Additive Cellular Automata Graded-Monadically
summary
Shows that cellular automata are graded comonadic by grading with finite neighborhoods (preordered monoid of subsets) and models uniform continuity via locally graded categories. Proves additive cellular automata are simultaneously graded comonadic and graded monadic by exhibiting an adjunction producing a graded monad (the monad of formal polynomials) right-adjoint to the graded co-writer comonad. Builds the bijection between graded coKleisli and graded Kleisli maps, gives resource-aware types for local/global rules, and demonstrates the constructions compose and recover the usual CA semantics.
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chapters (8)
- 0:00Cellular automata & graded monads
- 3:18Cellular automata rules
- 10:26Graded comonads & resource computation
- 17:39Locally graded coalgebra
- 20:27Additive cellular automata duality
- 23:47Graded monad-comonad adjunctions
- 29:26Additive cellular automata as monadic
- 31:55Summary of results
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