Completeness for categories of generalized automata
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866913840023732224 |
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| author | Boccali, Guido Laretto, Andrea Loregian, Fosco Luneia, Stefano |
| author_facet | Boccali, Guido Laretto, Andrea Loregian, Fosco Luneia, Stefano |
| contents | We present a slick proof of completeness and cocompleteness for categories of $F$-automata, where the span of maps $E\leftarrow E\otimes I \to O$ that usually defines a deterministic automaton of input $I$ and output $O$ in a monoidal category $(\mathcal K,\otimes)$ is replaced by a span $E\leftarrow F E \to O$ for a generic endofunctor $F : \mathcal K\to \mathcal K$ of a generic category $\mathcal K$: these automata exist in their `Mealy' and `Moore' version and form categories $F\text{-}\mathsf{Mly}$ and $F\text{-}\mathsf{Mre}$; such categories can be presented as strict 2-pullbacks in $\mathsf{Cat}$ and whenever $F$ is a left adjoint, both $F\text{-}\mathsf{Mly}$ and $F\text{-}\mathsf{Mre}$ admit all limits and colimits that $\mathcal K$ admits. We mechanize some of of our main results using the proof assistant Agda and the library `agda-categories`. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_03867 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Completeness for categories of generalized automata Boccali, Guido Laretto, Andrea Loregian, Fosco Luneia, Stefano Category Theory Formal Languages and Automata Theory We present a slick proof of completeness and cocompleteness for categories of $F$-automata, where the span of maps $E\leftarrow E\otimes I \to O$ that usually defines a deterministic automaton of input $I$ and output $O$ in a monoidal category $(\mathcal K,\otimes)$ is replaced by a span $E\leftarrow F E \to O$ for a generic endofunctor $F : \mathcal K\to \mathcal K$ of a generic category $\mathcal K$: these automata exist in their `Mealy' and `Moore' version and form categories $F\text{-}\mathsf{Mly}$ and $F\text{-}\mathsf{Mre}$; such categories can be presented as strict 2-pullbacks in $\mathsf{Cat}$ and whenever $F$ is a left adjoint, both $F\text{-}\mathsf{Mly}$ and $F\text{-}\mathsf{Mre}$ admit all limits and colimits that $\mathcal K$ admits. We mechanize some of of our main results using the proof assistant Agda and the library `agda-categories`. |
| title | Completeness for categories of generalized automata |
| topic | Category Theory Formal Languages and Automata Theory |
| url | https://arxiv.org/abs/2303.03867 |