Semiring-graded bidirectional processes for Haskell.

This library combines three existing constructions into a single Haskell type:

1. Semiring-graded effects — Katsumata (2014), dorchard/effect-monad.
2. Profunctor optics — Riley (2018), Clarke et al. (2024), mroman42/vitrea.
3. Categorical cybernetics — Capucci, Gavranović, Hedges, Rischel (2021).

The result: optics in Kl(Writer g) — concrete optics where each channel (forward/backward) accumulates a semiring grade. Forward grade = observation cost. Backward grade = intervention cost. The same bidirectional process, scored through different semirings, reveals different structure from both directions.

data GradedOptic g s t a b = forall c. GradedOptic
  (s -> (c, a, g))     -- forward: observe, grade the observation
  (c -> b -> (t, g))   -- backward: intervene, grade the intervention

One type. The existential c is the residual (the coend variable in the optic formula). The grade g is any semiring (Katsumata's parameter). This is a concrete optic in Kl(Writer g). Composition multiplies grades in each direction independently:

(|>>>|) :: Semiring g => GradedOptic g s t a b -> GradedOptic g a b x y -> GradedOptic g s t x y

Forward grades compose going in. Backward grades compose going out.

$ cabal run cybernetic-example

Pipeline: amount analysis |>>>| timing analysis
Observation: alice →100→ bob →95→ carol →90→ alice (round-trip)
Amount similarity: 0.9
Feedback: False (organic correction)

━━━ Semiring 1: Log Double (log-likelihood accumulation) ━━━
  Forward  (observation): 0.741
  Backward (feedback):    0.183
  Total bidirectional:    0.135

━━━ Semiring 2: GF(3) (ternary bidirectional classifier) ━━━
  Forward  (observation): Plus (1)
  Backward (feedback):    Minus (-1)
  Total bidirectional:    Minus (-1)
  → ORGANIC: feedback overrides detection

━━━ Semiring 3: Tropical (min-cost bidirectional path) ━━━
  Forward  (observation): 6.5 cost units
  Backward (feedback):    8.0 cost units
  Total bidirectional:    14.5 cost units

━━━ Semiring 4: Polynomial (bidirectional operation accounting) ━━━
  Forward  (observation): 3x³
  Backward (feedback):    x
  Total bidirectional:    3x⁴

The GF(3) result is the crux: Plus × Minus = Minus. Organic feedback overrides sybil detection in the ternary field. One semiring multiplication across the optic's backward channel.

Requires GHC ≥ 9.4.

cabal build
cabal test
cabal run bridge-example       # forward-only: three scenarios × four semirings
cabal run cybernetic-example   # bidirectional: four semirings, two directions

• Katsumata (2014), Parametric effect monads and semantics of effect systems
• Riley (2018), Categories of optics
• Clarke, Elkins, Gibbons, Loregian, Milewski, Pillmore, Román (2024), Profunctor optics: a categorical update
• Capucci, Gavranović, Hedges, Rischel (2021), Towards foundations of categorical cybernetics
• Gavranović (2024), Fundamental components of deep learning: a category-theoretic approach (PhD thesis)
• Hedges (2017), Coherence for lenses and open games

MIT