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Derivation Depth as an Information Metric: Axioms, Coding Theorems, and Storage--Computation Tradeoffs

Jianfeng Xu

TL;DR

This work defines and proves the computability of derivation depth-a computable metric of the reasoning effort needed to answer a query based on a given set of premises, and forms optimal cache allocation as a mathematical optimization problem solvable with approximation guarantees.

Abstract

We introduce derivation depth-a computable metric of the reasoning effort needed to answer a query based on a given set of premises. We model information as a two-layered structure linking abstract knowledge with physical carriers, and separate essential core facts from operational shortcuts. For any finite premise base, we define and prove the computability of derivation depth. By encoding reasoning traces and applying information-theoretic incompressibility arguments, we establish fundamental bounds linking depth to the descriptive complexity of queries. For frequently asked, information-rich queries, the minimal description length grows proportionally to depth times the logarithm of the knowledge base size. This leads to a practical storage-computation tradeoff: queries accessed beyond a critical threshold become cheaper to cache than recompute. We formulate optimal cache allocation as a mathematical optimization problem solvable with approximation guarantees and extend the framework to handle noisy or incomplete knowledge bases.

Derivation Depth as an Information Metric: Axioms, Coding Theorems, and Storage--Computation Tradeoffs

TL;DR

This work defines and proves the computability of derivation depth-a computable metric of the reasoning effort needed to answer a query based on a given set of premises, and forms optimal cache allocation as a mathematical optimization problem solvable with approximation guarantees.

Abstract

We introduce derivation depth-a computable metric of the reasoning effort needed to answer a query based on a given set of premises. We model information as a two-layered structure linking abstract knowledge with physical carriers, and separate essential core facts from operational shortcuts. For any finite premise base, we define and prove the computability of derivation depth. By encoding reasoning traces and applying information-theoretic incompressibility arguments, we establish fundamental bounds linking depth to the descriptive complexity of queries. For frequently asked, information-rich queries, the minimal description length grows proportionally to depth times the logarithm of the knowledge base size. This leads to a practical storage-computation tradeoff: queries accessed beyond a critical threshold become cheaper to cache than recompute. We formulate optimal cache allocation as a mathematical optimization problem solvable with approximation guarantees and extend the framework to handle noisy or incomplete knowledge bases.
Paper Structure (67 sections, 35 theorems, 92 equations, 2 algorithms)

This paper contains 67 sections, 35 theorems, 92 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem setting and motivating tension
  3. Related work and conceptual positioning
  4. Logical and semantic foundations (effective substrates).
  5. Information measures: Shannon entropy vs. algorithmic information.
  6. Time--space tradeoffs and caching/view materialization.
  7. Submodularity and approximation for system-wide allocation.
  8. Large-scale semantic systems and retrieval augmentation (motivation).
  9. Research approach: from axioms to coding theorems to optimization
  10. Axiomatize information instances and effective semantics
  11. Canonicalize intrinsic content and define depth relative to available premises
  12. Prove depth-as-information theorems via derivation trace encodings
  13. Turn information bounds into storage--computation tradeoffs and allocation principles
  14. Main contributions
  15. Organization
  16. ...and 52 more sections

Key Result

Lemma 2.1

Let $\mathcal{L}=\mathrm{FO(LFP)}$ be the underlying logical system over finite relational structures. For any state set $S(X,T)$ expressible in $\mathcal{L}$ over a sub-signature $\Sigma=\{R_1,\ldots,R_n\}$, and any arity-matching $\Sigma'=\{R'_1,\ldots,R'_n\}$, there exists $S'(Y,T')$ expressible

Theorems & Definitions (121)

  • Definition 2.1: Expressible and effectively representable state sets
  • Remark 2.1: On decidability of $\text{Cn}(\cdot)$
  • Definition 2.2: Computable enabling (realization) mechanism
  • Definition 2.3: Information instance
  • Definition 2.4: Compositional interpretation between sub-signatures
  • Definition 2.5: Synonymous state sets
  • Remark 2.2: Relation to interpretability
  • Definition 2.6: Ideal information
  • Definition 2.7: Noisy semantic base (set perturbation)
  • Lemma 2.1: Semantic transferability under signature isomorphism
  • ...and 111 more