Substitutional Complexity: A Language-Relative Measure of Representational Reduction

This preprint introduces substitutional complexity as a language-relative measure of representational reduction. Instead of measuring only the length of a final description, the framework measures the number or weighted cost of legitimate substitutions required to reduce a complete relational description of a system to a representative symbol within a chosen language. The paper defines unweighted and weighted substitutional complexity, introduces a language-relative representation predicate, and discusses basic properties such as monotonicity under language extension and the role of trivializing languages. It also distinguishes substitutional complexity from Kolmogorov complexity, minimum description length, term rewriting, proof complexity, and classical computational complexity. As a motivating application, the paper considers the verification–generation distinction familiar from the P versus NP problem. No claim is made that the classical P versus NP problem is resolved. Rather, the framework suggests that the gap between verification and generation may be studied as a language-relative substitutional gap.