Local Grammar-Based Coding Revisited
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866909580533956608 |
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| author | Dębowski, Łukasz |
| author_facet | Dębowski, Łukasz |
| contents | In the setting of minimal local grammar-based coding, the input string is represented as a grammar with the minimal output length defined via simple symbol-by-symbol encoding. This paper discusses four contributions to this field. First, we invoke a simple harmonic bound on ranked probabilities, which reminds Zipf's law and simplifies universality proofs for minimal local grammar-based codes. Second, we refine known bounds on the vocabulary size, showing its partial power-law equivalence with mutual information and redundancy. These bounds are relevant for linking Zipf's law with the neural scaling law for large language models. Third, we develop a framework for universal codes with fixed infinite vocabularies, recasting universal coding as matching ranked patterns that are independent of empirical data. Finally, we analyze grammar-based codes with finite vocabularies being empirical rank lists, proving that that such codes are also universal. These results extend foundations of universal grammar-based coding and reaffirm previously stated connections to power laws for human language and language models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_13636 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Local Grammar-Based Coding Revisited Dębowski, Łukasz Information Theory Computation and Language 94A29 In the setting of minimal local grammar-based coding, the input string is represented as a grammar with the minimal output length defined via simple symbol-by-symbol encoding. This paper discusses four contributions to this field. First, we invoke a simple harmonic bound on ranked probabilities, which reminds Zipf's law and simplifies universality proofs for minimal local grammar-based codes. Second, we refine known bounds on the vocabulary size, showing its partial power-law equivalence with mutual information and redundancy. These bounds are relevant for linking Zipf's law with the neural scaling law for large language models. Third, we develop a framework for universal codes with fixed infinite vocabularies, recasting universal coding as matching ranked patterns that are independent of empirical data. Finally, we analyze grammar-based codes with finite vocabularies being empirical rank lists, proving that that such codes are also universal. These results extend foundations of universal grammar-based coding and reaffirm previously stated connections to power laws for human language and language models. |
| title | Local Grammar-Based Coding Revisited |
| topic | Information Theory Computation and Language 94A29 |
| url | https://arxiv.org/abs/2209.13636 |