Towards Measurable Types for Dynamical Process Modeling Languages

Process modeling languages such as "Dynamical Grammars" are highly expressive in the processes they model using stochastic and deterministic dynamical systems, and can be given formal semantics in terms of an operator algebra. However such process languages may be more limited in the types of objects whose dynamics is easily expressible. For many applications in biology, the dynamics of spatial objects in particular (including combinations of discrete and continuous spatial structures) should be formalizable at a high level of abstraction. We suggest that this may be achieved by formalizing such objects within a type system endowed with type constructors suitable for complex dynamical objects. To this end we review and illustrate the operator algebraic formulation of heterogeneous process modeling and semantics, extending it to encompass partial differential equations and intrinsic graph grammar dynamics. We show that in the operator approach to heterogeneous dynamics, types require integration measures. From this starting point, "measurable" object types can be enriched with generalized metrics under which approximation can be defined. The resulting measurable and "metricated" types can be built up systematically by type constructors such as vectors, products, and labelled graphs. We find conditions under which functions and quotients can be added as constructors of measurable and metricated types.