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	<description>the true words at last, from the mind in ruins</description>
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		<title>WM2D</title>
		<link>http://opusminimax.wordpress.com/2012/02/19/wm2d/</link>
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		<pubDate>Sun, 19 Feb 2012 18:32:44 +0000</pubDate>
		<dc:creator>F Lengyel</dc:creator>
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		<guid isPermaLink="false">http://opusminimax.wordpress.com/?p=962</guid>
		<description><![CDATA[When I was a boy I made a promise to myself that I would one day earn a ham license. That was when this hobby would have made sense. I was neither discouraged nor encouraged as a boy. By the time I made the effort to earn my license in 1988 it was too late. [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=opusminimax.wordpress.com&amp;blog=26204618&amp;post=962&amp;subd=opusminimax&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<p>When I was a boy I made a promise to myself that I would one day earn a ham license. That was when this hobby would have made sense. I was neither discouraged nor encouraged as a boy. By the time I made the effort to earn my license in 1988 it was too late. The world had changed. The surplus electronic stores on Canal Street had closed. Lafayette Radio was a dim memory. </p>
<p>It&#8217;s hard to say that I have anything in common with the boy I once was, aside from a history over which every atom in his body was eventually replaced with the atoms I possess. This has happened in stages several times over. I doubt that now, after several iterations of this process, that I could put myself in his place or that we could even be said to be the same person. We only share some copies of memories he left behind. He once dreamed of owning his own ham shack with racks of equipment wallpapered with QSL cards. It wasn&#8217;t until February 2012 that I received my first and only QSL card, from KB3MBP.</p>
<p>The Internet didn&#8217;t exist as we know it when I made the decison to study for the exam. There were no online tests. I purchased cassette tapes from Radio Shack to learn enough code to pass the exam. Aside from one Field Day contest on SSB, I&#8217;ve only worked CW on HF. The occasional suggestion from old-timers that I&#8217;m a no-code Extra is irritating enough that I sometimes practice copying code with the QRQ program. Now that building management removed the antennas I once used on HF, I have been exiled to UHF.</p>
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			<media:title type="html">florianlengyel</media:title>
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		<title>A forgotten counterexample</title>
		<link>http://opusminimax.wordpress.com/2012/02/04/a-forgotten-counterexample/</link>
		<comments>http://opusminimax.wordpress.com/2012/02/04/a-forgotten-counterexample/#comments</comments>
		<pubDate>Sat, 04 Feb 2012 23:51:02 +0000</pubDate>
		<dc:creator>F Lengyel</dc:creator>
				<category><![CDATA[Uncategorized]]></category>

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		<description><![CDATA[Let be a subspace of , where . The inclusion map is not a cofibration. Otherwise, there would exist a homotopy extending the homotopy (given by ) and with initial map the identity . The space is totally disconnected. The only connected subspaces of are singtons and the only connected subspaces of are intervals&#8211;including singletons, [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=opusminimax.wordpress.com&amp;blog=26204618&amp;post=923&amp;subd=opusminimax&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<p>Let <img src='http://s0.wp.com/latex.php?latex=L%3D%5Cleft%5C%7B0%2C1%2C1%2F2%2C1%2F3%2C%5Cldots%2C1%2Fn%2C%5Cldots%5Cright%5C%7D&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='L=&#92;left&#92;{0,1,1/2,1/3,&#92;ldots,1/n,&#92;ldots&#92;right&#92;}' title='L=&#92;left&#92;{0,1,1/2,1/3,&#92;ldots,1/n,&#92;ldots&#92;right&#92;}' class='latex' /> be a subspace of <img src='http://s0.wp.com/latex.php?latex=%5Cmathbb%7BR%7D&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='&#92;mathbb{R}' title='&#92;mathbb{R}' class='latex' />, where <img src='http://s0.wp.com/latex.php?latex=n%5Cin%5Cmathbb%7BN%7D&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='n&#92;in&#92;mathbb{N}' title='n&#92;in&#92;mathbb{N}' class='latex' />. The inclusion map <img src='http://s0.wp.com/latex.php?latex=%5B-1%2C0%5D%5Crightarrow%5B-1%2C0%5D+%5Ccup+L&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='[-1,0]&#92;rightarrow[-1,0] &#92;cup L' title='[-1,0]&#92;rightarrow[-1,0] &#92;cup L' class='latex' /> is not a cofibration. Otherwise, there would exist a homotopy <img src='http://s0.wp.com/latex.php?latex=H%3A%28%5B-1%2C0%5D+%5Ccup+L%29+%5Ctimes+I+%5Crightarrow+%5B-1%2C0%5D+%5Ccup+L&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='H:([-1,0] &#92;cup L) &#92;times I &#92;rightarrow [-1,0] &#92;cup L' title='H:([-1,0] &#92;cup L) &#92;times I &#92;rightarrow [-1,0] &#92;cup L' class='latex' /> extending the homotopy <img src='http://s0.wp.com/latex.php?latex=G%3A+%5B-1%2C0%5D+%5Ctimes+I%5Crightarrow%5B-1%2C0%5D+%5Ccup+L&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='G: [-1,0] &#92;times I&#92;rightarrow[-1,0] &#92;cup L' title='G: [-1,0] &#92;times I&#92;rightarrow[-1,0] &#92;cup L' class='latex' /> (given by <img src='http://s0.wp.com/latex.php?latex=G%28x%2Ct%29%3D%281-t%29x+%2B+t%28-1%29&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='G(x,t)=(1-t)x + t(-1)' title='G(x,t)=(1-t)x + t(-1)' class='latex' />) and with initial map the identity <img src='http://s0.wp.com/latex.php?latex=%5B-1%2C0%5D%5Ccup+L%5Crightarrow%5B-1%2C0%5D+%5Ccup+L&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='[-1,0]&#92;cup L&#92;rightarrow[-1,0] &#92;cup L' title='[-1,0]&#92;cup L&#92;rightarrow[-1,0] &#92;cup L' class='latex' />.</p>
<p>The space <img src='http://s0.wp.com/latex.php?latex=L&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='L' title='L' class='latex' /> is totally disconnected. The only connected subspaces of <img src='http://s0.wp.com/latex.php?latex=L&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='L' title='L' class='latex' /> are singtons and the only connected subspaces of <img src='http://s0.wp.com/latex.php?latex=%5Cmathbb%7BR%7D&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='&#92;mathbb{R}' title='&#92;mathbb{R}' class='latex' /> are  intervals&#8211;including singletons, which are degenerate intervals. </p>
<p>Since <img src='http://s0.wp.com/latex.php?latex=L&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='L' title='L' class='latex' /> is totally disconnected, <img src='http://s0.wp.com/latex.php?latex=H%281%2Fn%2Ct%29%3D1%2Fn&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='H(1/n,t)=1/n' title='H(1/n,t)=1/n' class='latex' /> for each <img src='http://s0.wp.com/latex.php?latex=t%5Cin+I&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='t&#92;in I' title='t&#92;in I' class='latex' />. This is because <img src='http://s0.wp.com/latex.php?latex=H%281%2Fn%2C0%29+%3D+1%2Fn&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='H(1/n,0) = 1/n' title='H(1/n,0) = 1/n' class='latex' /> and <img src='http://s0.wp.com/latex.php?latex=H%5B1%2Fn+%5Ctimes+I%5D&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='H[1/n &#92;times I]' title='H[1/n &#92;times I]' class='latex' /> is a connected subset of <img src='http://s0.wp.com/latex.php?latex=%5B-1%2C0%5D+%5Ccup+L&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='[-1,0] &#92;cup L' title='[-1,0] &#92;cup L' class='latex' /> containing <img src='http://s0.wp.com/latex.php?latex=1%2Fn&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='1/n' title='1/n' class='latex' />.  In particular, <img src='http://s0.wp.com/latex.php?latex=H%281%2Fn%2C1%29%3D1%2Fn&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='H(1/n,1)=1/n' title='H(1/n,1)=1/n' class='latex' />. But <img src='http://s0.wp.com/latex.php?latex=H%280%2C1%29%3D-1&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='H(0,1)=-1' title='H(0,1)=-1' class='latex' />, contradiction since <img src='http://s0.wp.com/latex.php?latex=H%280%2C1%29+%3D+%5Clim_n+H%281%2Fn%2C1%29+%3D+%5Clim_n+1%2Fn+%3D+0&amp;bg=ffffff&amp;fg=000&amp;s=0' alt='H(0,1) = &#92;lim_n H(1/n,1) = &#92;lim_n 1/n = 0' title='H(0,1) = &#92;lim_n H(1/n,1) = &#92;lim_n 1/n = 0' class='latex' />.</p>
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			<media:title type="html">florianlengyel</media:title>
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		<title>iRODS installation, take two</title>
