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

I : STRUC [ σ ] → STRUC [ τ ] {\displaystyle 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 ( A ) ≡ I ( B ) {\displaystyle I({\mathfrak {A}})\equiv I({\mathfrak {B}})} for any isomorphic structures A {\displaystyle {\mathfrak {A}}} and B {\displaystyle {\mathfrak {B}}}.