Bridging Shannon and Turing: A Formal Entropy-Complexity Equivalence over Symbolic Structures

We develop a formal correspondence between Shannon entropy and Kolmogorov complexity in
structured symbolic domains governed by canonical rewrite grammars. We analyze how the expected
Kolmogorov complexity of syntactic forms relates to the Shannon entropy of their corresponding
irreducible canonical forms (semantics). We prove that these two quantities are linked precisely by
a conditional entropy term that measures the average ambiguity of the syntactic representations
under the grammar. We introduce the notion of symbolic degeneracy, ∆G(x), which quantifies
the structural redundancy of preimages and serves as an upper bound on this conditional entropy.
Our results clarify how grammatical structure impacts encoding costs and establish an equivalence
principle: when the conditional entropy is bounded, the complexity of syntax coincides with the
entropy of semantics up to an additive constant. We discuss implications for structured compression,
learning, and conceptual frameworks for structural cryptography.