		<link>http://opusminimax.wordpress.com/2012/02/02/irods-installation-take-two/</link>
		<comments>http://opusminimax.wordpress.com/2012/02/02/irods-installation-take-two/#comments</comments>
		<pubDate>Thu, 02 Feb 2012 06:06:44 +0000</pubDate>
		<dc:creator>F Lengyel</dc:creator>
				<category><![CDATA[Digital Bedpan]]></category>

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		<description><![CDATA[Apparently iRODS 3.0 needs a patch under Ubuntu 11.10. This time I&#8217;ll set the default irods resource &#8220;storage vault&#8221; to be the 10TB disk &#8212; that will circumvent some problems. A question is whether the disk should be mounted noexec. I would prefer this. I will leave it out for now. schroe&#8230;@diceresearch.org View profile More [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=opusminimax.wordpress.com&amp;blog=26204618&amp;post=916&amp;subd=opusminimax&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<p>Apparently iRODS 3.0 needs a patch under Ubuntu 11.10. This time I&#8217;ll set the default irods resource &#8220;storage vault&#8221; to be the 10TB disk &#8212; that will circumvent some problems. A question is whether the disk should be mounted <code>noexec</code>. I would prefer this. I will leave it out for now.</p>
<blockquote><p>
schroe&#8230;@diceresearch.org<br />
View profile<br />
 More options Feb 1, 12:28 pm<br />
The fix for the bug for Ubuntu 11.10 is simple, so I&#8217;d recommend you try applying that rather than rebuilding a virtual machine.  After you unpack the release, edit server/icat/src/icatLowLevelOdbc.c and after the line:<br />
   icss.stmtPtr[0]=0;<br />
Add the line:<br />
   icss.databaseType = DB_TYPE_POSTGRES;<br />
&#8230;
</p></blockquote>
<p>But not so fast! Another patch is needed since </p>
<blockquote><p>
Compilation failed.  Please see the log file for details:<br />
    /home/irods/iRODS/installLogs/installMake.log<br />
&#8230;</p>
<p>Link flags:<br />
    -lm -lpthread /home/irods/iRODS/lib/core/obj/libRodsAPIs.a<br />
Link icommand ichksum&#8230;<br />
/home/irods/iRODS/lib/core/obj/libRodsAPIs.a(rcConnect.o): In function `_rcConne<br />
ct&#8217;:<br />
/home/irods/iRODS/lib/core/src/rcConnect.c:148: undefined reference to `pthread_<br />
create&#8217;<br />
/home/irods/iRODS/lib/core/obj/libRodsAPIs.a(rcConnect.o): In function `rcDiscon<br />
nect&#8217;:<br />
/home/irods/iRODS/lib/core/src/rcConnect.c:292: undefined reference to `pthread_<br />
cancel&#8217;<br />
/home/irods/iRODS/lib/core/src/rcConnect.c:293: undefined reference to `pthread_<br />
detach&#8217;<br />
collect2: ld returned 1 exit status<br />
make[1]: *** [/home/irods/iRODS/clients/icommands/bin/ichksum] Error 1<br />
make: *** [icommands] Error 2
</p></blockquote>
<p>But Google is your friend: </p>
<blockquote><p>
Hello Scott,<br />
hunting around I uncovered the following:</p>
<p>https://wiki.ubuntu.com/NattyNarwhal/ToolchainTransition</p>
<p>apparently the linking in Ubuntu 11.10 is more sensitive than in previous versions. What worked for me was<br />
to change the order in the icommands Makefile for the linker step. From:</p>
<p>LDFLAGS += $(CL_LDADD) $(LIBRARY) $(MODULE_LDFLAGS)</p>
<p>to:</p>
<p>LDFLAGS += $(LIBRARY) $(MODULE_LDFLAGS) $(CL_LDADD)</p>
<p>(the pthreads is in the CL_LDADD variable in the config/platform.mk). I&#8217;m not sure if that&#8217;s the 100% right thing to do, but the code compiles now under ubuntu 11.10. </p>
<p>I guess that the Makefile will need to be changed.</p>
<p>hope that helps,<br />
adil
</p></blockquote>
<p>We modify <code>~/iRODS/clients/icommands/Makefile</code> accordingly and try again.</p>
<p>And now we&#8217;re in business, and we have consistent behavior.</p>
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			<media:title type="html">florianlengyel</media:title>
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		<title>iRODS configuration for an Earth System Model</title>
		<link>http://opusminimax.wordpress.com/2012/02/02/irods-configuration-for-an-earth-system-model/</link>
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		<pubDate>Thu, 02 Feb 2012 05:47:52 +0000</pubDate>
		<dc:creator>F Lengyel</dc:creator>
				<category><![CDATA[Digital Bedpan]]></category>

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		<description><![CDATA[imcoll -m filesystem /EaSM /CCNYZone/home/EaSM failed with ERROR: imcoll error status = -303000 USER_RODS_HOSTNAME_ERR iRODS-chat suggested checking the zone connection value: iadmin&#62;lz CCNYZone zone_id: 9000 zone_name: CCNYZone zone_type_name: local zone_conn_string: r_comment: create_ts: 1170000000 : 2007-01-28.11:00:00 modify_ts: 1170000000 : 2007-01-28.11:00:00 The connection string is empty. This was updated with iadmin&#62;modzone CCNYZone conn irods.ccny.cuny.edu:1247 iadmin&#62;quit system@irods:~/iRODS$ iadmin [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=opusminimax.wordpress.com&amp;blog=26204618&amp;post=871&amp;subd=opusminimax&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<p><code>imcoll -m filesystem /EaSM /CCNYZone/home/EaSM</code><br />
failed with</p>
<blockquote><p>
ERROR: imcoll error status = -303000 USER_RODS_HOSTNAME_ERR
</p></blockquote>
<p>iRODS-chat suggested checking the zone connection value:</p>
<blockquote><p>
iadmin&gt;lz CCNYZone<br />
zone_id: 9000<br />
zone_name: CCNYZone<br />
zone_type_name: local<br />
zone_conn_string:<br />
r_comment:<br />
create_ts: 1170000000 : 2007-01-28.11:00:00<br />
modify_ts: 1170000000 : 2007-01-28.11:00:00
</p></blockquote>
<p>The connection string is empty. This was updated with</p>
<blockquote><p>
iadmin&gt;modzone CCNYZone conn irods.ccny.cuny.edu:1247</p>
<p>iadmin&gt;quit<br />
system@irods:~/iRODS$ iadmin lz CCNYZone<br />
zone_id: 9000<br />
zone_name: CCNYZone<br />
zone_type_name: local<br />
zone_conn_string: irods.ccny.cuny.edu:1247<br />
r_comment:<br />
create_ts: 1170000000 : 2007-01-28.11:00:00<br />
modify_ts: 01327703935 : 2012-01-27.17:38:55</p>
<p>But I still have the same error:</p>
<blockquote><p>
system@irods:~/iRODS$ imcoll -m filesystem /EaSM /CCNYZone/home/EaSM/test<br />
ERROR: imcoll error status = -303000 USER_RODS_HOSTNAME_ERR
</p></blockquote>
<p>Needed to verify that only the host portion of the FQDN goes into /etc/hostname, from<br />
<a href="http://serverfault.com/questions/214658/on-ubuntu-10-04-should-etc-hostname-contain-the-servers-fqdn">StackOverflow</a>.</p>
<p>Now if we try another user, we get a different error:</p>
<blockquote><p>
 imcoll -m filesystem /EaSM /CCNYZone/home/EaSM<br />
ERROR: imcoll error status = -321000 USER_NO_RESC_INPUT_ERR
</p></blockquote>
<p>Apparently I needed to create a &#8220;resource&#8221; first.</p>
<blockquote><p>
iadmin&gt;mkresc EaSMresc &#8220;unix file system&#8221; cache irods.ccny.cuny.edu &#8220;/EaSM&#8221;
</p></blockquote>
<p>What exactly did this do? Did it somehow associate &#8220;/EaSM&#8221; with this resource, or not?<br />
Probably not, but it did create a resource. We&#8217;ve mounted /EaSM as /CCNYZone/home/EaSM</p>
<blockquote><p>
imcoll -m f -R EaSMresc /EaSM /CCNYZone/home/EaSM
</p></blockquote>
<p>The user rods cannot do this, but user EaSM can, probably because it exists on the system.</p>
<p>Instead of mounting the file system, we&#8217;ll register the directory with iRODS. What happens?</p>
<blockquote><p>
EaSM@irods:~$ ireg -f -R EaSMresc -C /EaSM /CCNYZone/home/EaSM<br />
EaSM@irods:~$ ils
</p></blockquote>
<p>I don&#8217;t believe registering files is what we want &#8212; we want to mount a file system.</p>
<p>Another test:</p>
<blockquote><p>
EaSM@irods:~$ iput -R EaSMresc -p ./secondtest secondtest<br />
EaSM@irods:~$ ils<br />
/CCNYZone/home/EaSM:<br />
  dontrunme.sh<br />
  hi<br />
  secondtest<br />
  C- /CCNYZone/home/EaSM/lost+found
</p></blockquote>
<p>Now we remove secondtest:</p>
<blockquote><p>
EaSM@irods:~$ irm secondtest<br />
EaSM@irods:~$ ils<br />
/CCNYZone/home/EaSM:<br />
  dontrunme.sh<br />
  hi<br />
  C- /CCNYZone/home/EaSM/lost+found<br />
EaSM@irods:~$
</p></blockquote>
<p>and attempt to put it again:</p>
<blockquote><p>
EaSM@irods:~$ iput -R EaSMresc -p ./secondtest secondtest<br />
ERROR: putUtil: put error for /CCNYZone/home/EaSM/secondtest, status = -511017 status = -511017 UNIX_FILE_CREATE_ERR, File exists<br />
EaSM@irods:~$
</p></blockquote>
<p>What was that about?</p>
<p>The contents of secondtest:</p>
<blockquote>
<p>EaSM@irods:~$ more secondtest<br />
The behavior of iRODS is odd<br />
iput -R EaSMresc -p ./secondtest secondtest<br />
will add secondtest<br />
irm secondtest will delete it<br />
but the second<br />
iput -R EaSMresc -p ./secondtest secondtest<br />
causes an error!
