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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.07690 |
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| _version_ | 1866915926380642304 |
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| author | Wu, Jiu Hui Tian, Hua Yuan, Mengqi Zhou, Kejiang |
| author_facet | Wu, Jiu Hui Tian, Hua Yuan, Mengqi Zhou, Kejiang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_07690 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bilattice-Catastrophe Isomorphism for Four-Valued Logic in Digital Systems Wu, Jiu Hui Tian, Hua Yuan, Mengqi Zhou, Kejiang Disordered Systems and Neural Networks 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. |
| title | Bilattice-Catastrophe Isomorphism for Four-Valued Logic in Digital Systems |
| topic | Disordered Systems and Neural Networks |
| url | https://arxiv.org/abs/2604.07690 |