[Autoplectic]

Summary

Adds capacity bounds for finite ε-transducers. A finite ε-transducer is a unifilar finite-state channel: (s, x, y) determines the next causal state. The bounds cover both the feedback and no-feedback settings. The PR also adds two classic channels with memory as examples.

Feedback capacity of finite-state channels is not computable in general (Grigorescu et al. 2024), so every method returns a CapacityBounds(lower, upper, ...) bracket rather than a single number.

Changes

• Examples (sofic/examples/processes.py): trapdoor() / trapdoor_et() and ising_channel() / ising_channel_et(). Each starts in state 0 or 1 with probability 1/2. _transducer gains an initials= keyword so a factory can declare several initial states.
• sofic.generators.capacity (new package):
  • Feedback:
    • Q-graph upper bound max I(X,S;Y|Q), solved as a convex program over the stationary joint (Sabag–Permuter–Pfister 2017; Sabag–Huleihel–Permuter 2020).
    • Certified lower bound from optimized graph-based encoders. Any input that depends on the current causal state and Q-graph node is a valid feedback input, and its directed-information rate is bounded below with block entropy bounds. When the input is also BCJR-invariant, the single-letter rate is credited as well.
    • Q-graph duality upper bound, which for unifilar channels is a finite MDP (Huleihel et al. 2025).
    • Belief-state value iteration (Permuter et al. 2008; Yang–Kavčić–Tatikonda 2005), returned with certified=False.
  • No feedback:
    • Lower bound from an optimized order-k Markov input. The rate is h(Y) - E H(Y|X,S), with h(Y) bounded below by block entropies. This is the objective of the generalized Blahut–Arimoto algorithm, optimized directly here.
    • Q-graph duality upper bound as a dynamic program over P(S,Q|x^t) (Huleihel et al. 2021). The value function is convex in that belief, so linear interpolation on the grid over-estimates it and the gridded bound stays valid at any resolution.
  • Shared pieces: a QGraph class, and a simplex grid with Freudenthal-triangulation interpolation plus relative value iteration.
• API: EpsilonTransducer.feedback_capacity(method=...) and EpsilonTransducer.capacity(method=...).
• Optional extra: capacity = ["cvxpy"]. Without cvxpy the convex step uses scipy SLSQP, and solutions from either solver are polished with SLSQP.
• Docs: new page docs/generators/channel_capacity.rst, 10 new bib entries (DOIs checked on Crossref), and a CHANGELOG entry.

Validation (tests/test_channel_capacity.py)

Channel	Feedback bracket	No-feedback bracket
Binary memoryless	equals the Blahut–Arimoto capacity from dit (agrees to about 1e-6)	equals the same value
Trapdoor	0.694236 – 0.694242; the true value is log2(φ) = 0.694242	0.5727 – 0.5936; best published is 0.572 – 0.584
Ising	0.5755215 – 0.5755216; the published value is 0.575522	—

Hypothesis tests check, on random 2-state channels, that every lower bound is at most the matching upper bound, that the no-feedback lower bound is at most the feedback upper bound, and that every bound lies in [0, 1]. DP-heavy tests are marked slow.

Notes

• Markov Q-graphs (nodes are the last k outputs) are not tight for the trapdoor or Ising channels. The tight graphs, [[0,1],[0,2],[1,2]] for the trapdoor and [[1,2],[0,2],[0,3],[0,2]] for Ising, were found by exhaustive search and must be passed explicitly. The docs show how.
• The no-feedback bounds require every input to be allowed in every state.
• Q-graph upper bounds are accurate to the optimizer's tolerance (about 1e-12 in the tests).

Test plan

• uv run pytest (full default suite): 2039 passed
• uv run pytest tests/test_channel_capacity.py -m "" (includes slow): 23 passed
• ruff check, ruff format --check, and ty check clean on the new code
• Sphinx build: citations resolve and the ipython doctest on the new page passes