Research · Papers · Symmetry, state encodings and search gauges · ML-033
Finite-horizon Nerode minimisation of the 22-state carry transducer
Under finite-horizon Nerode minimisation, the natural 22-state carry transducer maintains 22 classes through bit 29, followed by 16, four, and one terminal class.
Published 2026-08-29
For everyone
Plain summary
A carry calculation cannot shrink by merging partial histories that share future behavior until the very end. Testing a 22-state carry transducer with 2,048 raw symbols projected onto 128 actions shows that all 22 Nerode equivalence classes persist through bit 29. The count drops to 16, four, and one only at the terminal stages. At most 128 final occurrences can be merged—an optimistic upper bound that yields no significant savings. This closes ordinary full-alphabet recoding as a 1,000-row route. Restricted alphabets, larger relations, and defect automata remain open. The register records no prior-art position.
Result
Under finite-horizon Nerode minimisation, the natural 22-state carry transducer maintains 22 classes through bit 29, followed by 16, four, and one terminal class. Merging is limited to an optimistic ceiling of 128 terminal occurrences, which yields no automatic row saving. Ordinary full-alphabet deterministic recoding is closed as a 1,000-row route.
Setting and definitions
Transducer T has 22 states, 2,048 raw symbols, and 128 projected actions. Nerode classes partition histories by continuation equivalence. A terminal occurrence is an occurrence at the final stage.
Method
Exact finite-horizon Nerode minimisation was replayed independently following the 2,048-to-128 projection, confirming zero right-congruence violations and the 22, 16, 4, 1 profile.
Discussion
Scope is limited to ordinary full-alphabet deterministic recoding on the recorded 1,000-row route. The 128-occurrence ceiling provides no row reduction. Restricted alphabets, enlarged relations, and defect automata remain open. Corrections: NONE. The register records no prior-art position.
For everyone — the takeaway
What this means
The carry machine keeps almost all of its distinct states right up to the final bits. Recoding can only save work when two different histories lead to identical future behavior, but here those mergers happen only at the very end. Even with an optimistic ceiling of 128 merged occurrences, the standard full-alphabet approach cannot deliver the hoped-for 1,000-row shortcut. Alternative designs using smaller alphabets, larger relation sets, or defect tracking remain unaddressed.
Register references
- Entry: ML-033
- Receipt: CONT
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Changelog
Last reviewed 2026-08-29
- 2026-08-29Published on this site.