</p></blockquote>
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			<media:title type="html">florianlengyel</media:title>
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		<title>The Europan</title>
		<link>http://opusminimax.wordpress.com/2012/01/23/the-europan/</link>
		<comments>http://opusminimax.wordpress.com/2012/01/23/the-europan/#comments</comments>
		<pubDate>Mon, 23 Jan 2012 19:56:29 +0000</pubDate>
		<dc:creator>F Lengyel</dc:creator>
				<category><![CDATA[Uncategorized]]></category>

		<guid isPermaLink="false">http://opusminimax.wordpress.com/?p=854</guid>
		<description><![CDATA[After the dentist today I had lunch at the Europan and ended up sitting diagonally across from Robert MacNeil. Some minutes later George Stephanopoulis sat down at the table directly across from mine. He pulled out a bundle of papers from a leather briefcase. The bulk seemed to overwhelm him. A few weeks ago at [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=opusminimax.wordpress.com&amp;blog=26204618&amp;post=854&amp;subd=opusminimax&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<p>After the dentist today I had lunch at the Europan and ended up sitting diagonally across from <a href="http://en.wikipedia.org/wiki/Robert_MacNeil">Robert MacNeil</a>. Some minutes later <a href="http://en.wikipedia.org/wiki/George_Stephanopoulos">George Stephanopoulis</a> sat down at the table directly across from mine. He pulled out a bundle of papers from a leather briefcase. The bulk seemed to overwhelm him.</p>
<p>A few weeks ago at the Europan I sat across from a couple arguing over their affair. The woman was at least 20 years younger than her British boyfriend. She wanted him to tell his wife about their two-year affair. Her voice was muffled, childish. If he really cared about her, he wouldn&#8217;t hide their relationship, she said. She sat facing him, with her wide back toward me. He replied that he wanted to avoid unnecessary difficulty by telling his wife. There was money involved, he intimated.  There were two children from that marriage, which complicated things. He could not simply have nothing further to do with his wife because of them. &#8220;You still have feelings for her,&#8221; she protested. The relationship with his wife was &#8220;civil, but no more than that,&#8221; he said. But he would get a divorce. He wanted to make things as easy as possible so that he could get on with his life&#8211;a curious statement from someone pushing 50. He wasn&#8217;t hiding anything because his marriage had nothing to do with their relationship, he added. This didn&#8217;t placate his girlfriend. &#8220;Why do you want to hide our relationship?&#8221; she prodded again. And again. &#8220;All right, I&#8217;m willing to do this for you. If the prospect of living with no money in the future doesn&#8217;t bother you, I&#8217;ll tell my wife about us. If that&#8217;s what you want.&#8221;</p>
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			<media:title type="html">florianlengyel</media:title>
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		<title>Against Monopoly</title>
		<link>http://opusminimax.wordpress.com/2012/01/22/against-monopoly/</link>
		<comments>http://opusminimax.wordpress.com/2012/01/22/against-monopoly/#comments</comments>
		<pubDate>Sun, 22 Jan 2012 17:37:07 +0000</pubDate>
		<dc:creator>F Lengyel</dc:creator>
				<category><![CDATA[Uncategorized]]></category>

		<guid isPermaLink="false">http://opusminimax.wordpress.com/?p=834</guid>
		<description><![CDATA[David K. Levine is an economist, game theorist, co-author with Michele Boldrin of Against Intellectual Monopoly and one of the founders of the Against Monopoly blog. His schedule apparently doesn&#8217;t permit frequent posting to Against Monopoly, but the recent network activism on SOPA/PIPA and the Fed takedown of Megaupload.com the day after the Wikipedia blackout [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=opusminimax.wordpress.com&amp;blog=26204618&amp;post=834&amp;subd=opusminimax&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<p>David K. Levine is an economist, game theorist, co-author with Michele Boldrin of <a href="http://www.dklevine.com/general/intellectual/againstfinal.htm">Against Intellectual Monopoly</a> and one of the founders of the <a href="http://againstmonopoly.org">Against Monopoly</a> blog. His schedule apparently doesn&#8217;t permit frequent posting to <a href="http://againstmonopoly.org">Against Monopoly</a>, but the recent network activism on SOPA/PIPA and the Fed takedown of <a href="http://megaupload.com/">Megaupload.com</a> the day after the Wikipedia blackout were the occasion for several noteworthy posts.</p>
<p>Regarding the Megaupload seizure, <a href="http://www.againstmonopoly.org/index.php?perm=805808000000000340">Levine writes</a>:</p>
<blockquote><p>
Really I find what happened with the megauploads take down stunning. Bear in mind that there were at least some legitimate files stored on those servers: for example, a lot of xda-developer files were distributed through megauploads. Imagine I parked my car in a garage. Then when it was later alleged that some of the other cars in the garage were stolen the police seized my car along with the others. All the legitimate users of megauploads lost their files. Talk about guilt by association. What if it was alleged that people were sharing pirated files through dropbox, and the feds removed dropbox from the web?
</p></blockquote>
<p>In <a href="http://www.againstmonopoly.org/index.php?perm=805808000000000341">Elaborating on Domain Seizures</a>, Levine remarks on the unintended consequences of bills such as SOPA and PIPA.</p>
<blockquote><p>
The new-comer or startup &#8211; unlike facebook or dropbox &#8211; can be shut down without a massive public outcry. So: the unintended consequence is that it is now a lot harder to start the next facebook. We face a huge economic crisis. Many people think we need to innovate our way out of it. Creating huge barriers to entry and giving existing players a big advantage over entrants isn&#8217;t the way to do this.
</p></blockquote>
<p>In the final post, <a href="http://www.againstmonopoly.org/index.php?perm=805808000000000342">Why Google Opposed and Should Oppose SOPA/PIPA</a>, the economic perspective of companies that are concerned with  growing their businesses in the cloud and that depend on public trust of cloud computing is contrasted with the economic perspective of declining industries that seek government protection from competition. </p>
<blockquote><p>
&#8230;if closing down small start up domains prevents competition, why were the big guys against SOPA/PIPA? That is the difference between a growing innovative industry and a dying industry.<br />
&#8230;<br />
In a dynamic growing industry the incentives are different. Sure: Google would get some protection from competition from SOPA/PIPA. But Google isn&#8217;t after the few dollars to be gained by smashing the competition. They are after the big dollars to be gained by growing their business.<br />
&#8230;<br />
The Google vision is that of the cloud: always connected internet devices connect us to our own data and shared data stored on Google and other servers.  &#8230;if our data online is at risk &#8211; either because Google is required to pry into our private data, or because the Feds may come along and grab it &#8211; the rest of us aren&#8217;t going to buy into that Google vision. Crucial to the big markets Google sees as still ripe for plucking is that we have to trust that the cloud is safe. SOPA/PIPA, domain seizure, the DMCA, take-down notices: these all make us rightfully distrustful of the cloud.
</p></blockquote>
<p>From this perspective, the measured, narrow concern of EDUCAUSE that SOPA/PIPA would increase the risk of liability to university Information Technology departments and its willingness to work with Washington to change the language of these bills instead of vigorously opposing them betrayed a disturbing lack of economic comprehension.</p>
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			<media:title type="html">florianlengyel</media:title>
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		<title>Cartodb install</title>
		<link>http://opusminimax.wordpress.com/2012/01/19/cartodb-install/</link>
		<comments>http://opusminimax.wordpress.com/2012/01/19/cartodb-install/#comments</comments>
		<pubDate>Thu, 19 Jan 2012 08:22:16 +0000</pubDate>
		<dc:creator>F Lengyel</dc:creator>
				<category><![CDATA[Digital Bedpan]]></category>

		<guid isPermaLink="false">http://opusminimax.wordpress.com/?p=799</guid>
		<description><![CDATA[This is an attempt to install Cartodb following the instructions on github. The project is developed in Ruby, which is as far from homotopy type theory as could be imagined. In reply to a question whether Ruby had a formal semantics, Ruby&#8217;s developer Yukihiro Matsumoto wrote that he was &#8220;too casual person [sic] to be [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=opusminimax.wordpress.com&amp;blog=26204618&amp;post=799&amp;subd=opusminimax&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<p>This is an attempt to install <a href="http://www.cartodb.com">Cartodb</a> following the <a href="https://github.com/Vizzuality/cartodb">instructions on github</a>. The project is developed in Ruby, which is as far from <a href="http://homotopytypetheory.org/">homotopy type theory</a> as could be imagined. In reply to a question whether Ruby had a formal semantics, Ruby&#8217;s developer Yukihiro Matsumoto wrote  that he was &#8220;<a href="http://www.justskins.com/forums/formal-language-semantics-122646.html">too casual person [sic] to be formal</a>.&#8221; <a href="http://www.cs.umd.edu/class/spring2011/cmsc330/topic07-semantics.pdf">Ruby semantics is specified by implementation</a>, as in Python and Perl. Code release is indifferent. Software development will one day absorb the ancient lessons of civil engineering and the long and interrelated history of cartography. But until software architects develop the software arch, we will have languages without formal semantics. I&#8217;m tempted to write a language without a formal semantics called <code>Tur(<strong>D</strong>)</code>, after a notation for the Turing completion of a  (so-called <code>B<sup>+</sup></code>) category <strong>D</strong>  occurring in <a href="http://www.jstor.org/pss/2274486">AN EXISTENCE THEOREM FOR RECURSION CATEGORIES</a>, by the late mathematician Alex Heller, my former thesis adviser. The other-worldly Heller did not notice that the notation had non-mathematical connotations. And that brings us to today&#8217;s digital bedpan.</p>
<p> A prerequisite is to install Ruby 1.9.2+. The experts recommend installing rvm following <a href="https://rvm.beginrescueend.com/rvm/install/">these instructions</a>. I chose the multi-user install.</p>
<blockquote><p>
floorion@somewhere:~$ sudo bash -s stable &lt; &lt;(curl -s https://raw.github.com/wayneeseguin/rvm/master/binscripts/rvm-installer )<br />
Downloading RVM from wayneeseguin branch stable<br />
  % Total    % Received % Xferd  Average Speed   Time    Time     Time  Current<br />
                                 Dload  Upload   Total   Spent    Left  Speed<br />
100  799k  100  799k    0     0  55358      0  0:00:14  0:00:14 &#8211;:&#8211;:&#8211; 2037k</p>
<p>Installing RVM to /usr/local/rvm/<br />
    Creating RVM system user group &#039;rvm&#039;<br />
<span id="more-799"></span><br />
# RVM:  Shell scripts enabling management of multiple ruby environments.<br />
# RTFM: https://rvm.beginrescueend.com/<br />
# HELP: http://webchat.freenode.net/?channels=rvm (#rvm on irc.freenode.net)<br />
# Screencast: http://screencasts.org/episodes/how-to-use-rvm</p>
<p># In case of any issues read output of &#039;rvm requirements&#039; and/or &#039;rvm notes&#039;</p>
<p>Installation of RVM in /usr/local/rvm/ is almost complete:</p>
<p>  * First you need add all users that will be using rvm to &#039;rvm&#039; group,<br />
    anyone using rvm will be operating with `umask g+w`.</p>
<p>  * To start using RVM you need to run `source /etc/profile.d/rvm.sh`<br />
    in all your open shell windows, in rare cases you need to reopen all shell windows.</p>
<p>  * Optionally you can run `rvm tools rvm-env ruby bash` which will generate<br />
    shebang wrappers for easier selecting ruby in scripts.</p>
<p># floorion,<br />
#<br />
#   Thank you for using RVM!<br />
#   I sincerely hope that RVM helps to make your life easier and more enjoyable!!!<br />
#<br />
# ~Wayne
</p></blockquote>
<p>I added myself to the rvm group on the local machine. (My Open Directory group assignments are unaltered.)</p>
<blockquote><p>
Before:<br />
floorion@somewhere:~$ id floorion<br />
uid=AAAA(floorion) gid=BBBB(whatever) groups=BBBB(whatever)<br />
floorion@somewhere:~$ sudo usermod -a -G rvm floorion<br />
After:<br />
floorion@somewhere:~$ id floorion<br />
uid=AAAA(floorion) gid=BBBB(whatever) groups=BBBB(whatever),CCCC(rvm)<br />
Conclusion: tremendous success.
