Rate-Distortion Theory for Deductive Sources under Closure Fidelity
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914611163299840 |
|---|---|
| author | Xu, Jianfeng |
| author_facet | Xu, Jianfeng |
| contents | We study lossy compression of a finite statement source generated in a fixed deductive environment. The source symbols are statements in a knowledge base endowed with a shared proof system, and reconstruction fidelity is measured by preservation of deductive closure rather than by symbolwise equality. Fixing the proof system and a canonical scan order yields a decomposition of the source alphabet into an irredundant core and redundant stored consequences. At zero distortion, each core symbol induces a set of distortion-free reconstructions. In the nonconfusable (disjoint-core) regime, we show that the minimum zero-distortion rate equals the source mass of the core times the entropy of the source conditioned on that core. In the general confusable-core regime, we characterise the exact zero-distortion rate via a hypergraph-entropy quantity induced by jointly realisable core subsets, with a reduction to Korner-style graph entropy under a natural pairwise realisability condition. For reconstruction alphabets contained in the deductive closure of the source knowledge base, we further prove that the full rate-distortion function depends only on the core, so redundant states are invisible to both rate and distortion. Finally, when the decoder is limited to a bounded inference-depth budget (a bounded number of iterations of the immediate-consequence operator), we obtain an exact rate-depth-distortion characterisation. Under an additional order-robustness assumption identifying the chosen core with the order-free essential set, this characterisation interpolates between classical symbolwise compression and unconstrained deductive compression. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15698 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rate-Distortion Theory for Deductive Sources under Closure Fidelity Xu, Jianfeng Information Theory Logic in Computer Science We study lossy compression of a finite statement source generated in a fixed deductive environment. The source symbols are statements in a knowledge base endowed with a shared proof system, and reconstruction fidelity is measured by preservation of deductive closure rather than by symbolwise equality. Fixing the proof system and a canonical scan order yields a decomposition of the source alphabet into an irredundant core and redundant stored consequences. At zero distortion, each core symbol induces a set of distortion-free reconstructions. In the nonconfusable (disjoint-core) regime, we show that the minimum zero-distortion rate equals the source mass of the core times the entropy of the source conditioned on that core. In the general confusable-core regime, we characterise the exact zero-distortion rate via a hypergraph-entropy quantity induced by jointly realisable core subsets, with a reduction to Korner-style graph entropy under a natural pairwise realisability condition. For reconstruction alphabets contained in the deductive closure of the source knowledge base, we further prove that the full rate-distortion function depends only on the core, so redundant states are invisible to both rate and distortion. Finally, when the decoder is limited to a bounded inference-depth budget (a bounded number of iterations of the immediate-consequence operator), we obtain an exact rate-depth-distortion characterisation. Under an additional order-robustness assumption identifying the chosen core with the order-free essential set, this characterisation interpolates between classical symbolwise compression and unconstrained deductive compression. |
| title | Rate-Distortion Theory for Deductive Sources under Closure Fidelity |
| topic | Information Theory Logic in Computer Science |
| url | https://arxiv.org/abs/2604.15698 |