and Practice of Constraint Programming. Springer-Verlag, New York, pp. 181-195.]] Google Scholar
- AHO, A. V., AND ULLMAN, J.D. 1979. Universality of data retrieval languages. In Proceedings of the 6th Symposium on Principles of Programming Languages (Texas, Jan.). DIMACS, Rutgers Univ., New Brunswick, N.J., pp. 110-120.]] Google Scholar
- BELEGRADEK, 0. V., STOLBOUSHKIN, A. P., AND TAITSLIN, M.A. 1995. On order-generic queries. DIMACS Tech. Rep. 95-56. Dec.]] Google Scholar
- BENEDIKT, M., DONG, G., LIBKIN, L., AND WONG, L. 1996. Relational expressive power of constraint query languages. In Proceedings of the 15th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (Montreal, Que., Canada, June 3-5). ACM, New York, pp. 5-16.]] Google Scholar
- BENEDIKT, M., AND KEISLER, H.J. 1997. On the expressive power of unary counters. In Proceedings of the International Conference on Database Theory. Lecture Notes on Computer Science, vol. 1186. Springer-Verlag, New York, pp. 291-305.]] Google Scholar
- BENEDIKT, M., AND LIBKIN, L. 1996. On the structure of queries in constraint query languages. In Proceedings of IEEE Symposium on Logic in Computer Science. IEEE, New York, pp. 25-34.]] Google Scholar
- BENEDIKT, M., AND LIBKIN, L. 1997. Languages for relational databases over interpreted structures. In Proceedings of the 16th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (Tucson, Az., May 12-14). ACM, New York, pp. 87-98.]] Google Scholar
- CHANDRA, A., AND HAREL, D. 1980. Computable queries for relational databases. J. Comput. Syst. Sci. 21, 2, 156-178.]]Google Scholar
- CHANG, C. C., AND KEISLER, H.J. 1990. Model Theory. North-Holland, Amsterdam, The Netherlands.]]Google Scholar
- EBBINGHAUS, H.-D., AND FLUM, J. 1995. Finite Model Theory. Springer-Verlag, New York.]]Google Scholar
- FAGIN, R. 1976. Probabilities on finite models. J. Symb. Logic 41, 1, 50-58.]]Google Scholar
- GAIFMAN, H. 1982. On local and non-local properties. In Proceedings of the Herbrand Symposium. Logic Colloquium '81. North-Holland, Amsterdam, The Netherlands, pp. 105-135.]]Google Scholar
- GRAHAM, R. L., ROTHSCHILD, B. L., AND SPENCER, J.U. 1990. Ramsey Theory. Wiley, New York.]]Google Scholar
- GRUMBACH, S., AND SU, J. 1994. Finitely representable databases. In Proceedings of the 13th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (Minneapolis, Minn., May 24-26). ACM, New York, pp. 289-300.]] Google Scholar
- GRUMBACH, S., AND SU, J. 1995. First-order definability over constraint databases. In Proceedings of Principles and Practice of Constraint Programming. Springer-Verlag, New York, pp. 121-136.]] Google Scholar
- GRUMBACH, S., Su, J., AND TOLLU, C. 1994. Linear constraint databases. In Proceedings of Logic and Computational Complexity. Springer-Verlag, New York, pp. 426-446.]] Google Scholar
- HENSON, C.W. 1974. The isomorphism property in nonstandard analysis and its use in the theory of Banach spaces. J. Symb. Logic 39, 717-731.]]Google Scholar
- HULL, R., AND SU, J. 1994. Domain independence and the relational calculus. Acta Inf. 31, 513-524.]] Google Scholar
- HULL, R., AND YAP, C.K. 1984. The format model: A theory of database organization. J. ACM 31, 3 (July), 518-537.]] Google Scholar
- KANELLAKIS, P. 1995. Constraint programming and database languages: A tutorial. In Proceedings of the 14th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (San Jose, Calif., May 22-25). ACM, New York, pp. 46-53.]] Google Scholar
- KANELLAKIS, P., KUPER, G., AND REVESZ, P. 1990. Constraint query languages. J. Comput. Syst. Sci. 51, 26-52.]] Google Scholar
- KNIGHT, J. F., PILLAY, A., AND STEINHORN, C. I. 1986. Definable sets in ordered structures II. Trans. AMS 295, 593- 605.]]Google Scholar
