<p>Lecture 2 </p><p>1 Review </p><p>·<br>| | </p><p>1</p><p>·<br>| | </p><p>∗</p><p></p><ul style="display: flex;"><li style="flex:1">{</li><li style="flex:1">}<sup style="top: -0.3301em;">∗ </sup>→ { </li><li style="flex:1">}</li></ul><p></p><p>∗</p><p></p><ul style="display: flex;"><li style="flex:1">{</li><li style="flex:1">}</li></ul><p></p><ul style="display: flex;"><li style="flex:1">{</li><li style="flex:1">|</li><li style="flex:1">}</li></ul><p>∈</p><p>∈∈<br>⇒⇒</p><p></p><ul style="display: flex;"><li style="flex:1">• time </li><li style="flex:1">∈ time </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">·</li><li style="flex:1">·</li><li style="flex:1">| | </li></ul><p></p><p>• space </p><p><sup style="top: -0.3168em;">1</sup>Note that we do not count the space used on the input or output tapes; this allows us to meaningfully speak of sub-linear space machines (with linear- or superlinear-length output). </p><p>2</p><p>0</p><p></p><ul style="display: flex;"><li style="flex:1">∈ time </li><li style="flex:1">space </li><li style="flex:1">space </li></ul><p></p><p>0</p><p></p><ul style="display: flex;"><li style="flex:1">time </li><li style="flex:1">⊆ space </li></ul><p></p><p>2 P, NP, and NP-Completeness </p><p>2.1 The&nbsp;Class P </p><p>P</p><p>def </p><p>P</p><p>time </p><p>≥1 </p><p>P<br>| | <br>P</p><p>•</p><p>•</p><p>100 <br>100 8</p><p>P<br>P</p><p><sup style="top: -0.3176em;">2</sup>This decision is also motivated by “speedup theorems” which state that if a language can be decided in time <br>(resp., space)&nbsp;( )&nbsp;then it can be decided in time (resp., space)&nbsp;( )&nbsp;for any constant&nbsp;. (This assumes that&nbsp;( ) is a “reasonable” function, but the details need not concern us here.) </p><p>P</p><p>2.2 The&nbsp;Classes NP and coNP </p><p></p><ul style="display: flex;"><li style="flex:1">NP </li><li style="flex:1">NP </li></ul><p>∈ NP </p><p>3</p><p></p><ul style="display: flex;"><li style="flex:1">| | </li><li style="flex:1">∈</li></ul><p></p><ul style="display: flex;"><li style="flex:1">∈</li><li style="flex:1">P</li><li style="flex:1">∈ P </li></ul><p>| | <br>∈<br>P<br>NP </p><p>∈<br>P<br>NP </p><p>NP <br>P ⊆ NP ⊆&nbsp;<sub style="top: 0.2684em;">≥1 </sub>time </p><ul style="display: flex;"><li style="flex:1">P ⊆ NP </li><li style="flex:1">∈ NP </li></ul><p>| | </p><ul style="display: flex;"><li style="flex:1">| | </li><li style="flex:1">∈</li></ul><p>| | <br>| | </p><p><sup style="top: -0.3176em;">3</sup>It is essential that the running time of is to require the length of&nbsp;to be at most&nbsp;(| |) in condition (2). be measured in terms of the length of&nbsp;alone. An alternate approach </p><p>≤ (| |) </p><p></p><ul style="display: flex;"><li style="flex:1">∈ { </li><li style="flex:1">}</li></ul><p></p><p>(| |) </p><p>| | · </p><p>∈ time </p><p>NP </p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">1</li></ul><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">1</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">∈</li><li style="flex:1">⇔</li></ul><p>| | </p><p>ntime <br>∈ ntime </p><p>| | <br>| | </p><p>nspace </p><p>| | </p><ul style="display: flex;"><li style="flex:1">NP </li><li style="flex:1">P</li></ul><p>NP </p><p><sub style="top: 0.2692em;">≥1 </sub>ntime </p><p>?