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, $\text{STRUC}[\sigma ]$ and $\text{STRUC}[\tau ]$. A query is then any mapping

$I:\text{STRUC}[\sigma ]\to \text{STRUC}[\tau ]$

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 $\mathfrak{𝔅}$.