Theorem: CALC is included in QLOGSPACE.
Proof (outline):
Let j be a query in CALC over R. We will describe a TM Mj that solves the recognition problem for j, and uses a work tape with length logarithmic in the size of the input tape.
Mj is started with the input enca(I)#enca (u). Mj should accept the input iff uÎj(I). It is possible to show by induction on the number of quantifiers of j that the computation can be performed using k*log(|enca(I)#enca (u)|) cells of the work tape.
Remark: CALC does not express all of QLOGSPACE (e.g. even query which is in LOGSPACE but is not a first order query and thus can not be expressed by CALC)