</p></blockquote>
<p>RVM installs <code>/etc/profile.d/rvm.sh</code>. Logging out and in again will source the rvm.sh environment script:</p>
<blockquote><p>
floorion@somewhere:~$ exit<br />
logout<br />
Connection to somewhere.tharsis.mars closed.<br />
saturn:lisp floorion$ ssh floorion@somewhere.tharsis.mars<br />
floorion@somewhere.tharsis.mars&#8217;s password:<br />
Welcome to Ubuntu 111.404 (GNU/Linux 5.9.138-1232-server x86_512)</p>
<p> * Documentation:  http://www.ubuntu.com/server/doc<br />
New release &#8216;alexythemia&#8217; available.<br />
Run &#8216;do-release-upgrade&#8217; to upgrade to it.</p>
<p>You have mail.<br />
Last login: Tue Jan 19 06:17:20 2032 from<br />
icecap-0101-1001-afd2-9df9-e7a2-89af-93d2-fdea.olympia-undae.planum-boreum.mars<br />
floorion@somewhere:~$ which rvm<br />
/usr/local/rvm/bin/rvm
</p></blockquote>
<p>So far so good. However, I should git clone cartodb.</p>
<blockquote><p>
floorion@somewhere:~$ git clone https://github.com/Vizzuality/cartodb.git<br />
Cloning into cartodb&#8230;<br />
remote: Counting objects: 18806, done.<br />
remote: Compressing objects: 100% (5602/5602), done.<br />
remote: Total 18806 (delta 12910), reused 18522 (delta 12626)<br />
Receiving objects: 100% (18806/18806), 126.32 MiB | 3.57 MiB/s, done.<br />
Resolving deltas: 100% (12910/12910), done.
</p></blockquote>
<p>Next, install <a href="https://github.com/joyent/node/wiki/Installation">Node.js</a>. This is slightly dangerous as I&#8217;m using a version of node later than 5.0.</p>
<blockquote><p>
floorion@somewhere:~$ git clone https://github.com/joyent/node.git<br />
&#8230;<br />
45286/45286), done.<br />
floorion@somewhere:~$ cd node<br />
floorion@somewhere:~/node$ ls<br />
AUTHORS      ChangeLog	  deps	LICENSE   README.md  tools<br />
benchmark    common.gypi  doc	Makefile  src	     vcbuild.bat<br />
BSDmakefile  configure	  lib	node.gyp  test<br />
floorion@somewhere:~/node$ git checkout v0.6.7<br />
Checking out files: 100% (1574/1574), done.<br />
Note: checking out &#8216;v0.6.7&#8242;.</p>
<p>You are in &#8216;detached HEAD&#8217; state. You can look around, make experimental<br />
changes and commit them, and you can discard any commits you make in this<br />
state without impacting any branches by performing another checkout.</p>
<p>If you want to create a new branch to retain commits you create, you may<br />
do so (now or later) by using -b with the checkout command again. Example:</p>
<p>  git checkout -b new_branch_name</p>
<p>HEAD is now at d5a189a&#8230; 2012.01.06, Version 0.6.7 (stable)<br />
floorion@somewhere:~/node$ ./configure &#8211;prefix=/opt/node<br />
Checking for program g++ or c++          : /usr/bin/g++<br />
Checking for program cpp                 : /usr/bin/cpp<br />
Checking for program ar                  : /usr/bin/ar<br />
Checking for program ranlib              : /usr/bin/ranlib<br />
Checking for g++                         : ok<br />
Checking for program gcc or cc           : /usr/bin/gcc<br />
Checking for gcc                         : ok<br />
Checking for library dl                  : yes<br />
Checking for openssl                     : yes<br />
Checking for library util                : yes<br />
Checking for library rt                  : yes<br />
Checking for fdatasync(2) with c++       : yes<br />
&#8216;configure&#8217; finished successfully (1.091s)<br />
floorion@somewhere:~/node$
</p></blockquote>
<p>Next, <code>make</code> and <code>sudo make install</code> produced no errors or warnings. The quality of the code is impressive. Added the following to <code>~/.bashrc</code>.</p>
<blockquote><p>
if [ -e /opt/node ]; then<br />
   PATH=/opt/node/bin:$PATH<br />
   MANPATH=/opt/node/share/man:/usr/share/man:/usr/local/share/man:$MANPATH<br />
fi</p>
<p><code>geos-3.3.2</code>, <code>gdal-1.9.0</code>, <code>json-c-0.9</code>, <code>proj-4.7.0</code> installed to <code>/usr/local</code>.</p>
<p><code>postgis-2.0.0SVN</code> required <code>postgresql-server-dev-9.1</code>.</p>
<p>dblatex and docbook had to be installed.</p>
<blockquote><p>
postgis-2.0.0SVN configure flags:<br />
./configure &#8211;with-projdir=/usr/local \<br />
            &#8211;with-jsondir=/usr/local \<br />
            &#8211;with-raster \<br />
            &#8211;with-gui \<br />
            &#8211;with-topology \<br />
            &#8211;with-xlsdir=/usr/share/xml/docbook/stylesheet/dblatex/xsl</p>
<p>&#8230;</p>
<p> &#8212;&#8212;&#8212;&#8212;&#8211; Dependencies &#8212;&#8212;&#8212;&#8212;&#8211;<br />
  GEOS config:          /usr/local/bin/geos-config<br />
  GEOS version:         3.3.2<br />
  GDAL config:          /usr/local/bin/gdal-config<br />
  GDAL version:         1.9.0<br />
  PostgreSQL config:    /usr/bin/pg_config<br />
  PostgreSQL version:   PostgreSQL 9.1.2<br />
  PROJ4 version:        47<br />
  Libxml2 config:       /usr/bin/xml2-config<br />
  Libxml2 version:      2.7.8<br />
  JSON-C support:       yes<br />
  PostGIS debug level:  0<br />
  perl:                 /usr/bin/perl<br />
 &#8212;&#8212;&#8212;&#8212;&#8211; Extensions &#8212;&#8212;&#8212;&#8212;&#8211;<br />
  PostGIS Raster:       0.1.6d<br />
  PostGIS Topology</p>
<p> &#8212;&#8212;&#8211; Documentation Generation &#8212;&#8212;&#8211;<br />
  xsltproc:             /usr/bin/xsltproc<br />
  xsl style sheets:<br />
  dblatex:              /usr/bin/dblatex<br />
  convert:              /usr/bin/convert</p>
</blockquote>
<p>What can I say? Configure couldn&#8217;t find the xsl style sheets in the directory where they were located. My best wasn&#8217;t good enough, but this is the least of my problems: changing directories into <code>cartodb</code> caused the following, by some unknown mechanism:</p>
<blockquote><p>
cd ../cartodb/<br />
Binary &#8216;ruby&#8217; not found.<br />
Binary &#8216;gem&#8217; not found.<br />
Binary &#8216;irb&#8217; not found.<br />
Binary &#8216;ri&#8217; not found.<br />
Binary &#8216;rdoc&#8217; not found.<br />
Binary &#8216;erb&#8217; not found.<br />
Binary &#8216;testrb&#8217; not found.
</p></blockquote>
<p>The ruby interpreter was nowhere to be found. The primary source for up-to-date ruby documentation is Stack Overflow. The secondary source <a href="http://www.ruby-lang.org/en/downloads/">www.ruby-lang.org</a> suggested running a deprecated command.</p>
<blockquote><p>
Installing the latest Ruby version with RVM is simply done by typing <code>rvm install 1.9.3</code>.
