Query (complexity)

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
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 $$\mathfrak{B}$$.