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srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Straight-line_grammar" title="Special:EditPage/Straight-line grammar">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Straight-line+grammar%22">"Straight-line grammar"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Straight-line+grammar%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Straight-line+grammar%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Straight-line+grammar%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Straight-line+grammar%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Straight-line+grammar%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">August 2014</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>A <b>straight-line grammar</b> (sometimes abbreviated as SLG) is a <a href="/wiki/Formal_grammar" title="Formal grammar">formal grammar</a> that generates exactly one string.<sup id="cite_ref-GECCO_1-0" class="reference"><a href="#cite_note-GECCO-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Consequently, it does not branch (every non-terminal has only one associated production rule) nor loop (if non-terminal <i>A</i> appears in a derivation of <i>B</i>, then <i>B</i> does not appear in a derivation of <i>A</i>).<sup id="cite_ref-GECCO_1-1" class="reference"><a href="#cite_note-GECCO-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Areas_of_usefulness">Areas of usefulness</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Straight-line_grammar&action=edit&section=1" title="Edit section: Areas of usefulness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Straight-line grammars are widely used in the development of algorithms that execute directly on compressed structures (without prior decompression).<sup id="cite_ref-Lohrey.Maneth.Schmidt.2009_2-0" class="reference"><a href="#cite_note-Lohrey.Maneth.Schmidt.2009-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 212">: 212 </span></sup> </p><p>SLGs are of interest in fields like <a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a>, <a href="/wiki/Lossless_data_compression" class="mw-redirect" title="Lossless data compression">Lossless data compression</a>, <a href="/w/index.php?title=Structure_discovery&action=edit&redlink=1" class="new" title="Structure discovery (page does not exist)">Structure discovery</a> and <a href="/wiki/Compressed_data_structure" title="Compressed data structure">Compressed data structures</a>.<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="Indicate how SLGs are used in each of these fields. E.g. in the current version of the 'Kolmogorov complexity' article, they are mentioned nowhere. (October 2013)">clarification needed</span></a></i>]</sup> </p><p>The problem of finding a context-free grammar (equivalently: an SLG) of minimal size that generates a given string is called the <a href="/wiki/Smallest_grammar_problem" title="Smallest grammar problem">smallest grammar problem</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2014)">citation needed</span></a></i>]</sup> </p><p>Straight-line grammars (more precisely: straight-line context-free string grammars) can be generalized to <i>Straight-line context-free tree grammars</i>. The latter can be used conveniently to compress <a href="/wiki/Tree_(data_structure)" class="mw-redirect" title="Tree (data structure)">trees</a>.<sup id="cite_ref-Lohrey.Maneth.Schmidt.2009_2-1" class="reference"><a href="#cite_note-Lohrey.Maneth.Schmidt.2009-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 212">: 212 </span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Formal_Definition">Formal Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Straight-line_grammar&action=edit&section=2" title="Edit section: Formal Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Context-free_grammar" title="Context-free grammar">context-free grammar</a> <i>G</i> is an SLG if: </p><p>1. for every non-terminal <i>N</i>, there is at most one production rule that has <i>N</i> as its left-hand side, and </p><p>2. the <a href="/wiki/Directed_graph" title="Directed graph">directed graph</a> <i>G</i>=<<i>V</i>,<i>E</i>>, defined by <i>V</i> being the set of non-terminals and (<i>A</i>,<i>B</i>) ∈ <i>E</i> whenever <i>B</i> appears at the right-hand side of a production rule for <i>A</i>, is <a href="/wiki/Directed_acyclic_graph" title="Directed acyclic graph">acyclic</a>. </p><p>A mathematical definition of the more general formalism of straight-line context-free tree grammars can be found in Lohrey et al.