Structural Commutation Relations for Stochastic Labelled Graph Grammar Rule Operators

We show how to calculate the operator algebra and the operator Lie algebra of a stochastic labelled-graph grammar. More specifically, we carry out a generic calculation of the product (and therefore the commutator) of time-evolution operators for any two labelled-graph grammar rewrite rules, where the operator corresponding to each rule is defined in terms of elementary two-state creation/annihilation operators. The resulting graph grammar algebra has the following properties: (1) The product and commutator of two such operators is a sum of such operators with integer coefficients. Thus, the algebra and the Lie algebra occurs entirely at the structural (or graph-combinatorial) level of graph grammar rules, lifted from the level of elementary creation/annihilation operators (an improvement over [1], Propositions 1 and 2). (2) The product of the off-diagonal (state-changing) parts of two such graph rule operators is a sum of off-diagonal graph rule operators with non-negative integer coefficients. (3) These results apply whether the semantics of a graph grammar rule leaves behind hanging edges (Theorem 1), or removes hanging edges (Theorem 2). (4) The algebra is constructive in terms of elementary two-state creation/annihilation operators (Corollaries 3 and 8). These results are useful because dynamical transformations of labelled graphs comprise a general modeling framework, and algebraic commutators of time-evolution operators have many analytic uses including designing simulation algorithms and estimating their errors

Prospects for Declarative Mathematical Modeling of Complex Biological Systems

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

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 formalizating 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. 1 Measureable Types V25.nb