Exact Structural Abstraction and Tractability Limits

TL;DR

This paper investigates the limits of exact structural abstraction and classification, identifying fundamental obstructions and conditions for tractability across various problem types and structural criteria.

Contribution

It introduces a universal semantic framework for exact relevance certification and characterizes when exact classification is possible based on closure orbits and structural predicates.

Findings

Orbit gaps obstruct exact classification by closure-law-invariant predicates.

Exact classification succeeds when the target is constant on closure orbits.

Restricting to closure-closed domains can eliminate some obstructions.

Abstract

Any rigorously specified problem determines an admissible-output relation $R$. Here exact means exact agreement with $R$ itself; $R$ may encode approximation, randomization, statistical thresholds, failure states, or distributional guarantees. Exact relevance certification depends only on the induced decision quotient relation $s \sim_R s' \iff \operatorname{Adm}_R(s)=\operatorname{Adm}_R(s')$ and asks which coordinates recover those classes. Decision, counting, search, approximation, PAC/regret/risk, randomized-output guarantees, anytime or finite-horizon guarantees, and distributional guarantees all reduce to this quotient-recovery problem. Universal exact-semantics reduction identifies admissible-output quotient recovery as the canonical object. Optimizer-quotient realizability is maximal, so quotient shape alone cannot yield a tractability frontier. Orbit gaps are the exact obstruction to classification by closure-law-invariant structural predicates. Exact classification by closure-law-invariant predicates succeeds exactly when the target is constant on closure orbits; on a closure-closed domain, equivalently, when the positive and negative orbit hulls are disjoint, in which case there is a least exact closure-invariant classifier. Across four natural candidate structural tractability criteria, a uniform pair-targeted affine witness produces same-orbit disagreements and rules out exact structural classification on the full binary pairwise domain. Because that witness class already sits inside the universal semantic framework, any universal exact-certification theory whose structural tractability proxy restricts to these finite local predicates inherits the same obstruction on that witness class. On a closure-closed domain, restricting helps only by removing orbit gaps. Without explicit margin control, arbitrarily small utility perturbations can flip relevance and sufficiency.