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<title>LH (complexity)</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">LH (complexity)</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Computational_complexity_theory" title="Computational complexity theory">computational complexity</a>, the <b>logarithmic time hierarchy</b> (<b>LH</b>) is the <a href="Complexity_class" title="Complexity class">complexity class</a> of all <a href="Computational_problem" title="Computational problem">computational problems</a> solvable in a <a href="Logarithmic_growth" title="Logarithmic growth">logarithmic</a> amount of <a href="Computation_time" class="mw-redirect" title="Computation time">computation time</a> on an <a href="Alternating_Turing_machine" title="Alternating Turing machine">alternating Turing machine</a> with a bounded number of alternations. It is a particular case of a <a href="Alternating_Turing_machine#Bounded_alternation" title="Alternating Turing machine">bounded alternating Turing machine</a> hierarchy. It is equal to <a href="FO_(complexity)" class="mw-redirect" title="FO (complexity)">FO</a> and to FO-uniform <a href="AC0" title="AC0">AC<sup>0</sup></a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p>The <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
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</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>th level of the logarithmic time hierarchy is the set of languages recognised by alternating Turing machines in logarithmic time with <a href="Random-access_Turing_machine" title="Random-access Turing machine">random access</a> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i-1}">
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<annotation encoding="application/x-tex">{\displaystyle i-1}</annotation>
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</math></span><img src="./9d2ca5c639f26340e0e80f5883cc93a00254513c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.805ex; height:2.343ex;" alt="{\displaystyle i-1}" loading="lazy"></span> alternations, beginning with an <a href="Existential_state" class="mw-redirect" title="Existential state">existential state</a>. <b>LH</b> is the union of all levels.
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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFNeil_Immerman1999" class="citation book cs1"><a href="Neil_Immerman" title="Neil Immerman">Neil Immerman</a> (1999). <a href="Descriptive_Complexity" title="Descriptive Complexity"><i>Descriptive Complexity</i></a>. Springer. p. <a rel="nofollow" class="external text" href="https://archive.org/details/descriptivecompl00imme_115/page/n96">85</a>.</cite></span>
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</style><div id="Complexity_classes148" style="font-size:114%;margin:0 4em"><a href="Complexity_class" title="Complexity class">Complexity classes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Considered feasible</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="DLOGTIME" title="DLOGTIME">DLOGTIME</a></li>
<li><a href="AC0" title="AC0">AC<sup>0</sup></a></li>
<li><a href="ACC0" title="ACC0">ACC<sup>0</sup></a></li>
<li><a href="TC_(complexity)" title="TC (complexity)">TC</a>
<ul><li><a href="TC0" title="TC0">TC<sup>0</sup></a></li></ul></li>
<li><a href="L_(complexity)" title="L (complexity)">L</a></li>
<li><a href="SL_(complexity)" title="SL (complexity)">SL</a></li>
<li><a href="RL_(complexity)" title="RL (complexity)">RL</a></li>
<li><a href="FL_(complexity)" title="FL (complexity)">FL</a></li>
<li><a href="NL_(complexity)" title="NL (complexity)">NL</a>
<ul><li><a href="NL-complete" title="NL-complete">NL-complete</a></li></ul></li>
<li><a href="NC_(complexity)" title="NC (complexity)">NC</a></li>
<li><a href="SC_(complexity)" title="SC (complexity)">SC</a></li>
<li><a href="CC_(complexity)" title="CC (complexity)">CC</a></li>
<li><a href="P_(complexity)" title="P (complexity)">P</a>
<ul><li><a href="P-complete" title="P-complete">P-complete</a></li></ul></li>
<li><a href="ZPP_(complexity)" title="ZPP (complexity)">ZPP</a></li>
<li><a href="RP_(complexity)" title="RP (complexity)">RP</a></li>
<li><a href="BPP_(complexity)" title="BPP (complexity)">BPP</a></li>
<li><a href="BQP" title="BQP">BQP</a></li>
<li><a href="APX" title="APX">APX</a></li>
<li><a href="FP_(complexity)" title="FP (complexity)">FP</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Suspected infeasible</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="UP_(complexity)" title="UP (complexity)">UP</a></li>
<li><a href="NP_(complexity)" title="NP (complexity)">NP</a>
<ul><li><a href="NP-completeness" title="NP-completeness">NP-complete</a></li>
<li><a href="NP-hardness" title="NP-hardness">NP-hard</a></li>
<li><a href="Co-NP" title="Co-NP">co-NP</a></li>
<li><a href="Co-NP-complete" title="Co-NP-complete">co-NP-complete</a></li></ul></li>
<li><a href="TFNP" title="TFNP">TFNP</a></li>
<li><a href="FNP_(complexity)" title="FNP (complexity)">FNP</a></li>
<li><a href="Arthur%E2%80%93Merlin_protocol" title="Arthur–Merlin protocol">AM</a></li>
<li><a href="QMA" title="QMA">QMA</a></li>
<li><a href="Polynomial_hierarchy" title="Polynomial hierarchy">PH</a></li>
<li><a href="Parity_P" title="Parity P">⊕P</a></li>
<li><a href="PP_(complexity)" title="PP (complexity)">PP</a></li>
<li><a href="%E2%99%AFP" title="♯P">#P</a>
<ul><li><a href="%E2%99%AFP-complete" title="♯P-complete">#P-complete</a></li></ul></li>
<li><a href="IP_(complexity)" title="IP (complexity)">IP</a></li>
<li><a href="PSPACE" title="PSPACE">PSPACE</a>
<ul><li><a href="PSPACE-complete" title="PSPACE-complete">PSPACE-complete</a></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Considered infeasible</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="EXPTIME" title="EXPTIME">EXPTIME</a></li>
<li><a href="NEXPTIME" title="NEXPTIME">NEXPTIME</a></li>
<li><a href="EXPSPACE" title="EXPSPACE">EXPSPACE</a></li>
<li><a href="2-EXPTIME" title="2-EXPTIME">2-EXPTIME</a></li>
<li><a href="ELEMENTARY" title="ELEMENTARY">ELEMENTARY</a></li>
<li><a href="PR_(complexity)" title="PR (complexity)">PR</a></li>
<li><a href="R_(complexity)" title="R (complexity)">R</a></li>
<li><a href="RE_(complexity)" title="RE (complexity)">RE</a></li>
<li><a href="ALL_(complexity)" title="ALL (complexity)">ALL</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Class hierarchies</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Polynomial_hierarchy" title="Polynomial hierarchy">Polynomial hierarchy</a></li>
<li><a href="Exponential_hierarchy" title="Exponential hierarchy">Exponential hierarchy</a></li>
<li><a href="Grzegorczyk_hierarchy" title="Grzegorczyk hierarchy">Grzegorczyk hierarchy</a></li>
<li><a href="Arithmetical_hierarchy" title="Arithmetical hierarchy">Arithmetical hierarchy</a></li>
<li><a href="Boolean_hierarchy" title="Boolean hierarchy">Boolean hierarchy</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Families of classes</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="DTIME" title="DTIME">DTIME</a></li>
<li><a href="NTIME" title="NTIME">NTIME</a></li>
<li><a href="DSPACE" title="DSPACE">DSPACE</a></li>
<li><a href="NSPACE" title="NSPACE">NSPACE</a></li>
<li><a href="Probabilistically_checkable_proof" title="Probabilistically checkable proof">Probabilistically checkable proof</a></li>
<li><a href="Interactive_proof_system" title="Interactive proof system">Interactive proof system</a></li></ul>
</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><a href="List_of_complexity_classes" title="List of complexity classes">List of complexity classes</a></div></td></tr></tbody></table></div>
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