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First-order definable counting-only queries

  • Jelle Hellings
  • Marc GyssensEmail author
  • Dirk Van Gucht
  • Yuqing Wu
Article

Abstract

Many data sources can be represented easily by collections of sets of objects. For several practical queries on such collections of sets of objects, the answer does not depend on the precise composition of these sets, but only on the number of sets to which each object belongs. This is the case k= 1 for the more general situation where the query answer only depends on the number of sets to which each collection of at most k objects belongs. We call such queries k-counting-only. Here, we focus on k-SyCALC, i.e., k-counting-only queries that are first-order definable. As k-SyCALC is semantically defined, however, it is not surprising that it is already undecidable whether a first-order query is in 1-SyCALC. Therefore, we introduce SimpleCALC-k, a syntactically defined (strict) fragment of k-SyCALC. It turns out that many practical queries in k-SyCALC can already be expressed in SimpleCALCk. We also define the query language GCountk, which expresses counting-only queries directly by using generalized counting terms, and show that this language is equivalent to SimpleCALC-k. We prove that the k-counting-only queries form a non-collapsing hierarchy: for every k, there exist (k+ 1)-counting-only queries that are not k-counting-only. This result specializes to both SimpleCALCk and k-SyCALC. Finally, we establish a strong dichotomy between 1-SyCALC and SimpleCALCk on the one hand and 2-SyCALC on the other hand by showing that satisfiability, validity, query containment, and query equivalence are decidable for the former two languages, but not for the latter one.

Keywords

Bag of sets Counting-only query First-order definable query Satisfiability 

Mathematics Subject Classification (2010)

03C07 03C80 03D15 03D55 68P15 

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Notes

Acknowledgments

This is a revised and extended version of the conference paper ‘First-order definable counting-only queries’, presented at the 10th International Symposium on Foundations of Information and Knowledge Systems, Budapest, Hungary (FoIKS 2018) [15].

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Exploratory Systems Lab, Department of Computer ScienceUniversity of CaliforniaDavisUSA
  2. 2.Faculty of Sciences, Hasselt UniversityHasseltBelgium
  3. 3.School of Informatics, Computing, and Engineering, Indiana UniversityBloomingtonUSA
  4. 4.Pomona CollegeClaremontUSA

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