Intuitionistic logical framework for dimensionality reduction: a constructive approach to neural network optimization

Abstract

We study dimensionality reduction as the construction of a representation accompanied by checkable reconstruction and irredundancy evidence. A representation is logically sparse when no retained predicate is definable from the others in a declared language and semantic model. This notion is distinguished from predictive usefulness, statistical independence and minimum cardinality. The framework uses intuitionistic logic and Heyting semantics to specify admissible definitions, while a sequential deletion algorithm records explicit reconstruction formulas. We prove termination, compositional preservation of the target predicate bank and language-relative irredundancy. Complete finite-algebra search additionally yields an inclusion-minimal generating subset. An executable positive-fragment implementation checks threshold formulas by canonical antichains; a separate finite Kripke implementation tests implication and non-Boolean information states. We develop proof-theoretic adequacy, categorical preservation, certified refinement, neural realization and directional robustness, with explicit conditions for optimization, stability, PAC–Bayes and distribution-shift guarantees. Experiments on three classification benchmarks use twenty paired data splits, three levels of domain-aware descriptors, independent predictive diagnostics and dimension-matched baselines. They quantify exact decoder preservation and the empirical cost of fitting directly on the reduced coordinates. AUC bounds are stated for fixed scores and finite score classes, with their statistical scope made explicit. The resulting method connects representation design with the intuitionistic neural-symbolic programme through auditable proof obligations.