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How Many Variables are Needed to Express an Existential Positive Query?

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Abstract

The number of variables used by a first-order query is a fundamental measure which has been studied in numerous contexts, and which is known to be highly relevant to the task of query evaluation. In this article, we study this measure in the context of existential positive queries. Building on previous work, we present a combinatorial quantity defined on existential positive queries; we show that this quantity not only characterizes the minimum number of variables needed to express a given existential positive query by another existential positive query, but also that it characterizes the minimum number of variables needed to express a given existential positive query, over all first-order queries. Put differently and loosely, we show that for any existential positive query, no variables can ever be saved by moving out of existential positive logic to first-order logic. One component of this theorem’s proof is the construction of a winning strategy for a certain Ehrenfeucht-Fraïssé type game.

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Notes

  1. A class of queries has bounded arity if there is a constant upper bound on the arity of all relation symbols appearing in a query of the class.

  2. This can be justified as follows. We write g ∘ f = g(f) to denote the composition of g and f , with f applied first. We have that π1 is a homomorphism from B to A. Hence π1 ∘ h is an endomorphism of A, and since A is a core, π1 ∘ h is an automorphism of A. By associativity of function composition,

    $$\text{id}_{A} = (\pi_{1} \circ h) \circ (\pi_{1} \circ h)^{-1}=\pi_{1} \circ (h \circ (\pi_{1} \circ h)^{-1})=\pi_{1} \circ h^{\prime}$$

    that is, if h′ = h ∘ (π1 ∘ h)− 1, then π1(h′(a)) = a for all a ∈ A. Moreover, h′ is a homomorphism from A to B because (π1 ∘ h)− 1 is an automorphism of A and h is a homomorphism from A to B.

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Acknowledgments

The first author was supported by the FWF Austrian Science Fund (Parameterized Compilation, P26200). The second author was supported by the Spanish Project MINECO COMMAS TIN2013-46181-C2-R, Basque Project GIU15/30, and Basque Grant UFI11/45.

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  1. TU Wien (Austria), Wien, Austria

    Simone Bova

  2. Birkbeck, University of London, London, UK

    Hubie Chen

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  1. Simone Bova
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  2. Hubie Chen
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Bova, S., Chen, H. How Many Variables are Needed to Express an Existential Positive Query?. Theory Comput Syst 63, 1573–1594 (2019). https://doi.org/10.1007/s00224-018-9884-z

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