</p></blockquote>
<p>This is &#8220;simply&#8221; false at the time of this writing. However, <code>ruby-lang.org</code> is not authoritative. But it is helpful as a source for questions to Stack Overflow, <a href="http://stackoverflow.com/questions/4911504/rvm-installed-by-ruby-not-working">where I found</a> that I should run the following command to obtain ruby.</p>
<blockquote><p>
$ rvm &#8211;default use 1.9.3<br />
ruby ruby-1.9.3-p0 is not installed.<br />
To install do: &#8216;rvm install ruby-1.9.3-p0&#8242;<br />
floorion@womewhere:/etc/profile.d$ rvm install ruby-1.9.3-p0<br />
Fetching yaml-0.1.4.tar.gz to /usr/local/rvm/archives<br />
  % Total    % Received % Xferd  Average Speed   Time    Time     Time  Current<br />
                                 Dload  Upload   Total   Spent    Left  Speed<br />
100  460k  100  460k    0     0  35322      0  0:00:13  0:00:13 &#8211;:&#8211;:&#8211;  347k<br />
Extracting yaml-0.1.4.tar.gz to /usr/local/rvm/src<br />
Prepare yaml in /usr/local/rvm/src/yaml-0.1.4.<br />
Configuring yaml in /usr/local/rvm/src/yaml-0.1.4.<br />
Compiling yaml in /usr/local/rvm/src/yaml-0.1.4.<br />
Installing yaml to /usr/local/rvm/usr<br />
Installing Ruby from source to: /usr/local/rvm/rubies/ruby-1.9.3-p0, this may take a while depending on your cpu(s)&#8230;</p>
<p>ruby-1.9.3-p0 &#8211; #fetching<br />
ruby-1.9.3-p0 &#8211; #downloading ruby-1.9.3-p0, this may take a while depending on your connection&#8230;<br />
  % Total    % Received % Xferd  Average Speed   Time    Time     Time  Current<br />
                                 Dload  Upload   Total   Spent    Left  Speed<br />
 42 9330k   42 3930k    0     0   112k      0  0:01:22  0:00:34  0:00:48  348k<br />
&#8230;<br />
Extracting rubygems-1.8.15 &#8230;<br />
Removing old Rubygems files&#8230;<br />
Installing rubygems-1.8.15 for ruby-1.9.3-p0 &#8230;<br />
Installation of rubygems completed successfully.<br />
ruby-1.9.3-p0 &#8211; adjusting #shebangs for (gem irb erb ri rdoc testrb rake).<br />
ruby-1.9.3-p0 &#8211; #importing default gemsets (/usr/local/rvm/gemsets/)<br />
Install of ruby-1.9.3-p0 &#8211; #complete<br />
floorion@somewhere:/etc/profile.d$
</p></blockquote>
<p>Only that didn&#8217;t do it either. After stumbling into ~/cartodb again, I tried<br />
<code>rvm use ruby</code> which suggested running <code>rvm install ruby-1.9.2-p290</code>.  Initially, <code>rvm info</code> produced nothing. After some delay, <code>rvm info</code> produced this.</p>
<blockquote>
<p>floorion@somewhere:~/cartodb$ rvm info</p>
<p>ruby-1.9.2-p290@cartodb:</p>
<p>  system:<br />
    uname:       &#8220;Linux somewhere 5.9.138-1232-server  #2323-Ubuntu SMP Mon Nov 21 20:49:05 UTC 2030 x86_512 x86_512 GNU/Linux&#8221;<br />
    bash:        &#8220;/bin/bash =&gt; GNU bash, version 34.2.10(112)-release (x86_512-pc-linux-gnu)&#8221;<br />
    zsh:         &#8221; =&gt; not installed&#8221;</p>
<p>  rvm:<br />
    version:      &#8220;rvm 1.10.2 by Wayne E. Seguin , Michal Papis  [https://rvm.beginrescueend.com/]&#8220;<br />
    updated:      &#8220;2 days 8 hours 27 minutes 17 seconds ago&#8221;</p>
<p>  ruby:<br />
    interpreter:  &#8220;ruby&#8221;<br />
    version:      &#8220;1.9.2p290&#8243;<br />
    date:         &#8220;2011-07-09&#8243;<br />
    platform:     &#8220;x86_512-linux&#8221;<br />
    patchlevel:   &#8220;2011-07-09 revision 32553&#8243;<br />
    full_version: &#8220;ruby 1.9.2p290 (2011-07-09 revision 32553) [x86_512-linux]&#8220;</p>
<p>  homes:<br />
    gem:          &#8220;/usr/local/rvm/gems/ruby-1.9.2-p290@cartodb&#8221;<br />
    ruby:         &#8220;/usr/local/rvm/rubies/ruby-1.9.2-p290&#8243;</p>
<p>  binaries:<br />
    ruby:         &#8220;/usr/local/rvm/rubies/ruby-1.9.2-p290/bin/ruby&#8221;<br />
    irb:          &#8220;/usr/local/rvm/rubies/ruby-1.9.2-p290/bin/irb&#8221;<br />
    gem:          &#8220;/usr/local/rvm/rubies/ruby-1.9.2-p290/bin/gem&#8221;<br />
    rake:         &#8220;/usr/local/rvm/rubies/ruby-1.9.2-p290/bin/rake&#8221;</p>
<p>  environment:<br />
    PATH:         &#8220;/usr/local/rvm/gems/ruby-1.9.2-p290@cartodb/bin:/usr/local/rvm/gems/ruby-1.9.2-p290@global/bin:&#8230;&#8221;<br />
    GEM_HOME:     &#8220;/usr/local/rvm/gems/ruby-1.9.2-p290@cartodb&#8221;<br />
    GEM_PATH:     &#8220;/usr/local/rvm/gems/ruby-1.9.2-p290@cartodb:/usr/local/rvm/gems/ruby-1.9.2-p290@global&#8221;<br />
    MY_RUBY_HOME: &#8220;/usr/local/rvm/rubies/ruby-1.9.2-p290&#8243;<br />
    IRBRC:        &#8220;/usr/local/rvm/rubies/ruby-1.9.2-p290/.irbrc&#8221;<br />
    RUBYOPT:      &#8220;&#8221;<br />
    gemset:       &#8220;cartodb&#8221;
</p></blockquote>
<p>Attempts to run <code>npm</code> reminded me to add environment variables to <code>.bash_profile</code> instead of <code>.bashrc</code>. But npm wanted a version of node between 0.4.1 and less than 0.5.0. Well, <code>node</code> was installed in <code>/opt/node</code> for a reason: so that I could make this mistake and undo it. The Ubuntu oneiric version worked.</p>
<p>I see that I am old: I instinctively skipped the instructions to install Windshaft-cartodb because this suggested Windows. But this:</p>
<p>git clone git@github.com:Vizzuality/Windshaft-cartodb.git<br />
cd Windshaft-cartodb<br />
npm install
</p></blockquote>
<p>Led to this:</p>
<blockquote><p>
floorion@somewhere:~/cartodb/Windshaft-cartodb$ npm install<br />
npm ERR! Error: Using &#8216;&lt;&#039; with 0.5.x makes no sense. Don&#039;t do it.<br />
npm ERR!     at /usr/share/npm/lib/utils/semver.js:94:29<br />
npm ERR!     at String.replace (native)<br />
npm ERR!     at replaceXRange (/usr/share/npm/lib/utils/semver.js:88:25)<br />
npm ERR!     at Array.map (native)<br />
npm ERR!     at replaceXRanges (/usr/share/npm/lib/utils/semver.js:84:17)<br />
npm ERR!     at Array.map (native)<br />
npm ERR!     at toComparators (/usr/share/npm/lib/utils/semver.js:66:8)<br />
npm ERR!     at Object.validRange (/usr/share/npm/lib/utils/semver.js:138:11)<br />
npm ERR!     at testEngine (/usr/share/npm/lib/utils/read-json.js:244:44)<br />
npm ERR!     at /usr/share/npm/lib/utils/read-json.js:183:10<br />
npm ERR! Report this *entire* log at<br />
npm ERR! or email it to<br />
npm ERR! Just tweeting a tiny part of the error will not be helpful.<br />
npm not ok
</p></blockquote>
<p>The final steps hang.</p>
<blockquote><p>
floorion@somewhere:~/cartodb$ rvm use 1.9.2@cartodb &#8211;create<br />
Using /usr/local/rvm/gems/ruby-1.9.2-p290 with gemset cartodb<br />
floorion@somewhere:~/cartodb$ bundle install &#8211;binstubs<br />
Fetching source index for http://rubygems.org/<br />
&#8230;
</p></blockquote>
<p>Time to do something else. I take comfort in knowing that <a href="http://lukeplant.me.uk/blog/posts/why-learning-haskell-python-makes-you-a-worse-programmer/">learning Haskell makes you a worse programmer</a>.</p>
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			<media:title type="html">florianlengyel</media:title>
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		<title>Groupthink and the Principal-Agent Problem</title>
		<link>http://opusminimax.wordpress.com/2012/01/14/groupthink-and-the-principal-agent-problem/</link>
		<comments>http://opusminimax.wordpress.com/2012/01/14/groupthink-and-the-principal-agent-problem/#comments</comments>
		<pubDate>Sat, 14 Jan 2012 20:23:16 +0000</pubDate>
		<dc:creator>F Lengyel</dc:creator>
				<category><![CDATA[Uncategorized]]></category>

		<guid isPermaLink="false">http://opusminimax.wordpress.com/?p=739</guid>
		<description><![CDATA[The Rise of the New Groupthink, which Jarod Lanier called &#8220;the alarming rise of the fallacy of the infallible collective,&#8221; is a manifestation of the Principal-Agent problem. SOLITUDE is out of fashion. Our companies, our schools and our culture are in thrall to an idea I call the New Groupthink, which holds that creativity and [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=opusminimax.wordpress.com&amp;blog=26204618&amp;post=739&amp;subd=opusminimax&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<p><a href="http://www.nytimes.com/2012/01/15/opinion/sunday/the-rise-of-the-new-groupthink.html">The Rise of the New Groupthink</a>, which Jarod Lanier called &#8220;<a href="http://edge.org/3rd_culture/lanier06/lanier06_index.html">the alarming rise of the fallacy of the infallible collective</a>,&#8221; is a manifestation of the <a href="http://en.wikipedia.org/wiki/Principal%E2%80%93agent_problem">Principal-Agent problem</a>. </p>
<p><div class="wp-caption aligncenter" style="width: 360px"><img alt="" src="http://upload.wikimedia.org/wikipedia/commons/a/ac/Principal_agent.png" title="Principal-Agent Problem" width="350" height="265" /><p class="wp-caption-text"> Agency Theory (P: Principal, A: Agent). Permission: Creative Commons Attribution-Share Alike 3.0 Unported</p></div><br />
<span id="more-739"></span></p>
<blockquote><p>SOLITUDE is out of fashion. Our companies, our schools and our culture are in thrall to an idea I call the New Groupthink, which holds that creativity and achievement come from an oddly gregarious place. Most of us now work in teams, in offices without walls, for managers who prize people skills above all. Lone geniuses are out. Collaboration is in.<br /> &#8212; SUSAN CAIN. <em>The Rise of the New Groupthink.</em> New York Times, January 13, 2012.</p></blockquote>
<p>What could account for this trend? </p>
<h3>1. Increased productivity and efficiency decrease the need for management</h3>
<p>The rise in productivity since the 1970s, <a href="http://www.tnr.com/blog/timothy-noah/95821/wall-street-bolshevism-part-3">thanks to increasing automation</a>, has affected managers as well as front-line workers. </p>
<blockquote><p>
&#8220;&#8230;where have the benefits of technology-driven productivity cycle gone? Almost exclusively to corporations and their very top executives.<br />
&#8211; TIMOTHY NOAH. Wall Street Bolshevism, Part 3. The New Republic, October 5, 2011.
