Sequential decision problems, dependent types and generic solutions

TL;DR

This paper introduces a formal, computer-verified framework for solving a broad class of finite-horizon sequential decision problems using dependent types, enabling flexible modeling of uncertainties and formal proofs of optimality principles.

Contribution

It provides a generic, dependently-typed implementation for sequential decision problems, including formal proofs of Bellman's principle and backwards induction in Idris and Agda.

Findings

Formalization of Bellman's principle of optimality

Implementation handles time-dependent control and state spaces

Framework applicable to various uncertainty models

Abstract

We present a computer-checked generic implementation for solving finite-horizon sequential decision problems. This is a wide class of problems, including inter-temporal optimizations, knapsack, optimal bracketing, scheduling, etc. The implementation can handle time-step dependent control and state spaces, and monadic representations of uncertainty (such as stochastic, non-deterministic, fuzzy, or combinations thereof). This level of genericity is achievable in a programming language with dependent types (we have used both Idris and Agda). Dependent types are also the means that allow us to obtain a formalization and computer-checked proof of the central component of our implementation: Bellman's principle of optimality and the associated backwards induction algorithm. The formalization clarifies certain aspects of backwards induction and, by making explicit notions such as viability and reachability, can serve as a starting point for a theory of controllability of monadic dynamical systems, commonly encountered in, e.g., climate impact research.