[Autoplectic]

Summary

Adds sofic.algebra, a new subpackage for Krohn-Rhodes theory on any deterministic presentation in sofic: DFAs, unifilar automata, unifilar HMMs / ε-machines, and sofic shifts.

Core

• TransformationSemigroup: enumerated only when needed, a shortest word for every element, a size cap, and a SINK state to complete partial maps.
• Green's relations, egg boxes, maximal subgroups and Schützenberger groups (as sympy groups), the minimal ideal, and synchronization.
• syntactic_monoid, is_star_free, and language varieties: piecewise testable, locally testable, definite, group, and commutative languages.
• holonomy_decomposition (Egri-Nagy & Nehaniv), computed from the generators alone. Its CascadeProduct checks that it emulates the original machine.
• krohn_rhodes_complexity, returning ComplexityBounds(lower, upper, exact), and prime_decomposition.
• transition_semigroup() / krohn_rhodes_complexity() on automata, MealyHMM, and SoficShift. New shift invariants factor_syntactic_monoid and factor_krohn_rhodes_complexity. Wheeler-presentation failures now say whether the language is star-free.

DeDeo, "Effective Theories for Circuits and Automata" (Chaos 2011)

• effective_theory(k) / CascadeProduct.truncate(k): coarse-grained machines on overlapping superstates. This reproduces Fig. 4b, using Fig. 4a rebuilt from the arXiv TeX source.
• independent_random_maps (Eq. 2), coupled_random_maps (correlated failures), and random_maps: probabilistic automata rewritten as random choices of deterministic maps.
• element_walk and classify_elements, which reproduce Figs. 5, 6, and 8. In Fig. 8 the walk on A_4 has period 3, so it is uniform only when averaged over a period.
• walk_information: splits the information lost about the initial state into irreversibility and dissipation. This split is original and uncited.

Other

• holonomy_information: splits C_mu across the holonomy levels. Original, uncited.
• projective_semigroup / mixed_state_semigroup: belief-map semigroups of HMMs.

Limitations

• Complexity lower bound: 0 for aperiodic semigroups, otherwise at least 1. It is raised to k - 1 when the full transformation monoid on k points appears acting on a k-element image set. The full transformation monoid itself is recognized from its generators, without enumerating it. The Rhodes-Tilson chain bounds are not implemented.
• Nonabelian simple factors are reported by order only.
• projective_semigroup enumerates up to a cap. When it closes, that proves finiteness; hitting the cap is inconclusive.

Test plan

• uv run pytest: 2169 passed, 17 skipped
• New tests in tests/test_algebra_*.py: hypothesis properties (cascade emulation, truncation commutes with every input, Green's classes partition, the information split adds up) and known values (T_n, Z_n, Egri-Nagy & Nehaniv Examples 7.1 and 7.2, DeDeo figures)
• One-off check: holonomy emulates 300 random semigroups on 6-8 states
• ruff check, ruff format --check, ty (clean on new code), module doctests
• Serial Sphinx build with ipython doctests and no warnings (the -j auto build hits an unrelated EOFError)