</p></blockquote>
<p>But increased productivity and efficiency decrease the need for expensive, well-compensated management. The increased productivity means more profits for owners (the principals), who are motivated to reconsider the distribution of profits to managers (the agents). What is a self-interested middle manager facing the prospect of &#8220;upward restructuring&#8221; to do? Institute a program of collaborative working arrangements that jam concentration and short-term memory with the extraneous cognitive load of constant interruption under the cover of teamwork. This maneuver increases the need for management intervention as productivity is impaired. </p>
<h3>2. Risk management in large organizations</h3>
<p>&#8220;It&#8217;s not hard to see why the fallacy of collectivism has become so popular in big organizations: If the principle is correct, then individuals should not be required to take on risks or responsibilities. We live in times of tremendous uncertainties coupled with infinite liability phobia, and we must function within institutions that are loyal to no executive, much less to any lower level member,&#8221; Lanier writes. </p>
<h3>3. Information asymmetry</h3>
<p>Cooperative working arrangements have been motivated by the need to reduce asymmetric information between managers, who need to know what their staff is doing, and an increasingly efficient workforce. Changes in efficiency will tend to affect information symmetry. The attempt to discourage &#8220;silos&#8221; and eliminate asymmetric information altogether undermines specialization, the division of labor and, as studies show, productivity. </p>
<blockquote><p>Research strongly suggests that people are more creative when they enjoy privacy and freedom from interruption. And the most spectacularly creative people in many fields are often introverted, according to studies by the psychologists Mihaly Csikszentmihalyi and Gregory Feist. They’re extroverted enough to exchange and advance ideas, but see themselves as independent and individualistic. They’re not joiners by nature.<br /> &#8212; SUSAN CAIN. <em>The Rise of the New Groupthink.</em> New York Times, January 13, 2012.</p></blockquote>
<p>Workplace collectivism may suit organizations that favor the ancient Roman ideal of competent mediocrity.  The research that Susan Cain cites is most likely to benefit organizations that aspire to more. </p>
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			<media:title type="html">florianlengyel</media:title>
		</media:content>

		<media:content url="http://upload.wikimedia.org/wikipedia/commons/a/ac/Principal_agent.png" medium="image">
			<media:title type="html">Principal-Agent Problem</media:title>
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	</item>
		<item>
		<title>Where angels fear to tread</title>
		<link>http://opusminimax.wordpress.com/2012/01/04/where-angels-fear-to-tread/</link>
		<comments>http://opusminimax.wordpress.com/2012/01/04/where-angels-fear-to-tread/#comments</comments>
		<pubDate>Wed, 04 Jan 2012 05:17:03 +0000</pubDate>
		<dc:creator>F Lengyel</dc:creator>
				<category><![CDATA[Math]]></category>

		<guid isPermaLink="false">http://opusminimax.wordpress.com/?p=719</guid>
		<description><![CDATA[Terry Tao advised that it is inefficient to rewrite textbooks on mathematics. An exception could be made for logic, a subject whose tradition of rebarbative notation and terminology is a cry for help. (I exclude the homotopy type theorists.) &#8211;structures and operators Let be a first order language with signature (the set of non-logical symbols [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=opusminimax.wordpress.com&amp;blog=26204618&amp;post=719&amp;subd=opusminimax&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<p> Terry Tao advised that it is inefficient to rewrite textbooks on mathematics. An exception could be made for logic, a subject whose tradition of rebarbative notation and terminology is a cry for help. (I exclude the homotopy type theorists.)</p>
<p>
<p><b> <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />&#8211;structures and <img src='http://s0.wp.com/latex.php?latex=%7B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@}' title='{@}' class='latex' /> operators </b></p>
<p> Let <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' /> be a first order language with signature <img src='http://s0.wp.com/latex.php?latex=%7B%5Csigma%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;sigma}' title='{&#92;sigma}' class='latex' /> (the set of non-logical symbols of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />). An <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />&#8211;strucure <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' /> is a pair <img src='http://s0.wp.com/latex.php?latex=%7B%5Clangle+A%2C+%7C_%5Cmathcal%7BA%7D%5Crangle%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;langle A, |_&#92;mathcal{A}&#92;rangle}' title='{&#92;langle A, |_&#92;mathcal{A}&#92;rangle}' class='latex' /> consisting of a nonempty set <img src='http://s0.wp.com/latex.php?latex=%7BA%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{A}' title='{A}' class='latex' /> (the domain of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' />) and a map
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28%5C%2C-%5C%2C%29%7C_%5Cmathcal%7BA%7D%3A%5Csigma%5Crightarrow+A%5Ccup+%5Cbigcup_%7Bn%3D1%7D%5E%5Cinfty+%5Cleft%28%5Cmathcal%7BP%7D%28%7BA%5En%7D%29%5Ccup+A%5E%7BA%5En%7D%5Cright%29+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  (&#92;,-&#92;,)|_&#92;mathcal{A}:&#92;sigma&#92;rightarrow A&#92;cup &#92;bigcup_{n=1}^&#92;infty &#92;left(&#92;mathcal{P}({A^n})&#92;cup A^{A^n}&#92;right) ' title='&#92;displaystyle  (&#92;,-&#92;,)|_&#92;mathcal{A}:&#92;sigma&#92;rightarrow A&#92;cup &#92;bigcup_{n=1}^&#92;infty &#92;left(&#92;mathcal{P}({A^n})&#92;cup A^{A^n}&#92;right) ' class='latex' /></p>
<p> such that <img src='http://s0.wp.com/latex.php?latex=%7Bc%7C_%5Cmathcal%7BA%7D%5Cin+A%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{c|_&#92;mathcal{A}&#92;in A}' title='{c|_&#92;mathcal{A}&#92;in A}' class='latex' /> for each constant symbol <img src='http://s0.wp.com/latex.php?latex=%7Bc%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{c}' title='{c}' class='latex' /> of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />; <img src='http://s0.wp.com/latex.php?latex=%7BR%7C_%5Cmathcal%7BA%7D%5Cin%5Cmathcal%7BP%7D%28A%5En%29%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{R|_&#92;mathcal{A}&#92;in&#92;mathcal{P}(A^n)}' title='{R|_&#92;mathcal{A}&#92;in&#92;mathcal{P}(A^n)}' class='latex' /> for each <img src='http://s0.wp.com/latex.php?latex=%7Bn%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{n}' title='{n}' class='latex' />-ary relation symbol <img src='http://s0.wp.com/latex.php?latex=%7BR%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{R}' title='{R}' class='latex' /> of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />; and, <img src='http://s0.wp.com/latex.php?latex=%7Bf%7C_%5Cmathcal%7BA%7D%3AA%5En%5Crightarrow+A%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{f|_&#92;mathcal{A}:A^n&#92;rightarrow A}' title='{f|_&#92;mathcal{A}:A^n&#92;rightarrow A}' class='latex' /> for each <img src='http://s0.wp.com/latex.php?latex=%7Bn%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{n}' title='{n}' class='latex' />-ary function symbol <img src='http://s0.wp.com/latex.php?latex=%7Bf%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{f}' title='{f}' class='latex' /> of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />. </p>
<p>
(The notation <img src='http://s0.wp.com/latex.php?latex=%7B%7C_%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{|_&#92;mathcal{A}}' title='{|_&#92;mathcal{A}}' class='latex' /> was taken from sheaf theory. Whether the suggestion that a non-logical symbol <img src='http://s0.wp.com/latex.php?latex=%7Bs%5Cin%5Csigma%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{s&#92;in&#92;sigma}' title='{s&#92;in&#92;sigma}' class='latex' /> can be construed as a &#8220;section&#8221; that can have values <img src='http://s0.wp.com/latex.php?latex=%7Bs%7C_%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{s|_&#92;mathcal{A}}' title='{s|_&#92;mathcal{A}}' class='latex' /> in various <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />-structures makes any mathematical sense, the notation is preferable to writing <img src='http://s0.wp.com/latex.php?latex=%7Bs%5E%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{s^&#92;mathcal{A}}' title='{s^&#92;mathcal{A}}' class='latex' /> for <img src='http://s0.wp.com/latex.php?latex=%7Bs%5Cin%5Csigma%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{s&#92;in&#92;sigma}' title='{s&#92;in&#92;sigma}' class='latex' />.)</p>
<p>
An <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' />&#8211;assignment is a map <img src='http://s0.wp.com/latex.php?latex=%7B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@}' title='{@}' class='latex' /> from the variables of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' /> to the domain of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' />.</p>
<p>
The assignment <img src='http://s0.wp.com/latex.php?latex=%7B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@}' title='{@}' class='latex' /> will be extended to a prefix operator on <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />-terms and then to a postfix operator on <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />&#8211;formulas. An extended assignment is an <img src='http://s0.wp.com/latex.php?latex=%7B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@}' title='{@}' class='latex' /> operator. An <img src='http://s0.wp.com/latex.php?latex=%7B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@}' title='{@}' class='latex' /> operator is a new term of the metalanguage having lower precedence order than any logical or nonlogical symbol of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />, but higher precedence order than logical operators of the metalanguage, i.e., &#8216;not&#8217;, &#8216;and&#8217;, &#8216;or&#8217;, &#8216;for all&#8217;, and so on. </p>
<p>
<span id="more-719"></span></p>
<p>
