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Stream: learning: questions

Topic: ✔ Definition of Categorical Automaton on Planet Physics.

Ignat Insarov (Jan 25 2023 at 17:50):

Hello!

I have found this definition of Categorical Automaton on the Planet Physics archive: <https://en.wikiversity.org/wiki/PlanetPhysics/Category_of_Automata>. Their markup is broken, but I repaired it: CategoryOfAutomata.pdf.

They say an automaton is the commuting square o ∘ δ = λ ∘ t. I understand the δ arrow, and I can accept the λ arrow, but what are the o and t arrows? I tried to find anything similar in literature but I have not. The o could be what Michael Arbib and Ernest Manes call «output map» in Machines in a Category…

Please help!

Spencer Breiner (Jan 25 2023 at 18:08):

Good question. The map on the left $t$ seems to be $\delta$ paired with the projection $(i,s) \mapsto (\delta(i,s),s)$.

The other map, $o$ is unspecified, and in general you can't derive it from what is given. If you're not tied to this definition, I'd suggest using Mealy machines instead:
$o: S \to O \hspace{3ex} u:S\times I \to S$.

Alternatively, you could specialize a class of mealy machines where states are of the form $S\times S$, and then use $\lambda$ as your output function under these restrictions.

Ignat Insarov (Jan 27 2023 at 20:00):

Thank you!

Notification Bot (Jan 27 2023 at 20:00):

Ignat Insarov has marked this topic as resolved.