</p><p></p><ul style="display: flex;"><li style="flex:1">P</li><li style="flex:1">NP </li></ul><p></p><ul style="display: flex;"><li style="flex:1">P</li><li style="flex:1">NP </li></ul><p></p><ul style="display: flex;"><li style="flex:1">P</li><li style="flex:1">NP </li></ul><p></p><ul style="display: flex;"><li style="flex:1">P</li><li style="flex:1">NP </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">ntime </li><li style="flex:1">time </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">3</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">time </li><li style="flex:1">⊆ ntime </li></ul><p></p><p>coNP </p><p></p><ul style="display: flex;"><li style="flex:1">def </li><li style="flex:1">def </li></ul><p></p><p>C</p><p></p><ul style="display: flex;"><li style="flex:1">coC </li><li style="flex:1">coC </li></ul><p></p><p>NP </p><ul style="display: flex;"><li style="flex:1">|</li><li style="flex:1">∈ C </li><li style="flex:1">{</li><li style="flex:1">}<sup style="top: -0.3301em;">∗ </sup>\ </li></ul><p>coNP <br>∈ coNP </p><p>4</p><p></p><ul style="display: flex;"><li style="flex:1">| | </li><li style="flex:1">∈</li></ul><p></p><p><sup style="top: -0.3176em;">4</sup>See footnote 3. </p><p>∈</p><ul style="display: flex;"><li style="flex:1">coNP </li><li style="flex:1">NP </li></ul><p></p><ul style="display: flex;"><li style="flex:1">coNP </li><li style="flex:1">∈</li></ul><p></p><ul style="display: flex;"><li style="flex:1">NP </li><li style="flex:1">coNP </li></ul><p></p><p>SAT { | </p><p>}</p><p>SAT <br>SAT ∈ coNP </p><p>SAT ∈ NP </p><p>TAUT <br>TAUT </p><p></p><ul style="display: flex;"><li style="flex:1">{</li><li style="flex:1">}</li></ul><p></p><p>TAUT </p><p>coNP coNP </p><p></p><ul style="display: flex;"><li style="flex:1">P ⊆ NP ∩ coNP </li><li style="flex:1">NP </li><li style="flex:1">coNP </li></ul><p>SAT ∈ NP <br>NP coNP </p><p>2.3 NP-Completeness </p><p>NP </p><p>0</p><p>5</p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul><p>0</p><p></p><ul style="display: flex;"><li style="flex:1">Karp reducible </li><li style="flex:1">many-to-one reducible </li></ul><p></p><p>0</p><p></p><ul style="display: flex;"><li style="flex:1">∈</li><li style="flex:1">∈</li></ul><p></p><p>0</p><p>≤</p><p><sup style="top: -0.3176em;">5</sup>Technically speaking, I mean “at least as hard as”. </p><p>0</p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul><p></p><p>def </p><p>0<br>0</p><p>≤</p><ul style="display: flex;"><li style="flex:1">≤</li><li style="flex:1">≤</li><li style="flex:1">≤</li></ul><p>≤≤</p><ul style="display: flex;"><li style="flex:1">∈ P </li><li style="flex:1">∈ P </li></ul><p></p><ul style="display: flex;"><li style="flex:1">∈ NP </li><li style="flex:1">∈ NP </li></ul><p>NP NP </p><ul style="display: flex;"><li style="flex:1">NP </li><li style="flex:1">NP </li></ul><p>NP </p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li><li style="flex:1">0</li></ul><p></p><p>NP-hard </p><p>∈ NP <br>NP <br>≤</p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul><p></p><p>NP-complete </p><p></p><ul style="display: flex;"><li style="flex:1">∈ NP </li><li style="flex:1">NP </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">NP </li><li style="flex:1">≤</li></ul><p></p><p>0</p><p></p><ul style="display: flex;"><li style="flex:1">coNP </li><li style="flex:1">coNP </li></ul><p>coNP <br>∈ coNP </p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li><li style="flex:1">0</li><li style="flex:1">0</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">≤</li><li style="flex:1">coNP </li><li style="flex:1">∈ coNP </li></ul><p>NP <br>NP <br>NP NP <br>NP <br>NP </p><p></p><ul style="display: flex;"><li style="flex:1">∃</li><li style="flex:1">∈ { </li><li style="flex:1">}</li></ul><p>NP </p><p>References </p>