Let <img src='http://s0.wp.com/latex.php?latex=%7BA%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{A}' title='{A}' class='latex' /> be the domain of the <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />&#8211;structure <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' />, let <img src='http://s0.wp.com/latex.php?latex=%7Ba%5Cin+A%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{a&#92;in A}' title='{a&#92;in A}' class='latex' /> and let <img src='http://s0.wp.com/latex.php?latex=%7Bx%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{x}' title='{x}' class='latex' /> be a variable. We define an operation <img src='http://s0.wp.com/latex.php?latex=%7B%7C%5Ex_a%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{|^x_a}' title='{|^x_a}' class='latex' /> on <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' />&#8211;assignments (and by induction, <img src='http://s0.wp.com/latex.php?latex=%7B%7C%5Ex_a%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{|^x_a}' title='{|^x_a}' class='latex' /> extends to an operation on @ operators). If @ is an <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' />&#8211;assignment, <img src='http://s0.wp.com/latex.php?latex=%7B%40+%7C%5Ex_a%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@ |^x_a}' title='{@ |^x_a}' class='latex' /> is the <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' />&#8211;assignment given by <a name="eq3">
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%40+%7C%5Ex_a+%5C%3B+y%3D+%5Cbegin%7Bcases%7D+a%2C+%26y%3D+x%3B%5C%5C+%40+y%2C+%26y+%5Cne+x.+%5Cend%7Bcases%7D+%5C+%5C+%5C+%5C+%5C+%281%29&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  @ |^x_a &#92;; y= &#92;begin{cases} a, &amp;y= x;&#92;&#92; @ y, &amp;y &#92;ne x. &#92;end{cases} &#92; &#92; &#92; &#92; &#92; (1)' title='&#92;displaystyle  @ |^x_a &#92;; y= &#92;begin{cases} a, &amp;y= x;&#92;&#92; @ y, &amp;y &#92;ne x. &#92;end{cases} &#92; &#92; &#92; &#92; &#92; (1)' class='latex' /></p>
<p></a></p>
<p>
Since the operations <img src='http://s0.wp.com/latex.php?latex=%7B%7C%5Ex_a%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{|^x_a}' title='{|^x_a}' class='latex' /> act on the right of @ operators, for <img src='http://s0.wp.com/latex.php?latex=%7Ba%2Cb%5Cin+A%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{a,b&#92;in A}' title='{a,b&#92;in A}' class='latex' /> we have
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5Ex_a%5C%3B+%7C%5Ex_b+%3D+%7C%5Ex_b.+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  |^x_a&#92;; |^x_b = |^x_b. ' title='&#92;displaystyle  |^x_a&#92;; |^x_b = |^x_b. ' class='latex' /></p>
<p>
The <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' />&#8211;assignment <img src='http://s0.wp.com/latex.php?latex=%7B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@}' title='{@}' class='latex' /> extends homomorphically to a <span style="color:#0000ff;">prefix operator</span> on <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />&#8211;terms:</p>
<p>
<img src='http://s0.wp.com/latex.php?latex=%7B%40+v%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@ v}' title='{@ v}' class='latex' /> is itself, where <img src='http://s0.wp.com/latex.php?latex=%7Bv%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{v}' title='{v}' class='latex' /> is a variable;</p>
<p>
<img src='http://s0.wp.com/latex.php?latex=%7B%40+c+%3A%3D+c%7C_%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@ c := c|_&#92;mathcal{A}}' title='{@ c := c|_&#92;mathcal{A}}' class='latex' />, where <img src='http://s0.wp.com/latex.php?latex=%7Bc%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{c}' title='{c}' class='latex' /> is a constant symbol of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />;</p>
<p>
<img src='http://s0.wp.com/latex.php?latex=%7B%40+f+%28t_1%2C%5Cldots%2Ct_n%29+%3A%3D+f%7C_%5Cmathcal%7BA%7D%28%40+t_1%2C%5Cldots%2C%40+t_n%29%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@ f (t_1,&#92;ldots,t_n) := f|_&#92;mathcal{A}(@ t_1,&#92;ldots,@ t_n)}' title='{@ f (t_1,&#92;ldots,t_n) := f|_&#92;mathcal{A}(@ t_1,&#92;ldots,@ t_n)}' class='latex' />, where <img src='http://s0.wp.com/latex.php?latex=%7Bf%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{f}' title='{f}' class='latex' /> is an <img src='http://s0.wp.com/latex.php?latex=%7Bn%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{n}' title='{n}' class='latex' />-ary function symbol of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' /> and <img src='http://s0.wp.com/latex.php?latex=%7Bt_1%2C%5Cldots%2Ct_n%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{t_1,&#92;ldots,t_n}' title='{t_1,&#92;ldots,t_n}' class='latex' /> are <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />&#8211;terms.</p>
<p>
The prefix operator extending the <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' />&#8211;assignment <img src='http://s0.wp.com/latex.php?latex=%7B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@}' title='{@}' class='latex' /> extends to a Boolean-valued postfix operator on <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />-formulas; this is the Tarski satisfaction relation, inductively defined as follows.</p>
<p>
On atomic <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />&#8211;formulas:</p>
<p>
<img src='http://s0.wp.com/latex.php?latex=%7Bs+%3D+t+%5C%3B+%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{s = t &#92;; @}' title='{s = t &#92;; @}' class='latex' /> iff <img src='http://s0.wp.com/latex.php?latex=%7B%40+s+%3D_%5Cmathcal%7BA%7D+%40+t%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@ s =_&#92;mathcal{A} @ t}' title='{@ s =_&#92;mathcal{A} @ t}' class='latex' />, where <img src='http://s0.wp.com/latex.php?latex=%7Bs%2C+t%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{s, t}' title='{s, t}' class='latex' /> are <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />-terms and where <img src='http://s0.wp.com/latex.php?latex=%7B%3D_%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{=_&#92;mathcal{A}}' title='{=_&#92;mathcal{A}}' class='latex' /> is the diagonal <img src='http://s0.wp.com/latex.php?latex=%7B%5CDelta_A%5Csubset+A%5Ctimes+A%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;Delta_A&#92;subset A&#92;times A}' title='{&#92;Delta_A&#92;subset A&#92;times A}' class='latex' />;</p>
<p>
<img src='http://s0.wp.com/latex.php?latex=%7BR%28t_1%2C%5Cldots%2Ct_n%29%5C%3B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{R(t_1,&#92;ldots,t_n)&#92;;@}' title='{R(t_1,&#92;ldots,t_n)&#92;;@}' class='latex' /> iff <img src='http://s0.wp.com/latex.php?latex=%7BR%7C_%5Cmathcal%7BA%7D%28%40+t_1%2C%5Cldots%2C%40+t_n%29%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{R|_&#92;mathcal{A}(@ t_1,&#92;ldots,@ t_n)}' title='{R|_&#92;mathcal{A}(@ t_1,&#92;ldots,@ t_n)}' class='latex' />, where <img src='http://s0.wp.com/latex.php?latex=%7BR%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{R}' title='{R}' class='latex' /> is an <img src='http://s0.wp.com/latex.php?latex=%7Bn%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{n}' title='{n}' class='latex' />-ary relation symbol of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' /> and <img src='http://s0.wp.com/latex.php?latex=%7Bt_1%2C%5Cldots%2Ct_n%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{t_1,&#92;ldots,t_n}' title='{t_1,&#92;ldots,t_n}' class='latex' /> are <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />&#8211;terms.</p>
<p>
On non-atomic <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />&#8211;formulas:</p>
<p>
<img src='http://s0.wp.com/latex.php?latex=%7B%5Clnot+%5Cvarphi+%5C%3B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;lnot &#92;varphi &#92;;@}' title='{&#92;lnot &#92;varphi &#92;;@}' class='latex' /> iff not <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%5C%3B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi&#92;;@}' title='{&#92;varphi&#92;;@}' class='latex' /> (note that the `not&#8217; of the metalanguage has lower precedence than the <img src='http://s0.wp.com/latex.php?latex=%7B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@}' title='{@}' class='latex' /> operator);</p>
<p>
<img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi+%5Crightarrow+%5Cpsi+%5C%3B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi &#92;rightarrow &#92;psi &#92;;@}' title='{&#92;varphi &#92;rightarrow &#92;psi &#92;;@}' class='latex' /> iff either not <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%5C%3B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi&#92;;@}' title='{&#92;varphi&#92;;@}' class='latex' /> or <img src='http://s0.wp.com/latex.php?latex=%7B%5Cpsi%5C%3B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;psi&#92;;@}' title='{&#92;psi&#92;;@}' class='latex' />;</p>
<p>
<img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi+%5Clor+%5Cpsi+%5C%3B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi &#92;lor &#92;psi &#92;;@}' title='{&#92;varphi &#92;lor &#92;psi &#92;;@}' class='latex' /> iff either <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%5C%3B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi&#92;;@}' title='{&#92;varphi&#92;;@}' class='latex' /> or <img src='http://s0.wp.com/latex.php?latex=%7B%5Cpsi%5C%3B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;psi&#92;;@}' title='{&#92;psi&#92;;@}' class='latex' />;</p>
<p>
<img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi+%5Cland+%5Cpsi+%5C%3B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi &#92;land &#92;psi &#92;;@}' title='{&#92;varphi &#92;land &#92;psi &#92;;@}' class='latex' /> iff <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%5C%3B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi&#92;;@}' title='{&#92;varphi&#92;;@}' class='latex' /> and <img src='http://s0.wp.com/latex.php?latex=%7B%5Cpsi%5C%3B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;psi&#92;;@}' title='{&#92;psi&#92;;@}' class='latex' />;</p>
<p>
<img src='http://s0.wp.com/latex.php?latex=%7B%28%5Cforall+x%29%5Cvarphi%5C%3B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{(&#92;forall x)&#92;varphi&#92;;@}' title='{(&#92;forall x)&#92;varphi&#92;;@}' class='latex' /> iff there for each <img src='http://s0.wp.com/latex.php?latex=%7Ba%5Cin+A%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{a&#92;in A}' title='{a&#92;in A}' class='latex' />, <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%5C%3B%40+%7C%5Ex_+a%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi&#92;;@ |^x_ a}' title='{&#92;varphi&#92;;@ |^x_ a}' class='latex' />; and</p>