- KUPER, G. 1990. On the expressive power of the relational calculus with arithmetic constraints. In Proceedings of the International Conference on Database Theory. Springer-Verlag, New York, pp. 201-211.]] Google Scholar
- MARKER, D. 1996. Model theory and exponentiation. Not. AMS 43, 753-759.]]Google Scholar
- NARASIMHA, R. 1971. Several Complex Variables. University of Chicago, Press, Chicago, Ill.]]Google Scholar
- OTTO, M., AND VAN DEN BUSSCHE, J. 1996. First-order queries on databases embedded in an infinite structure. Inf. Proc. Lett. 14, 37-41.]] Google Scholar
- PAREDAENS, J., VAN DEN BUSCHE, J., AND VAN GUCHT, D. 1994. Towards a theory of spatial database queries. In Proceedings of the 13th ACM Symposium on Principles of Database Systems (Minneapolis, Minn., May 24-26). ACM, New York, pp. 279-288.]] Google Scholar
- PAREDAENS, J., VAN DEN BUSCHE, J., AND VAN GUCHT, D. 1995. First-order queries on finite structures over the reals. In Proceedings of the lOth IEEE Symposium on Logic in Computer Science (San Diego, Calif.). IEEE, New York, pp. 79-87. (Full paper to appear in SIAM J. Comput.)]] Google Scholar
- PILLAY, A., AND STEINHORN, C. 1984. Definable sets in ordered structures. Bull. AMS 11,159-162.]]Google Scholar
- PILLAY, A., AND STEINHORN, C. 1988. Definable sets in ordered structures. III. Trans. AMS 309, 469-476.]]Google Scholar
- REVESZ, P. 1998. Constraint databases: A survey. In Proceedings of the Workshop on Semantics in Databases (Prague, Czech Republic). Springer-Verlag, New York.]] Google Scholar
- ROSENSTEIN, J.G. 1982. Linear Orderings. Academic Press, Orlando, Fla.]]Google Scholar
- SPEISSINGER, P. 1996. Order Minimality of the Gamma Function. Personal communication.]]Google Scholar
- STOLBOUSHKIN, A. P., AND TAITSLIN, M.A. 1996. Linear vs. order constraint queries over rational databases. In Proceedings of the 15th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (Montreal, Que., Canada, June 3-5). ACM, New York, pp. 17-27.]] Google Scholar
- TARSKI, A. 1951. /i Decision Method for Elementary Algebra and Geometry. University of California Press, Berkeley, Calif.]]Google Scholar
- VAN DEN DRIES, L. 1988. Alfred Tarski's elimination theory for real closed fields. J. Symb. Logic 53, 7-19.]]Google Scholar
- VAN DEN DRIES, L., MACINTYRE, A., AND MARKER, D. 1994. The elementary theory of restricted analytic fields with exponentiation. #Inn. Math. 85, 19-56.]]Google Scholar
- WILKIE, A. J. 1996. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential functions. J./IMS 9, 1051-1094.]]Google Scholar
Index Terms
- Relational expressive power of constraint query languages
Recommendations
On the expressive power of query languages for relational databases
POPL '82: Proceedings of the 9th ACM SIGPLAN-SIGACT symposium on Principles of programming languagesThe query languages used in relational database systems are a special class of programming languages. The majority, based on first-order logic, lend themselves to analysis using formal methods. First, we provide a definition of relational query ...
Towards more expressive ontology languages: The query answering problem
Ontology reasoning finds a relevant application in the so-called ontology-based data access, where a classical extensional database (EDB) is enhanced by an ontology, in the form of logical assertions, that generates new intensional knowledge which ...
On the expressive power of query languages
Two main topics are addressed. First, an algebraic approach is presented to define a general notion of expressive power. Heterogeneous algebras represent information systems and morphisms represent the correspondences between the instances of databases, ...
Comments