<sup id="cite_ref-Lohrey.Maneth.Schmidt.2009_2-2" class="reference"><a href="#cite_note-Lohrey.Maneth.Schmidt.2009-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 215">: 215 </span></sup> </p><p>An SLG in <a href="/wiki/Chomsky_normal_form" title="Chomsky normal form">Chomsky normal form</a> is equivalent to a <a href="/wiki/Straight-line_program" title="Straight-line program">straight-line program</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (May 2018)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="A_list_of_algorithms_using_SLGs">A list of algorithms using SLGs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Straight-line_grammar&action=edit&section=3" title="Edit section: A list of algorithms using SLGs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The <a href="/wiki/Sequitur_algorithm" title="Sequitur algorithm">Sequitur algorithm</a> constructs a straight-line grammar for a given string.</li> <li>The <a href="/wiki/Lempel-Ziv-Welch_algorithm" class="mw-redirect" title="Lempel-Ziv-Welch algorithm">Lempel-Ziv-Welch algorithm</a> creates a context-free grammar in such a deterministic way that it is necessary to store only the start rule of the generated grammar.</li> <li><a href="/wiki/Byte_pair_encoding" title="Byte pair encoding">Byte pair encoding</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Straight-line_grammar&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Grammar-based_code" title="Grammar-based code">Grammar-based code</a> – Lossless data compression algorithm</li> <li><a href="/wiki/Non-recursive_grammar" class="mw-redirect" title="Non-recursive grammar">Non-recursive grammar</a> - a grammar that does not loop, but may branch; generating a finite rather than a singleton language</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Straight-line_grammar&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-GECCO-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-GECCO_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-GECCO_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Florian Benz and Timo Kötzing, “An effective heuristic for the smallest grammar problem,” Proceedings of the fifteenth annual conference on Genetic and evolutionary computation conference - GECCO ’13, 2013. <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4503-1963-8" title="Special:BookSources/978-1-4503-1963-8">978-1-4503-1963-8</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F2463372.2463441">10.1145/2463372.2463441</a>, p. 488</span> </li> <li id="cite_note-Lohrey.Maneth.Schmidt.2009-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lohrey.Maneth.Schmidt.2009_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lohrey.Maneth.Schmidt.2009_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Lohrey.Maneth.Schmidt.2009_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarkus_LohreySebastian_ManethManfred_Schmidt-Schauß2009" class="citation book cs1">Markus Lohrey; Sebastian Maneth; Manfred Schmidt-Schauß (2009). "Parameter Reduction in Grammar-Compressed Trees". <a rel="nofollow" class="external text" href="http://www.eti.uni-siegen.de/ti/veroeffentlichungen/09-fossacs.pdf"><i>Proc. FOSSACS</i></a> <span class="cs1-format">(PDF)</span>. LNCS. Vol. 5504. Springer. pp. 212–226.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Parameter+Reduction+in+Grammar-Compressed+Trees&rft.btitle=Proc.+FOSSACS&rft.series=LNCS&rft.pages=212-226&rft.pub=Springer&rft.date=2009&rft.au=Markus+Lohrey&rft.au=Sebastian+Maneth&rft.au=Manfred+Schmidt-Schau%C3%9F&rft_id=http%3A%2F%2Fwww.eti.uni-siegen.de%2Fti%2Fveroeffentlichungen%2F09-fossacs.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStraight-line+grammar" class="Z3988"></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Straight-line_grammar&oldid=1183533793">https://en.wikipedia.org/w/index.php?title=Straight-line_grammar&oldid=1183533793</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Formal_languages" title="Category:Formal languages">Formal languages</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Use_American_English_from_January_2019" title="Category:Use American English from January 2019">Use American English from January 2019</a></li><li><a href="/wiki/Category:All_Wikipedia_articles_written_in_American_English" title="Category:All Wikipedia articles written in American English">All Wikipedia articles written in American English</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_August_2014" title="Category:Articles needing additional references from August 2014">Articles needing additional references from August 2014</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_clarification_from_October_2013" title="Category:Wikipedia articles needing clarification from October 2013">Wikipedia articles needing clarification from October 2013</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_August_2014" title="Category:Articles with unsourced statements from August 2014">Articles with unsourced statements from August 2014</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_May_2018" title="Category:Articles with unsourced statements from May 2018">Articles with unsourced statements from May 2018</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 4 November 2023, at 22:23<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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