Accelerating Fuzzy Formal Concept Analysis via Thermometer-Coded Bit-Slice Transformations and Parallel Search

Exhaustive concept enumeration in Fuzzy Formal Concept Analysis (FFCA) suffers from severe scalability bottlenecks due to repetitive, pointwise scalar evaluations over the unit interval during search space traversal. To address this, we introduce a structural shift from traditional scalar operations to a tensor-domain algebraic formulation. We reformulate the foundational residuated lattice structure ⟨L, ∧, ∨, ⊗, →, 0, 1⟩ by mapping the monoidal minimum t-norm ⊗ and its residuated implication → into a coordinate-free multi-layer Boolean lattice embedding across an orthogonal rank space, formally proving that all residuated lattice axioms remain structurally invariant under this higher-dimensional embedding. Specifically, the work introduces an algebraic embedding ϕ that maps finite-chain minimum-based residuated lattices into a coordinate-free multi-layer Boolean lattice structure, formally proving that the fundamental residuated lattice axioms remain structurally invariant under this higher-dimensional tensor mapping, enabling fuzzy derivation and closure operations to reduce completely to progressive implication cascades and word-level bitwise conjunctions. Building upon this theoretical foundation, we present Minimum Thermometer-Coded Enumeration (MTCE), a framework that encodes finite-chain minimum-based fuzzy formal concepts into nested thermometer layers while strictly preserving the original object-attribute dimensions without context expansion. Under this representation, fuzzy derivation, closure, and dual rank-domain canonicity checks reduce entirely to layer-wise Boolean conjunctions and progressive implication cascades via word-level bitwise operations. To demonstrate the practical viability of this algebraic formulation, we develop P^MTCE, a multi-threaded, bit-parallel concurrent implementation. Empirical evaluation across benchmark datasets demonstrates order-of-magnitude acceleration over sequential minimum-based CbO baselines (e.g., reducing Iris processing time from 99.1s to 1.0s). These results confirm that the proposed tensor-domain formulation establishes both a rigorous theoretical bridge and a highly efficient computational foundation for finite-chain fuzzy formal concept mining on modern multi-core architectures.