{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T13:58:26Z","timestamp":1780408706719,"version":"3.54.1"},"reference-count":18,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,17]],"date-time":"2024-12-17T00:00:00Z","timestamp":1734393600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NSF VIGRE graduate fellowship","award":["9060-000709"],"award-info":[{"award-number":["9060-000709"]}]},{"name":"NSF VIGRE graduate fellowship","award":["W911NF-12-1-0234-0"],"award-info":[{"award-number":["W911NF-12-1-0234-0"]}]},{"name":"NSF REU internship at the Santa Fe Institute","award":["9060-000709"],"award-info":[{"award-number":["9060-000709"]}]},{"name":"NSF REU internship at the Santa Fe Institute","award":["W911NF-12-1-0234-0"],"award-info":[{"award-number":["W911NF-12-1-0234-0"]}]},{"name":"DOE GAANN graduate fellowship","award":["9060-000709"],"award-info":[{"award-number":["9060-000709"]}]},{"name":"DOE GAANN graduate fellowship","award":["W911NF-12-1-0234-0"],"award-info":[{"award-number":["W911NF-12-1-0234-0"]}]},{"DOI":"10.13039\/100000185","name":"Defense Advanced Research Projects Agency (DARPA) Physical Intelligence Subcontract","doi-asserted-by":"publisher","award":["9060-000709"],"award-info":[{"award-number":["9060-000709"]}],"id":[{"id":"10.13039\/100000185","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000185","name":"Defense Advanced Research Projects Agency (DARPA) Physical Intelligence Subcontract","doi-asserted-by":"publisher","award":["W911NF-12-1-0234-0"],"award-info":[{"award-number":["W911NF-12-1-0234-0"]}],"id":[{"id":"10.13039\/100000185","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000183","name":"ARO","doi-asserted-by":"publisher","award":["9060-000709"],"award-info":[{"award-number":["9060-000709"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000183","name":"ARO","doi-asserted-by":"publisher","award":["W911NF-12-1-0234-0"],"award-info":[{"award-number":["W911NF-12-1-0234-0"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We show how to efficiently enumerate a class of finite-memory stochastic processes using the causal representation of \u03f5-machines. We characterize \u03f5-machines in the language of automata theory and adapt a recent algorithm for generating accessible deterministic finite automata, pruning this over-large class down to that of \u03f5-machines. 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(1965). Finite Markov Chains, D. Van Nostrand Company."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Kemeny, J.G., Snell, J.L., Knapp, A.W., and Griffeath, D.S. (1976). Denumerable Markov Chains, Springer.","DOI":"10.1007\/978-1-4684-9455-6"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1109\/MASSP.1986.1165342","article-title":"An introduction to hidden Markov models","volume":"3","author":"Rabiner","year":"1986","journal-title":"IEEE Assp. Mag."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1518","DOI":"10.1109\/TIT.2002.1003838","article-title":"Hidden Markov processes","volume":"48","author":"Ephraim","year":"2002","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1038\/nphys2190","article-title":"Between Order and Chaos","volume":"8","author":"Crutchfield","year":"2012","journal-title":"Nat. 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Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.tcs.2007.04.001","article-title":"Enumeration and random generation of accessible automata","volume":"381","author":"Bassino","year":"2007","journal-title":"Theor. Comput. Sci."},{"key":"ref_16","unstructured":"Hopcroft, J.E., and Ullman, J.D. (1979). Introduction to Automata Theory, Languages, and Computation, Addison-Wesley, Reading."},{"key":"ref_17","unstructured":"Johnson, B.D., Crutchfield, J.P., Ellison, C.J., and McTague, C.S. (2024, October 16). A181554: The Number of n-State Topological Epsilon-Machines Over a Binary Alphabet. On-Line Encyclopedia of Integer Sequences. Available online: http:\/\/oeis.org\/A181554."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"042135","DOI":"10.1103\/PhysRevE.89.042135","article-title":"Many roads to synchrony: Natural time scales and their algorithms","volume":"89","author":"James","year":"2014","journal-title":"Phys. Rev. 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