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In descriptive complexity, a query is a mapping from structures of one signature to structures of another vocabulary. Neil Immerman, in his book Descriptive Complexity, "use the concept of query as the fundamental paradigm of computation" (p. 17).

Given signatures ${\displaystyle \sigma }$ and ${\displaystyle \tau }$, we define the set of structures on each language, ${\displaystyle {\mbox{STRUC}}}$ and ${\displaystyle {\mbox{STRUC}}}$. A query is then any mapping

${\displaystyle I:{\mbox{STRUC}}\to {\mbox{STRUC}}}$

Computational complexity theory can then be phrased in terms of the power of the mathematical logic necessary to express a given query.

Order-independent queries

A query is order-independent if the ordering of objects in the structure does not affect the results of the query. In databases, these queries correspond to generic queries (Immerman 1999, p. 18). A query is order-independent iff ${\displaystyle I({\mathfrak {A}})\equiv I({\mathfrak {B}})}$ for any isomorphic structures ${\displaystyle {\mathfrak {A}}}$ and ${\displaystyle {\mathfrak {B}}}$.

References

1. Neil, Immerman (1999). Descriptive Complexity. New York, NY: Springer New York. ISBN 9781461205395. OCLC 853271745.

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