huffmanCoding
Huffman coding is defined as the optimal prefix-free scheme achieving average code length L at most one bit above the source entropy H. Researchers deriving compression bounds from J-cost in Recognition Science cite this when linking Shannon limits to minimized representation cost. The entry is a direct string assignment with notes on arithmetic coding approaching the entropy bound as message length grows.
claimThe optimal prefix-free code for a discrete source satisfies $L ≤ H + 1$, where $L$ is the average codeword length and $H$ is the Shannon entropy.
background
The module INFO-003 derives data compression limits from J-cost. J-cost is the function satisfying the Recognition Composition Law, with the shifted cost $H(x) = J(x) + 1$ obeying the d'Alembert equation $H(xy) + H(x/y) = 2 H(x) H(y)$. Shannon entropy is $H(X) = -∑ p(x) log₂ p(x)$, which sets the lower bound on average code length. In Recognition Science, this bound is the minimum J-cost for faithful message representation, and compression is J-cost minimization.
proof idea
This is a definition that directly assigns the descriptive string outlining optimality for symbol-by-symbol coding and the bound $L ≤ H + 1$, followed by a comment block on arithmetic coding convergence to entropy.
why it matters in Recognition Science
This definition anchors the compression discussion in the module by connecting Shannon entropy to J-cost minimization. It supports the reinterpretation of the source coding theorem inside the Recognition framework, where entropy equals minimum J-cost. The entry sits within the information domain and draws on the cost algebra and recognition structures imported from upstream.
scope and limits
- Does not prove the bound $L ≤ H + 1$ from the Recognition Composition Law.
- Does not implement the coding algorithm or specify an alphabet.
- Does not derive entropy as a J-cost minimum.
- Does not address lossy compression or continuous sources.
formal statement (Lean)
184def huffmanCoding : String :=
proof body
Definition body.
185 "Optimal prefix-free code, L ≤ H + 1"
186
187/-- Arithmetic coding:
188 - Near-optimal for any distribution
189 - L → H as message length → ∞
190 - Encodes message as a single number -/