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Dynamic Complexity of Parity Exists Queries

Authors Nils Vortmeier, Thomas Zeume

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ACM Subject Classification
  • Theory of computation → Logic and databases
  • Theory of computation → Complexity theory and logic
Keywords
  • Dynamic complexity
  • parity quantifier
  • arity hierarchy

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Abstract

Given a graph whose nodes may be coloured red, the parity of the number of red nodes can easily be maintained with first-order update rules in the dynamic complexity framework DynFO of Patnaik and Immerman. Can this be generalised to other or even all queries that are definable in first-order logic extended by parity quantifiers? We consider the query that asks whether the number of nodes that have an edge to a red node is odd. Already this simple query of quantifier structure parity-exists is a major roadblock for dynamically capturing extensions of first-order logic.
We show that this query cannot be maintained with quantifier-free first-order update rules, and that variants induce a hierarchy for such update rules with respect to the arity of the maintained auxiliary relations. Towards maintaining the query with full first-order update rules, it is shown that degree-restricted variants can be maintained.

Cite As Get BibTex

Nils Vortmeier and Thomas Zeume. Dynamic Complexity of Parity Exists Queries. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 37:1-37:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.CSL.2020.37

Author Details

Nils Vortmeier
  • TU Dortmund University, Dortmund, Germany
Thomas Zeume
  • TU Dortmund University, Dortmund, Germany

Funding

The authors acknowledge the financial support by DFG grant SCHW 678/6-2.

Acknowledgements

We are grateful to Samir Datta, Raghav Kulkarni, Anish Mukherjee and Thomas Schwentick for illuminating discussions.

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