Skip to content

Theory

Mathematical foundations for BE-DDSL and the modular dynamic programming framework.

Contents

A. Modular MDP Foundations

Defines the stage as the basic unit of decision-making, develops the factored Bellman operator and its contraction properties, shows equivalence with the standard MDP, and introduces forward operators and stage composition into periods.

  • MDP Foundations — Stages, perches, Bellman recursion, operators \(\mathrm{T}_{\prec}\)/\(\mathrm{T}_{\succ}\), connectors, branching

B. Foundations of Bellman calculus

Formal specification of the Bellman-Euler Domain-Specific Declarative Language. Covers primitive objects, symbols, stages, calibration, Bellman theory, representation maps (\(\Upsilon\) and \(\rho\)), and the SYM/T core architecture.

Document Description
Index Section overview and reading order
Mathematical Definition Core formal definitions
Primitive Objects Base objects: spaces, fields, operators
Symbol System Syntax and symbol resolution
Stages & Calibration Stage structure and parameter binding
Bellman Theory Value function operators
Representation Maps Υ and ρ maps between layers
Execution Pipeline From YAML to solution
SYM/T core Architecture Two-layer design
Composition as Twister Period and nest composition theory

E. Backus FFP Concepts

Functional programming foundations following Backus (1978), defining the algebraic structure underlying DDSL: objects, application, functional forms, and the algebra of programs.

C. Unified Framework

Extended mathematical reference providing a unified treatment of dynamic optimization with modular stage composition, including worked examples (consumption, portfolio, port-with-shocks) and computational framework.

  • Unified Framework Overview — Recursive problem definition, theoretical propositions, modular examples, computational framework

D. Modular MDPs (Dyn-X)

Graph-theoretic perspective on modular dynamic programs: factored Bellman operators, perch/mover architecture, and stage composition as directed graphs.