A relational database can be considered as a finite structure for a finite relational signature in first-order logic, i.e., there are no function symbols. Interpreting the logic over this signature in such structures allows the expressiveness and complexity of queries to be studied in detail. This is the starting point for finite model theory which has proven to be a viable tool to study relational database theory. In particular, it is known that computable queries expressed as isomorphism-preserving partial recursive functions can be formalized by Reflective Relational Machines. These are extended Turing Machines with an additional relational store, a query tape and the facility to evaluate the query on the tape against the database in the store in a single step.

In this paper we start to generalize the theory to post-relational databases. We first consider the case of having set-based complex values and references such that the semantics can still be expressed in finite sets. Following the approach that object oriented databases in general including those, where the underlying type systems does no longer allow the semantics defined by sets, can be expressed as theories in higher-order intuitionistic logic, we use such a logic instead of first-order logic. However, as we are not yet exploiting the full power of such logics, we can interpret the logic in the category FINSET of finite sets, i.e., again in a structure defined by a database.

Having done this the definition of computable queries and the model of Reflective Relational Machines carry over easily. We can show that the new model of Reflective Object Machines guarantees completeness, i.e., all computable queries can be expressed by the model.