Document Zbl 0262.94037

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Syst. Theory</a> <a href="/?q=in%3A177303" title="Articles in this Issue">7 (1973), 353-359 (1974)</a>. </div> <div class="abstract">Es ist möglich, einen Satz von <span class="zbmathjax-textit">P. Turakainen</span> [Proc. Am. Math. Soc. 21, 303–309 (1969; <a href="/0184.02802">Zbl 0184.02802</a>)] so zu interpretieren: Eine formale Sprache $S$ über dem Alphabet $X$ ist stochastisch im Sinne von Rabin dann und nur dann, wenn es eine rationale formale Potenzreihe $r$ in den nichtkommutativen Variablen $x\in X$ und mit reellen Koeffizienten gibt, so daß<span class="zbmathjax-space"></span>ein Wort zu $S$ gehört, wenn sein Koeffizient in $r$ positiv ist. So sieht man, daß<span class="zbmathjax-space"></span>das Studium der stochastischen Sprachen eine einfache Verallgemeinerung der klassischen Analysis ist, wo es üblich ist, die Folge der Zeichen der Taylorschen Entwicklung einer analytischen Funktion zu studieren (siehe z.B. <span class="zbmathjax-textit">L. Bieberbach</span> [Analytische Fortsetzung. Berlin etc.: Springer-Verlag (1955; <a href="/0064.06902">Zbl 0064.06902</a>)]). Diese Analogie erlaubt dem Verf. die folgenden Fragen zu lösen: Das Komplement einer stochastischen Sprache über einen Buchstaben ist stochastisch. Weder die Vereinigung noch der Durchschnitt zweier stochastischer Sprachen sind notwendig stochastisch. Die Beweise benutzen die analytische Zahlentheorie. <br class="zbmathjax-paragraph">Als Folgesatz wird gezeigt, daß<span class="zbmathjax-space"></span>Produkte und Homomorphismen aus der Menge der stochastischen Sprachen herausführen. Das war schon bekannt (<span class="zbmathjax-textit">P. Turakainen</span> [Inf. Control 18, 253–256 (1971; <a href="/0218.68013">Zbl 0218.68013</a>)] und <span class="zbmathjax-textit">M. Nasu</span> und <span class="zbmathjax-textit">N. Honda</span> [ibid. 18, 233–236 (1971; <a href="/0218.68012">Zbl 0218.68012</a>)]), aber man braucht hier Alphabete mit nur zwei Buchstaben. <br class="zbmathjax-paragraph">Am Ende dieses Artikels zeigt man, daß<span class="zbmathjax-space"></span>ein Satz Turakainens über stochastische Automaten über einem einzigen Buchstaben [<span class="zbmathjax-textit">P. Turakainen</span>, Ann. Acad. Sci. Fenn., Ser. A I 439, 10 p. (1969; <a href="/0174.29003">Zbl 0174.29003</a>)] ein Korollar eines Satzes über rationale Funktionen [<span class="zbmathjax-textit">Z. A. Melzak</span>, Can. J. Math. 12, 20–26 (1960; <a href="/0090.03201">Zbl 0090.03201</a>)] ist. <br class="zbmathjax-paragraph">Einige dieser Resultate wurden schon in dem guten Buch von <span class="zbmathjax-textit">V. Claus</span> [Stochastische Automaten. Stuttgart: B. G. Teubner (1971; <a href="/0226.94046">Zbl 0226.94046</a>)] erwähnt. Für Komplemente über formale Sprachen und nichtkommutative Potenzreihen, siehe die unten referierte Arbeit des Autors [Automata, Languages, Programming, Proc. Sympos. Inst. Rech. Informatique Automatique (IRIA), Rocquencourt 1972, 397–411 (1973; <a href="/0262.94038">Zbl 0262.94038</a>)].