Abstract
Fragments of Tarski’s relation algebra form the basis of many versatile graph and tree query languages including the regular path queries, XPath, and SPARQL. Surprisingly, however, a systematic study of the relative expressive power of relation algebra fragments on trees has not yet been undertaken. Our approach is to start from a basic fragment which only allows composition and union. We then study how the expressive power of the query language changes if we add diversity, converse, projections, coprojections, intersections, and/or difference, both for path queries and Boolean queries. For path queries, we found that adding intersection and difference yields more expressive power for some fragments, while adding one of the other operators always yields more expressive power. For Boolean queries, we obtain a similar picture for the relative expressive power, except for a few fragments where adding converse or projection yields no more expressive power. One challenging problem remains open, however, for both path and Boolean queries: does adding difference yields more expressive power to fragments containing at least diversity, coprojections, and intersection?
This material is based on work supported by the National Science Foundation under Grant No. NSF 1438990.
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Notes
- 1.
The basic atoms and operators, \(\emptyset \), \(\ell \in \varSigma \), \(\mathrm {id}\), \(\circ \), and \(\cup \) are left implicit because they are assumed to be present in every fragment.
- 2.
The 1-subtree reductions bear a close relationship to the F+B-index and the F&B-index used for indexing the structure of tree data [20].
- 3.
Notice that in the formalism of Hellings et al. [18], projection is considered to be a standard operator.
- 4.
Observe that, in relation algebra expressions, unions can always be pushed out to the outermost level.
- 5.
Recall from Sect. 2 that we need to consider converse only at the level of edges.
- 6.
Observe that the composition of node expressions is associative and that this composition is path-equivalent to the intersection of node expressions.
- 7.
Recall that the tree queries in Wu et al. [26] are essentially the same as the condition-free condition tree queries in this paper.
- 8.
Strictly speaking, they deal with union-free expressions, but since unions can always be pushed out to the outermost level, this is not a real restriction.
- 9.
These are generally referred to as the downward fragments of the relation algebra.
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Hellings, J., Wu, Y., Gyssens, M., Van Gucht, D. (2018). The Power of Tarski’s Relation Algebra on Trees. In: Ferrarotti, F., Woltran, S. (eds) Foundations of Information and Knowledge Systems. FoIKS 2018. Lecture Notes in Computer Science(), vol 10833. Springer, Cham. https://doi.org/10.1007/978-3-319-90050-6_14
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