Stochastic Process Semantics for Dynamical Grammar Syntax: An Overview

TL;DR

This paper introduces a unified operator algebra framework for probabilistic models, linking diverse fields like data clustering, logic, differential equations, and chemical kinetics through stochastic process semantics.

Contribution

It presents a novel mathematical formulation that maps grammatical syntax to operator-based semantics, connecting various stochastic models and fields.

Findings

Unified operator algebra framework for stochastic processes

Representation of diverse models like clustering and differential equations

Connections to quantum field theory and operator algebra

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 mapped to semantics given in terms of a ring of operators, so that grammatical composition corresponds to operator addition or multiplication. The operators are generators for the time-evolution of stochastic processes. Within this modeling framework one can express data clustering models, logic programs, ordinary and stochastic differential equations, graph grammars, and stochastic chemical reaction kinetics. This mathematical formulation connects these apparently distant fields to one another and to mathematical methods from quantum field theory and operator algebra.