<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A299">Michel Fliess</a></div> <div class="clearfix"></div><div class="scan"> <span> Page: <a class="goto" href="/scans/262/558.gif">−5</a> <a class="goto" href="/scans/262/559.gif">−4</a> <a class="goto" href="/scans/262/560.gif">−3</a> <a class="goto" href="/scans/262/561.gif">−2</a> <a class="goto" href="/scans/262/562.gif">−1</a> <a class="goto" href="/scans/262/563.gif">±0</a> <a class="goto" href="/scans/262/564.gif">+1</a> <a class="goto" href="/scans/262/565.gif">+2</a></span> <a class="scan" href="/scans/262/563.gif">Show Scanned Page</a><br> <img src="/scans/262/563.gif" alt="scan of Zbl 0262.94037" onclick="$(this).siblings('.scan').click();"> </div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A3414216">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A3414216">Cited in <strong>4</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A68Q45" title="MSC2020">68Q45</a> </td> <td class="space"> Formal languages and automata </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A16W60" title="MSC2020">16W60</a> </td> <td class="space"> Valuations, completions, formal power series and related constructions (associative rings and algebras) </td> </tr></table> </div><div class="keywords"> <h3>Citations:</h3><a href="/0184.02802">Zbl 0184.02802</a>; <a href="/0064.06902">Zbl 0064.06902</a>; <a href="/0218.68013">Zbl 0218.68013</a>; <a href="/0218.68012">Zbl 0218.68012</a>; <a href="/0174.29003">Zbl 0174.29003</a>; <a href="/0090.03201">Zbl 0090.03201</a>; <a href="/0226.94046">Zbl 0226.94046</a>; <a href="/0262.94038">Zbl 0262.94038</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0262.94037" data-ciurl="/ci/03414216" data-biburl="/bibtex/03414216.bib" data-amsurl="/amsrefs/03414216.bib" data-xmlurl="/xml/03414216.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/03414216.pdf" title="Zbl 0262.94037 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/BF01890611" aria-label="DOI for “Propriétés booléennes des langages stochastiques”" title="10.1007/BF01890611">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">B. Benzaghou, Algèbres de Hadamard,Bull. Soc. Math. France 98 (1970), 209–252. · <a href="/0206.33203" class="nowrap">Zbl 0206.33203</a></td> </tr><tr> <td>[2]</td> <td class="space">J. Berstel, Sur les pôles et le quotient de Hadamard de séries N-rationnelles,C.R. Acad. Sci. Paris 272 (1971), Série A, 1079–1081 (voir aussi: Contribution à l’étude des propriétés arithmétiques des langages formels, Thèse Sc. Math., Univ. Paris VII, Paris, 1972).</td> </tr><tr> <td>[3]</td> <td class="space">S. Eilenberg,Automata, Languages and Machines, Academic Press, New York, à paraître. · <a href="/0317.94045" class="nowrap">Zbl 0317.94045</a></td> </tr><tr> <td>[4]</td> <td class="space">M. Fliess, Formal languages and formal power series, Séminaire IRIA Logique et Automates, Le Chesnay, 1971, pp. 77–85.</td> </tr><tr> <td>[5]</td> <td class="space">M. Fliess, Sur certaines familles de séries formelles, Thèse Sc. Math., Univ. Paris VII, Paris, 1972.</td> </tr><tr> <td>[6]</td> <td class="space">M. Fliess, Séries rationnelles positives et processus stochastiques,Ann. Inst. H. Poincaré, Sect. B, à paraître. (Voir aussi: Automates stochastiques et séries rationnelles non commutatives,Automata, Languages and Programming (M. Nivat, ed.), North Holland, Amsterdam, 1973, pp. 397–411.)</td> </tr><tr> <td>[7]</td> <td class="space">G. H. Hardy etE. M. Wright,An Introduction to the Theory of Numbers, Clarendon Press, Oxford, 1938. · <a href="/0020.29201" class="nowrap">Zbl 0020.29201</a></td> </tr><tr> <td>[8]</td> <td class="space">C. Lech, A note on recurring series,Arkiv Mat. 2 (1953), 417–421. · <a href="/0051.27801" class="nowrap">Zbl 0051.27801</a> · <a href="https://doi.org/10.1007/BF02590997" class="nowrap">doi:10.1007/BF02590997</a></td> </tr><tr> <td>[9]</td> <td class="space">K. Mahler, On the Taylor coefficients of rational functions,Proc. Cambridge Philos. Soc. 52 (1956), 39–48. · <a href="/0070.04004" class="nowrap">Zbl 0070.04004</a> · <a