First-order definable counting-only queries

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.

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

[2]  Willard Van Orman Quine,et al.  Selected Logic Papers , 1966 .

[3]  Jian Pei,et al.  Mining Frequent Patterns without Candidate Generation: A Frequent-Pattern Tree Approach , 2006, Sixth IEEE International Conference on Data Mining - Workshops (ICDMW'06).

[4]  Jan Paredaens,et al.  On the Expressive Power of the Relational Algebra , 1978, Inf. Process. Lett..

[5]  Marc Gyssens,et al.  First-order definable counting-only queries , 2019, Annals of Mathematics and Artificial Intelligence.

[6]  Jef Wijsen,et al.  Calculi for symmetric queries , 2019, J. Comput. Syst. Sci..

[7]  John C. Smart,et al.  Mapping intellectual structure of a scientific subfield through author cocitations , 1990, J. Am. Soc. Inf. Sci..

[8]  Serge Abiteboul,et al.  Foundations of Databases: The Logical Level , 1995 .

[9]  J. Väänänen.  Generalized quantifiers and computation : 9th European Summer School in Logic, Language, and Information, ESSLLI '97 Workshop, Aix-en-Provence, France, August 11-22, 1997 : revised lectures , 1999 .

[10]  Jan Van den Bussche,et al.  Towards a theory of search queries , 2009, ICDT '09.

[11]  Martin Otto,et al.  On Logics with Two Variables , 1999, Theor. Comput. Sci..

[12]  Dirk Van Gucht,et al.  Differential constraints , 2005, PODS '05.

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

[14]  Harald Ganzinger,et al.  Set constraints are the monadic class , 1993, [1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science.

[15]  Dirk Van Gucht,et al.  Computationally Complete Relational Query Languages , 2009, Encyclopedia of Database Systems.

[16]  Jouko A. Väänänen,et al.  Generalized Quantifiers , 1997, Bull. EATCS.

[17]  Toon Calders,et al.  Non-derivable itemset mining , 2007, Data Mining and Knowledge Discovery.

[18]  Jouko A. Väänänen,et al.  Generalized Quantifiers, an Introduction , 1997, ESSLLI.

[19]  Stanley Peters,et al.  Quantifiers in language and logic , 2006 .

[20]  Nicole Schweikardt,et al.  First-order logic with counting , 2017, 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[21]  Martin Grohe,et al.  Finite Variable Logics in Descriptive Complexity Theory , 1998, Bulletin of Symbolic Logic.

[22]  Antonio Badia,et al.  Querying with Generalized Quantifiers , 1993, Workshop on Programming with Logic Databases , ILPS.

[23]  François Bancilhon,et al.  On the Completeness of Query Languages for Relational Data Bases , 1978, MFCS.

[24]  Harry R. Lewis,et al.  Complexity Results for Classes of Quantificational Formulas , 1980, J. Comput. Syst. Sci..

[25]  Bart Goethals,et al.  Survey on Frequent Pattern Mining , 2003 .

[26]  Jef Wijsen,et al.  An Approach towards the Study of Symmetric Queries , 2013, Proc. VLDB Endow..

[27]  Leonid Libkin,et al.  Elements of Finite Model Theory , 2004, Texts in Theoretical Computer Science.

[28]  I. Anderson.  Combinatorics of Finite Sets , 1987 .