Stochastic process semantics for dynamical grammars

Abstract

We define a class of probabilistic models in terms of an operator algebra of stochastic processes, and a representation for this class in terms of stochastic parameterized grammars. A syntactic specification of a grammar is formally mapped to semantics given in terms of a ring of operators, so that composition of grammars corresponds to operator addition or multiplication. The operators are generators for the time-evolution of stochastic processes. The dynamical evolution occurs in continuous time but is related to a corresponding discrete-time dynamics. An expansion of the exponential of such time-evolution operators can be used to derive a variety of simulation algorithms. Within this modeling framework one can express data clustering models, logic programs, ordinary and stochastic differential equations, branching processes, graph grammars, and stochastic chemical reaction kinetics. The mathematical formulation connects these apparently distant fields to one another and to mathematical methods from quantum field theory and operator algebra. Such broad expressiveness makes the framework particularly suitable for applications in machine learning and multiscale scientific modeling.

Static and dynamic models of biological networks

“…Note that (14) allows one to use a direct sampling method to estimate the energy function from the relation…”

A Calculus of Chemical Systems

“…These calculi generally fall into one of two camps: ones based on process calculi, such as Milner's pi-calculus [24], and rule-based ones. Examples of the former include [31,32,30]; examples of the latter include BIOCHAM, κ, BioNetGen, and Dynamical Grammars [2,6,15,26]. One positive feature of the rule-based approach is that rules correspond naturally to biological events, with a main example being chemical reactions.…”

“…In giving these semantics we take up a suggestion of Pedersen [29] to incorporate the various semantics within a general framework, here using commutative monoids. (This is strongly connected to the use of rings of operators by Mjolsness and Yosiphon [26]. )…”

“…It should be possible to usefully add modularity to any rule-based calculus. Indeed the grammatical approach in [26] provides what amounts to a parameterised module mechanism. One can enrich our calculus with facilities for modifications and simple complexes along the general lines of BIOCHAM, and with a construction for (static) compartments: all of these are omnipresent in cellular networks.…”

A Language for Biochemical Systems

“…the pi calculus [26], PEPA [2], Petri nets [25] and P-systems [19], and some are designed from scratch, e.g. Kappa [4], BioNetGen [5], BIOCHAM [3], Bioambients [27], Beta binders [7,24], Dynamical Grammars [15] and the Continuous Pi Calculus [14].…”