First-Order Definable Counting-Only Queries

  • Jelle HellingsEmail author
  • Marc Gyssens
  • Dirk Van Gucht
  • Yuqing Wu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10833)


For several practical queries on bags 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 group of at most k objects belongs. We call such queries k-counting-only. Here, we focus on \(k\)-SyCALC, 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 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 SimpleCALC-\(k\) and \(k\)-SyCALC. Finally, we establish a strong dichotomy between 1-SyCALC and SimpleCALC-\(k\) 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.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Jelle Hellings
    • 1
    Email author
  • Marc Gyssens
    • 1
  • Dirk Van Gucht
    • 2
  • Yuqing Wu
    • 3
  1. 1.Hasselt UniversityHasseltBelgium
  2. 2.Indiana UniversityBloomingtonUSA
  3. 3.Pomona CollegeClaremontUSA

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