{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,6,21]],"date-time":"2023-06-21T04:07:54Z","timestamp":1687320474339},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2022,3,1]],"date-time":"2022-03-01T00:00:00Z","timestamp":1646092800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"We formalise the undecidability of solvability of Diophantine equations, i.e.\npolynomial equations over natural numbers, in Coq's constructive type theory.\nTo do so, we give the first full mechanisation of the\nDavis-Putnam-Robinson-Matiyasevich theorem, stating that every recursively\nenumerable problem -- in our case by a Minsky machine -- is Diophantine. We\nobtain an elegant and comprehensible proof by using a synthetic approach to\ncomputability and by introducing Conway's FRACTRAN language as intermediate\nlayer. Additionally, we prove the reverse direction and show that every\nDiophantine relation is recognisable by $\\mu$-recursive functions and give a\ncertified compiler from $\\mu$-recursive functions to Minsky machines.<\/jats:p>","DOI":"10.46298\/lmcs-18(1:35)2022","type":"journal-article","created":{"date-parts":[[2022,3,3]],"date-time":"2022-03-03T17:40:33Z","timestamp":1646329233000},"source":"Crossref","is-referenced-by-count":0,"title":["Hilbert's Tenth Problem in Coq (Extended Version)"],"prefix":"10.46298","volume":"Volume 18, Issue 1","author":[{"given":"Dominique","family":"Larchey-Wendling","sequence":"first","affiliation":[]},{"given":"Yannick","family":"Forster","sequence":"additional","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2022,3,1]]},"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/lmcs.episciences.org\/9153\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/lmcs.episciences.org\/9153\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,20]],"date-time":"2023-06-20T20:17:42Z","timestamp":1687292262000},"score":1,"resource":{"primary":{"URL":"https:\/\/lmcs.episciences.org\/6195"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,1]]},"references-count":0,"URL":"https:\/\/doi.org\/10.46298\/lmcs-18(1:35)2022","relation":{"has-preprint":[{"id-type":"arxiv","id":"2003.04604v3","asserted-by":"subject"},{"id-type":"arxiv","id":"2003.04604v2","asserted-by":"subject"},{"id-type":"arxiv","id":"2003.04604v1","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"2003.04604","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.2003.04604","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"value":"1860-5974","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,3,1]]}}}