finite-reliability-representations

name	finite-reliability-representations
description	Finite Reliability Representations (FRR) methodology for noise-calibrated belief-space covers in decision-making systems. Provides certified suboptimality bounds based on sensing, process, and actuation noise. Use when designing reliable decision systems, POMDP policies, or safety-critical control.
version	1.0.0
author	Hermes Agent
license	MIT
source	arXiv:2607.04019
tags	["systems-engineering","reliability","POMDP","belief-space","noise-calibration","decision-making","control-theory"]
activation	reliability representations, belief space covers, noise-calibrated decision making, POMDP reliability, certified suboptimality, belief complexity, reliability entropy

Finite Reliability Representations (FRR)

Source: arXiv:2607.04019 - "Finite Reliability Representations: Noise-Calibrated Belief-Space Covers for Reliable Decision-Making"

Core Theory

Physical sensing and actuation noise floors should inform how much belief resolution a decision-making system can reliably use. FRR covers belief spaces by reliability cells: regions within which the optimal action-value function Q*(b,u) varies by at most a tolerance ε, uniformly over actions.

Key Insight

Noisy Bayesian updates should not be treated as globally contractive on arbitrary beliefs. Separate three objects:

1. Fixed-observation filter map
2. Predictive observation law
3. Controlled belief-transition kernel

Methodology

Step 1: Construct Reliability Cells

For a given tolerance ε:

• Partition the belief space into cells where Q*(b,u) varies by ≤ ε
• Use reachable-set Lipschitz modulus for belief-transition kernel
• Cells are NOT equivalence classes (approximate decision-closeness is not transitive)

Step 2: Certification Conditions

For nonlinear continuous-state systems:

• FRR obtained under reachable-set Lipschitz modulus for belief-transition kernel

For finite-state POMDPs:

• Prediction is linear
• Bayesian correction is normalized positive linear map
• Sensor noise enters through observation-distribution distinguishability
• Actuation uncertainty enters through action-execution channel

Step 3: Cell-Constant Policy

Under action-value Lipschitz condition:

• Construct policy constant on each reliability cell
• Suboptimality bound: 2ε/(1-γ) where γ is discount factor

Step 4: Reliability Entropy

H = log(N_min) where N_min = minimal number of reliability cells

• Measures certified decision-relevant belief complexity
• Distinguishes representation sufficiency from fundamental performance floors

Application Domains

1. Finite POMDPs - Exact construction on belief simplex
2. Linear-Gaussian filters - Analytic certification
3. Locally linearized nonlinear filters - Empirical certification
4. Particle-filter implementations - Empirical certification of reliability cells