User holger - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T11:17:28Z http://mathoverflow.net/feeds/user/12147 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57965/oracle-relativization-and-p-vs-np-philosophical/57979#57979 Answer by Holger for Oracle, Relativization, and P vs NP, [Philosophical] Holger 2011-03-09T18:27:10Z 2011-03-09T18:44:03Z <p>$P^A$ and $NP^A$ are shorthand notation to be used with care. What we mean by $\mathcal C^A$ depends heavily on the syntactical definition of the class $\mathcal C$ and is not well-defined if it's not clear which syntactical definition of $\mathcal C$ we are referring to. $\mathcal C^A$ is more an invariant of $\mathcal C$-machines than of the set of languages decided by $\mathcal C$-machines.</p> <p>While $P=NP$ means that $P$-machines can decide exactly the same languages as $NP$-machines, $P^B\neq NP^B$ for some oracle $B$ as proven by <a href="http://scholar.google.com/scholar?q=Baker+Gill+Solovay+Relativizations+of+the+P+NP+question" rel="nofollow">Baker, Gill, and Solovay (1975)</a> means that $P$-machines with additional access to the oracle $B$ can decide <em>strictly fewer</em> languages than $NP$-machines with access to $B$. Baker, Gill, and Solovay also give an oracle $A$ with respect to which these two types of machines can decide exactly the same languages.</p> http://mathoverflow.net/questions/1785/how-do-you-keep-your-research-notes-organized/51780#51780 Answer by Holger for How do you keep your research notes organized? Holger 2011-01-11T17:42:02Z 2011-01-11T17:42:02Z <p>I have been using <a href="http://www.dokuwiki.org/" rel="nofollow">dokuwiki</a> with a <a href="http://www.dokuwiki.org/plugin%3ajsmath" rel="nofollow">tex plugin</a> for quite a while now to organize my projects, open problems, things I want to read, and a daily research diary. That works quite well and should be usable on most university webspaces (PHP only; no SQL or fancy progams needed on the server).</p>