href="https://doi.org/10.1017/S0305004100030966" class="nowrap">doi:10.1017/S0305004100030966</a></td> </tr><tr> <td>[10]</td> <td class="space">Z. A. Melzak, Power series representing certain rational functions,Canadian J. Math. 12 (1960), 20–26. · <a href="/0090.03201" class="nowrap">Zbl 0090.03201</a> · <a href="https://doi.org/10.4153/CJM-1960-002-8" class="nowrap">doi:10.4153/CJM-1960-002-8</a></td> </tr><tr> <td>[11]</td> <td class="space">M. Nasu etN. Honda, A context-free language which is not acceptable by a probabilistic automaton,Information and Control 18 (1971), 233–236. · <a href="/0218.68012" class="nowrap">Zbl 0218.68012</a> · <a href="https://doi.org/10.1016/S0019-9958(71)90373-1" class="nowrap">doi:10.1016/S0019-9958(71)90373-1</a></td> </tr><tr> <td>[12]</td> <td class="space">A. Paz, Some aspects of probabilistic automata,Information and Control 9 (1966), 26–60. · <a href="/0151.24703" class="nowrap">Zbl 0151.24703</a> · <a href="https://doi.org/10.1016/S0019-9958(66)90092-1" class="nowrap">doi:10.1016/S0019-9958(66)90092-1</a></td> </tr><tr> <td>[13]</td> <td class="space">M. P. Schützenberger, On the definition of a family of automata,Information and Control 4 (1961), 245–270. · <a href="/0104.00702" class="nowrap">Zbl 0104.00702</a> · <a href="https://doi.org/10.1016/S0019-9958(61)80020-X" class="nowrap">doi:10.1016/S0019-9958(61)80020-X</a></td> </tr><tr> <td>[14]</td> <td class="space">M. P. Schützenberger, On a theorem of R. Jungen,Proc. Amer. Math. Soc. 13 (1962), 885–890. · <a href="/0107.03102" class="nowrap">Zbl 0107.03102</a> · <a href="https://doi.org/10.2307/2034080" class="nowrap">doi:10.2307/2034080</a></td> </tr><tr> <td>[15]</td> <td class="space">M. P. Schützenberger, Parties rationnelles d’un monoïde libre, Actes Congrès Internat. Math. Nice (1970), vol. 3, pp. 281–282, Gauthier-Villars, Paris, 1971.</td> </tr><tr> <td>[16]</td> <td class="space">T. Skolem, Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen und diophantischer Gleichungen, Comptes Rendus 8e Congrès Math. Scandin. Stockholm (1934), pp. 163–188, Lund, 1935. · <a href="/61.1080.01" class="nowrap">JFM 61.1080.01</a></td> </tr><tr> <td>[17]</td> <td class="space">P. Turakainen, On probabilistic automata and their generalizations,Annales Acad. Sci. Fennicae, Ser. A.I., no.429, 1968. · <a href="/0179.02502" class="nowrap">Zbl 0179.02502</a></td> </tr><tr> <td>[18]</td> <td class="space">P. Turakainen, Generalized automata and stochastic languages,Proc. Amer. Math. Soc. 21 (1969), 303–309. · <a href="/0184.02802" class="nowrap">Zbl 0184.02802</a> · <a href="https://doi.org/10.1090/S0002-9939-1969-0242596-1" class="nowrap">doi:10.1090/S0002-9939-1969-0242596-1</a></td> </tr><tr> <td>[19]</td> <td class="space">P. Turakainen, On languages represented in rational probabilistic automata,Annales Acad. Sci. Fennicae, Ser. A.I., no.439, 1969. · <a href="/0174.29003" class="nowrap">Zbl 0174.29003</a></td> </tr><tr> <td>[20]</td> <td class="space">P. Turakainen, The family of stochastic languages is closed neither under catenation nor under homomorphism,Annal. Univ. Turku, Ser. A.I. no.133, 1970. · <a href="/0188.33303" class="nowrap">Zbl 0188.33303</a></td> </tr><tr> <td>[21]</td> <td class="space">P. Turakainen, Some closure properties of the family of stochastic languages,Information and Control 18 (1971), 253–256. · <a href="/0218.68013" class="nowrap">Zbl 0218.68013</a> · <a href="https://doi.org/10.1016/S0019-9958(71)90391-3" class="nowrap">doi:10.1016/S0019-9958(71)90391-3</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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