  Declarative modeling uses symbolic expressions to represent models. With such
expressions one can formalize high-level mathematical computations on models
that would be difficult or impossible to perform directly on a lower-level
simulation program, in a general-purpose programming language. Examples of such
computations on models include model analysis, relatively general-purpose
model-reduction maps, and the initial phases of model implementation, all of
which should preserve or approximate the mathematical semantics of a complex
biological model. The potential advantages are particularly relevant in the
case of developmental modeling, wherein complex spatial structures exhibit
dynamics at molecular, cellular, and organogenic levels to relate genotype to
multicellular phenotype. Multiscale modeling can benefit from both the
expressive power of declarative modeling languages and the application of model
reduction methods to link models across scale. Based on previous work, here we
define declarative modeling of complex biological systems by defining the
operator algebra semantics of an increasingly powerful series of declarative
modeling languages including reaction-like dynamics of parameterized and
extended objects; we define semantics-preserving implementation and
semantics-approximating model reduction transformations; and we outline a
"meta-hierarchy" for organizing declarative models and the mathematical methods
that can fruitfully manipulate them.
