We present a slick proof of completeness and cocompleteness for categories of F-automata, where the span of maps E ←d E⊗ I s→ O that usually defines a deterministic automaton of input I and output O in a monoidal category (K,⊗) is replaced by a span E ← FE → O for a generic endofunctor F : K → K of a generic category K: these automata exist in their "Mealy" and "Moore" version and form categories F-Mly and F-Mre; such categories can be presented as strict 2-pullbacks in Cat and whenever F is a left adjoint, both F-Mly and F-Mre admit all limits and colimits that K admits. We mechanize our main results using the proof assistant Agda and the library https://github.com/agda/agda-categories.
@InProceedings{boccali_et_al:LIPIcs.CALCO.2023.20, author = {Boccali, Guido and Laretto, Andrea and Loregian, Fosco and Luneia, Stefano}, title = {{Completeness for Categories of Generalized Automata}}, booktitle = {10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023)}, pages = {20:1--20:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-287-7}, ISSN = {1868-8969}, year = {2023}, volume = {270}, editor = {Baldan, Paolo and de Paiva, Valeria}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2023.20}, URN = {urn:nbn:de:0030-drops-188174}, doi = {10.4230/LIPIcs.CALCO.2023.20}, annote = {Keywords: Deterministic automata, Moore machines, Mealy machines, coalgebras, cocomplete category} }
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