Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Decision Quotient: A Regime-Sensitive Complexity Theory of Exact Relevance Certification

Tristan Simas

Abstract

Which coordinates of a decision problem can be hidden without changing the decision, and what is the coarsest exact abstraction that preserves all decision-relevant distinctions? We study this as an exact relevance-certification problem organized around the optimizer quotient. We classify how hard it is to certify this structure across three settings: static (counterexample exclusion), stochastic (conditioning and expectation), and sequential (temporal structure). In the static regime, sufficiency collapses to relevance containment, so minimum sufficiency is coNP-complete. In the stochastic regime, preservation and decisiveness separate: preservation is polynomial-time under explicit-state encoding with bridge theorems to static sufficiency and the optimizer quotient, while decisiveness is PP-hard under succinct encoding with anchor and minimum variants in $\textsf{NP}^{\textsf{PP}}$. In the sequential regime, all queries are PSPACE-complete. We also prove an encoding-sensitive contrast between explicit-state tractability and succinct-encoding hardness, derive an integrity-competence trilemma, and isolate twelve tractable subcases. A Lean 4 artifact mechanically verifies the optimizer-quotient universal property, main reductions, and finite decider core.

Decision Quotient: A Regime-Sensitive Complexity Theory of Exact Relevance Certification

Abstract

Which coordinates of a decision problem can be hidden without changing the decision, and what is the coarsest exact abstraction that preserves all decision-relevant distinctions? We study this as an exact relevance-certification problem organized around the optimizer quotient. We classify how hard it is to certify this structure across three settings: static (counterexample exclusion), stochastic (conditioning and expectation), and sequential (temporal structure). In the static regime, sufficiency collapses to relevance containment, so minimum sufficiency is coNP-complete. In the stochastic regime, preservation and decisiveness separate: preservation is polynomial-time under explicit-state encoding with bridge theorems to static sufficiency and the optimizer quotient, while decisiveness is PP-hard under succinct encoding with anchor and minimum variants in . In the sequential regime, all queries are PSPACE-complete. We also prove an encoding-sensitive contrast between explicit-state tractability and succinct-encoding hardness, derive an integrity-competence trilemma, and isolate twelve tractable subcases. A Lean 4 artifact mechanically verifies the optimizer-quotient universal property, main reductions, and finite decider core.
Paper Structure (97 sections, 71 theorems, 63 equations)

This paper contains 97 sections, 71 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Informal interpretive guide.
  3. Probability splits the certification problem.
  4. Connection to backdoor tractability in CSPs.
  5. Contributions
  6. Quantifier Structure
  7. Paper Structure
  8. Artifact Availability
  9. Formal Setup
  10. Decision Problems with Coordinate Structure
  11. Sufficiency and Relevance
  12. Optimizer Quotient
  13. Computational Model and Input Encoding
  14. Succinct encoding (primary for hardness).
  15. Explicit-state encoding.
  16. ...and 82 more sections

Key Result

Proposition 2.5

The empty set $\emptyset$ is sufficient if and only if $\text{Opt}(s)$ is constant over the entire state space.

Theorems & Definitions (183)

  • Definition 2.1: Decision Problem
  • Definition 2.2: Projection
  • Definition 2.3: Optimizer Map
  • Definition 2.4: Sufficient Coordinate Set
  • Proposition 2.5: Empty-Set Sufficiency Equals Constant Decision Boundary
  • proof
  • Proposition 2.6: Insufficiency Equals Counterexample Witness
  • proof
  • Definition 2.7: Minimal Sufficient Set
  • Definition 2.8: Relevant Coordinate
  • ...and 173 more