On the structure of the space of complexity partial functions

Dual complexity spaces were introduced by Romaguera and Schellekens in order to obtain a robust mathematical model for the complexity analysis of algorithms and programs. This model is based on the notions of a cone and of a quasi-metric space. Later on, the structure of the dual complexity spaces was modified with the purpose of giving quantitative measures of the improvements in the complexity of programs. This new complexity structure was presented as an ordered cone endowed with an invariant extended quasi-metric. Here we construct a general dual complexity space by using (complexity) partial functions. This new complexity structure is a pointed ordered cone endowed with a subinvariant bicomplete extended quasi-metric as complexity distance. We apply this approach to modelling certain processes that arise, in a natural way, in symbolic computation.