Rate-Distortion Theory for Deductive Sources under Closure Fidelity

1. Problem Statement

The paper addresses the fundamental limits of lossy compression for deductive sources. Unlike classical source coding, where the source is a flat alphabet of symbols and fidelity is measured by symbol-wise equality (e.g., Hamming distortion), this work considers a source generated within a fixed deductive environment.

• Source Model: The source is a finite knowledge base ($S_O$) equipped with a fixed proof system and a deductive closure operator $Cn(\cdot)$. The random source symbol is a statement drawn from this knowledge base.
• Fidelity Criterion: Instead of requiring exact symbol reproduction, the paper proposes Closure Fidelity. A reconstruction is considered faithful if it preserves the deductive closure of the original statement. That is, the set of all logical consequences derivable from the reconstruction must match the set of consequences derivable from the original source.
• Core Question: How does the shared inference structure (the ability of the decoder to re-derive consequences) change the fundamental rate-distortion limits? Specifically, can redundant stored consequences be compressed away if the decoder can re-derive them from a smaller "core" set of premises?

2. Methodology and Framework

The author develops a rigorous information-theoretic framework based on the following components:

• Irredundant Core Decomposition:
  The source alphabet $S_O$ is decomposed into two disjoint sets based on a fixed canonical order and the proof system:
  1. Irredundant Core ($A = \text{Atom}(S_O)$): The minimal set of premises required to generate the entire closure $Cn(S_O)$. No element in $A$ can be derived from the others in $A$.
  2. Redundant Part ($J = S_O \setminus A$): Stored consequences that can be re-derived from $A$.
• Distortion Metrics:
  • Closure Distortion ($d_{Cn}$): A Jaccard-type distance measuring the difference between the closure sets of the source and reconstruction.
  • $\delta$-Constrained Distortion: A variant where the decoder is limited to a bounded number of inference steps ($\delta$). This introduces a rate-delay-distortion tradeoff.
• Key Assumptions:
  • Disjointness Assumption (Assumption 3.1): The zero-distortion reconstruction sets for distinct core elements are disjoint. This prevents "confusability" where one reconstruction symbol could satisfy the closure requirement for two different core premises.
  • Order-Robustness: Under certain conditions, the chosen core $A$ coincides with the order-free essential set, ensuring the results are intrinsic to the deductive structure rather than the arbitrary ordering of symbols.

3. Key Contributions and Results

A. Exact Zero-Distortion Rate

The paper proves that under closure fidelity, the minimum rate required for zero distortion depends only on the irredundant core, not the full source.

• Theorem 3.1: The zero-distortion rate is given by:
  $$R_{sem}(0) = P_A H(\pi_A)$$
  Where $P_A$ is the probability mass of the core $A$, and $H(\pi_A)$ is the entropy of the source distribution conditioned on $A$.
• Implication: Redundant states ($J$) are "invisible" to the rate-distortion function. They can be transmitted with zero cost because the decoder can reconstruct them from the core. This yields a strict compression gain over classical Hamming fidelity ($H(S_O)$).

B. Full Rate-Distortion Decomposition

The result extends to non-zero distortion levels.

• Theorem 4.1: The full rate-distortion function decomposes entirely into a scaled version of the rate-distortion function of the core sub-source:
  $$R_{sem}(D) = P_A \cdot R^{(A)}\left(\frac{D}{P_A}\right)$$
  This confirms that the redundant part $J$ contributes neither to the rate nor to the distortion constraint; the entire problem reduces to compressing the core $A$ under the closure distortion metric.

C. Rate-Delay-Distortion Tradeoff (Limited Inference)

The paper analyzes the scenario where the decoder has a limited inference budget (maximum $\delta$ inference steps).

• $\delta$-Irredundant Core ($A_\delta$): As $\delta$ increases, more states become "redundant" because they can be derived within $\delta$ steps.
• Theorem 5.1 & 5.2: The paper derives an exact characterization for the zero-distortion rate and the full rate-distortion surface as a function of $\delta$:
  $$R_{sem}(0, \delta) = P_\delta H(\pi_\delta)$$
  • At $\delta=0$, the theory collapses to classical symbol-wise compression ($R = H(S_O)$).
  • At $\delta \ge D_d$ (maximum derivation depth), the theory converges to the unconstrained deductive compression result ($R = P_A H(\pi_A)$).
  • This provides a continuous interpolation between classical and semantic/deductive compression.

D. Extensions and Consequences

• Source-Channel Separation: The paper establishes that the classical separation theorem holds for these structured sources. Reliable communication is possible if the deductive rate is less than the channel capacity scaled by the block length.
• Heterogeneous Receivers: The theory is extended to receivers with restricted vocabularies. The zero-distortion rate remains unchanged as long as the receiver's alphabet contains at least one valid zero-distortion representative for every element of the core.
• Converse Bounds: New Fano-type converse bounds are derived that are tighter than classical bounds because they account for the "free" nature of redundant states.

4. Significance and Impact

• Theoretical Shift: The paper moves beyond "semantic communication" as a vague architectural concept, providing a Shannon-theoretic formulation for deductive sources. It quantifies exactly how shared inference capability reduces communication rates.
• Quantifying Redundancy: It demonstrates that in knowledge-rich environments (e.g., databases, logic programming, AI knowledge bases), a significant portion of the data is structurally redundant. The "effective" information content is determined by the minimal generating set (the core), not the total stored size.
• Practical Relevance: The results offer a theoretical basis for optimizing data transmission in systems where the receiver has access to a shared knowledge base or inference engine (e.g., IoT devices with local rule sets, database synchronization, or AI agents sharing context).
• Future Directions: The paper identifies the removal of the "disjointness assumption" as a key open problem, suggesting that the general case involves graph-entropy-like structures due to potential confusability of core elements.

In summary, this work establishes that deductive redundancy is a compressible resource. By shifting the fidelity criterion from symbol equality to closure preservation, the fundamental limits of communication are governed by the entropy of the irredundant core, offering substantial rate savings proportional to the amount of stored logical consequences.