Bilattice-Catastrophe Isomorphism for Four-Valued Logic in Digital Systems

TL;DR

This paper establishes a categorical isomorphism linking four-valued logic, bilattice structures, and catastrophe theory, providing a foundational framework for understanding robustness and discontinuities in digital systems.

Contribution

It rigorously proves the bilattice-catastrophe isomorphism theorem and demonstrates the mathematical necessity of certain variables in modeling continuous-discrete interfaces.

Findings

Proves the bilattice-catastrophe isomorphism theorem.

Shows the four-valued algebra FOUR is minimal for describing interfaces.

Identifies X and Z as topological invariants of discretized systems.

Abstract

Belnap's four-valued logic, distinguished by its inherent bilattice structure, provides a natural algebraic bridge between discrete Four-valued logic (4VL) in circuit and continuous catastrophe theory (CT). Building on the rigorous verification of the bilattice-catastrophe isomorphism theorem, we establish a categorical correspondence spanning the catastrophe category, interlaced bilattice category, and 4VL category, with the cusp catastrophe emerging as the canonical CT counterpart to 4VL.This unification provides a foundational framework for explaining 4VL's robustness. Crucially, we demonstrate that the four-valued algebra FOUR is the minimal complete algebraic structure capable of describing continuous-discrete interfaces with involution symmetry. Unlike the empirical adoption of X and Z in engineering practice, our work reveals their mathematical necessity: X and Z are topological invariants of discretized continuous dynamical systems, encoding fundamental properties of catastrophe-induced discontinuities. The work enables cross-disciplinary extensions to uncertainty propagation, complex system modeling, and fault-tolerant design.