{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T12:19:00Z","timestamp":1769084340959,"version":"3.49.0"},"reference-count":32,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,6]],"date-time":"2025-01-06T00:00:00Z","timestamp":1736121600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"ARO","award":["W911NF-12-1-0234-0"],"award-info":[{"award-number":["W911NF-12-1-0234-0"]}]},{"name":"NSF VIGRE","award":["W911NF-12-1-0234-0"],"award-info":[{"award-number":["W911NF-12-1-0234-0"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>\u03f5-Machines are minimal, unifilar presentations of stationary stochastic processes. They were originally defined in the history machine sense as hidden Markov models whose states are the equivalence classes of infinite pasts with the same probability distribution over futures. In analyzing synchronization though, an alternative generator definition was given as follows: unifilar, edge-emitting hidden Markov models with probabilistically distinct states. The key difference is that history \u03f5-machines are defined by a process, whereas generator \u03f5-machines define a process. We show here that these two definitions are equivalent in the finite-state case.<\/jats:p>","DOI":"10.3390\/sym17010078","type":"journal-article","created":{"date-parts":[[2025,1,6]],"date-time":"2025-01-06T10:40:52Z","timestamp":1736160052000},"page":"78","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Equivalence of History and Generator \u03f5-Machines"],"prefix":"10.3390","volume":"17","author":[{"given":"Nicholas F.","family":"Travers","sequence":"first","affiliation":[{"name":"Complexity Sciences Center, University of California at Davis, One Shields Avenue, Davis, CA 95616, USA"},{"name":"Mathematics Department, University of California at Davis, One Shields Avenue, Davis, CA 95616, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4466-5410","authenticated-orcid":false,"given":"James P.","family":"Crutchfield","sequence":"additional","affiliation":[{"name":"Complexity Sciences Center, University of California at Davis, One Shields Avenue, Davis, CA 95616, USA"},{"name":"Mathematics Department, University of California at Davis, One Shields Avenue, Davis, CA 95616, USA"},{"name":"Physics Department, University of California at Davis, One Shields Avenue, Davis, CA 95616, USA"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1103\/PhysRevLett.63.105","article-title":"Inferring statistical complexity","volume":"63","author":"Crutchfield","year":"1989","journal-title":"Phys. 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