Query (complexity)

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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[s] the concept of query as the fundamental paradigm of computation" (p. 17).

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

I : \mbox{STRUC}[\sigma] \to \mbox{STRUC}[\tau]

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[edit]

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  I(\mathfrak{A}) \equiv I(\mathfrak{B}) for any isomorphic structures \mathfrak{A} and \mathfrak{B}.