The process of designing an object-oriented database schema consists of several phases. During the phase of abstract logical formalisation one of many possible abstract object-oriented database schemas must be chosen. This choice can be driven by the costs of the ultimately implemented schema: How much space is needed? How long does it take to compute queries and updates including enforcement of semantic constraints? Because abstract logical formalisation is done independently of an actual database management system, we need an abstract database machine. Queries and updates are formulated as programs for this database machine. Such programs are composed of steps which are connected by channels for typed streams of value lists. In each step, a basic or compound operation is executed, accepting input streams and further parameters, delivering output streams for subsequent steps, and accessing the persistent database state. The abstract database machine is designed to meet two goals: to be expressive enough to implement queries and updates, as considered for schema design, and to be simple enough to allow cost estimations.