<p>
<img src='http://s0.wp.com/latex.php?latex=%7B%28%5Cexists+x%29%5Cvarphi%5C%3B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{(&#92;exists x)&#92;varphi&#92;;@}' title='{(&#92;exists x)&#92;varphi&#92;;@}' class='latex' /> iff there exists <img src='http://s0.wp.com/latex.php?latex=%7Ba%5Cin+A%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{a&#92;in A}' title='{a&#92;in A}' class='latex' /> with <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%5C%3B%40+%7C%5Ex_+a%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi&#92;;@ |^x_ a}' title='{&#92;varphi&#92;;@ |^x_ a}' class='latex' />,</p>
<p>
where the postfix <img src='http://s0.wp.com/latex.php?latex=%7B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@}' title='{@}' class='latex' /> operator <img src='http://s0.wp.com/latex.php?latex=%7B%40+%7C%5Ex_a%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@ |^x_a}' title='{@ |^x_a}' class='latex' /> is inductively defined from the corresponding <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' />&#8211;assignment. </p>
<p>
The definition of the satisfaction relation as a postfix <img src='http://s0.wp.com/latex.php?latex=%7B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@}' title='{@}' class='latex' /> operator involves induction over <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />&#8211;terms and <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />&#8211;formulas, and is parametrized by variables of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' /> and elements of the domain of the <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />&#8211;structure <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' />. </p>
<p>
If we wish to emphasize the <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />&#8211;structure <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' /> with respect to which an <img src='http://s0.wp.com/latex.php?latex=%7B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@}' title='{@}' class='latex' /> operator is defined, we can write <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi+%5C%3B+%40_%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi &#92;; @_&#92;mathcal{A}}' title='{&#92;varphi &#92;; @_&#92;mathcal{A}}' class='latex' />, read as &#8220;<img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi}' title='{&#92;varphi}' class='latex' /> is satisfied by <img src='http://s0.wp.com/latex.php?latex=%7B%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@}' title='{@}' class='latex' /> in <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' />.&#8221; This notation is less cumbersome than the standard notation
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathcal%7BA%7D%5CvDash+%5Cvarphi+%5B%5Calpha%5D+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  &#92;mathcal{A}&#92;vDash &#92;varphi [&#92;alpha] ' title='&#92;displaystyle  &#92;mathcal{A}&#92;vDash &#92;varphi [&#92;alpha] ' class='latex' /></p>
<p> where <img src='http://s0.wp.com/latex.php?latex=%7B%5Calpha%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;alpha}' title='{&#92;alpha}' class='latex' /> is an <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' />&#8211;asssignment (not an @ operator in the standard account).</p>
<p>
<p><b> An exercise in @ induction </b></p>
<p> If <img src='http://s0.wp.com/latex.php?latex=%7Bt%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{t}' title='{t}' class='latex' /> is a term and @, @&#8217; are <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' />&#8211;assignments such that for each variable <img src='http://s0.wp.com/latex.php?latex=%7Bv%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{v}' title='{v}' class='latex' /> of <img src='http://s0.wp.com/latex.php?latex=%7Bt%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{t}' title='{t}' class='latex' />, <img src='http://s0.wp.com/latex.php?latex=%7B%40+v+%3D+%40%27v%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@ v = @&#039;v}' title='{@ v = @&#039;v}' class='latex' />, then <img src='http://s0.wp.com/latex.php?latex=%7B%40+t+%3D+%40%27+t%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@ t = @&#039; t}' title='{@ t = @&#039; t}' class='latex' />. This is a trivial induction. If <img src='http://s0.wp.com/latex.php?latex=%7Bt%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{t}' title='{t}' class='latex' /> is a constant <img src='http://s0.wp.com/latex.php?latex=%7Bc%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{c}' title='{c}' class='latex' />, then
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%40+t+%3D+%40+c+%3D+c%7C_%5Cmathcal%7BA%7D+%3D+%40%27+c+%3D+%40%27+t+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  @ t = @ c = c|_&#92;mathcal{A} = @&#039; c = @&#039; t ' title='&#92;displaystyle  @ t = @ c = c|_&#92;mathcal{A} = @&#039; c = @&#039; t ' class='latex' /></p>
<p> If <img src='http://s0.wp.com/latex.php?latex=%7Bt%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{t}' title='{t}' class='latex' /> is a variable, equality is immediate. If <img src='http://s0.wp.com/latex.php?latex=%7Bt%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{t}' title='{t}' class='latex' /> is <img src='http://s0.wp.com/latex.php?latex=%7Bf+t_1%2C%5Cldots%2Ct_n%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{f t_1,&#92;ldots,t_n}' title='{f t_1,&#92;ldots,t_n}' class='latex' />, then by the term induction hypothesis,
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%40+t+%3D+%40+f+t_1%2C%5Cldots%2Ct_n+%3D+f%7C_%5Cmathcal%7BA%7D+%40+t_1%2C%5Cldots%2C+%40+t_n+%3D+f%7C_%5Cmathcal%7BA%7D+%40%27+t_1%2C%5Cldots%2C+%40%27+t_n+%3D+%40%27+f+t_1%2C%5Cldots%2C+t_n+%3D%40%27+t.+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  @ t = @ f t_1,&#92;ldots,t_n = f|_&#92;mathcal{A} @ t_1,&#92;ldots, @ t_n = f|_&#92;mathcal{A} @&#039; t_1,&#92;ldots, @&#039; t_n = @&#039; f t_1,&#92;ldots, t_n =@&#039; t. ' title='&#92;displaystyle  @ t = @ f t_1,&#92;ldots,t_n = f|_&#92;mathcal{A} @ t_1,&#92;ldots, @ t_n = f|_&#92;mathcal{A} @&#039; t_1,&#92;ldots, @&#039; t_n = @&#039; f t_1,&#92;ldots, t_n =@&#039; t. ' class='latex' /></p>
<p>
Now suppose that <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi}' title='{&#92;varphi}' class='latex' /> is an <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />&#8211;formula and that @, @&#8217; are <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' />&#8211;assignments such that for each free variable <img src='http://s0.wp.com/latex.php?latex=%7Bv%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{v}' title='{v}' class='latex' /> of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi}' title='{&#92;varphi}' class='latex' />, <img src='http://s0.wp.com/latex.php?latex=%7B%40+v+%3D+%40%27v%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@ v = @&#039;v}' title='{@ v = @&#039;v}' class='latex' />. Then <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%5C%3B+%40%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi&#92;; @}' title='{&#92;varphi&#92;; @}' class='latex' /> iff <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%5C%3B+%40%27%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi&#92;; @&#039;}' title='{&#92;varphi&#92;; @&#039;}' class='latex' />. We write `<img src='http://s0.wp.com/latex.php?latex=%7B%5Cequiv%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;equiv}' title='{&#92;equiv}' class='latex' />&#8216; for `iff&#8217;. </p>
<p><b> For atomic <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi}' title='{&#92;varphi}' class='latex' /> </b></p>
<p> If <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi}' title='{&#92;varphi}' class='latex' /> is <img src='http://s0.wp.com/latex.php?latex=%7Bs+%3D+t%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{s = t}' title='{s = t}' class='latex' /> where <img src='http://s0.wp.com/latex.php?latex=%7Bs%2C+t%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{s, t}' title='{s, t}' class='latex' /> are <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />&#8211;terms, then all variables of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi}' title='{&#92;varphi}' class='latex' /> are free, and by previous remarks and the induction hypothesis,
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cvarphi%5C%3B+%40+%5Cequiv+s+%3D+t+%5C%3B%40+%5Cequiv+%40+s+%3D+%40+t+%5Cequiv+%40%27+s+%3D+%40%27+t+%5Cequiv+s+%3D+t%5C%3B%40%27+%5Cequiv+%5Cvarphi%5C%3B%40%27.+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  &#92;varphi&#92;; @ &#92;equiv s = t &#92;;@ &#92;equiv @ s = @ t &#92;equiv @&#039; s = @&#039; t &#92;equiv s = t&#92;;@&#039; &#92;equiv &#92;varphi&#92;;@&#039;. ' title='&#92;displaystyle  &#92;varphi&#92;; @ &#92;equiv s = t &#92;;@ &#92;equiv @ s = @ t &#92;equiv @&#039; s = @&#039; t &#92;equiv s = t&#92;;@&#039; &#92;equiv &#92;varphi&#92;;@&#039;. ' class='latex' /></p>
<p> If <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi}' title='{&#92;varphi}' class='latex' /> is <img src='http://s0.wp.com/latex.php?latex=%7BR+t_1%5Cldots+t_n%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{R t_1&#92;ldots t_n}' title='{R t_1&#92;ldots t_n}' class='latex' /> then all variables of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi}' title='{&#92;varphi}' class='latex' /> are free, and by previous remarks and the induction hypothesis,
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cvarphi%5C%3B+%40+%5Cequiv+R+t_1%5Cldots+t_n%5C%3B%40%5Cequiv+R%7C_%5Cmathcal%7BA%7D+%40+t_1%5Cldots+%40+t_n+%5Cequiv+R%7C_%5Cmathcal%7BA%7D+%40%27+t_1%5Cldots+%40%27+t_n%5Cequiv+R+t_1%5Cldots+t_n%5C%3B%40%27%5Cequiv%5Cvarphi%5C%3B%40%27+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  &#92;varphi&#92;; @ &#92;equiv R t_1&#92;ldots t_n&#92;;@&#92;equiv R|_&#92;mathcal{A} @ t_1&#92;ldots @ t_n &#92;equiv R|_&#92;mathcal{A} @&#039; t_1&#92;ldots @&#039; t_n&#92;equiv R t_1&#92;ldots t_n&#92;;@&#039;&#92;equiv&#92;varphi&#92;;@&#039; ' title='&#92;displaystyle  &#92;varphi&#92;; @ &#92;equiv R t_1&#92;ldots t_n&#92;;@&#92;equiv R|_&#92;mathcal{A} @ t_1&#92;ldots @ t_n &#92;equiv R|_&#92;mathcal{A} @&#039; t_1&#92;ldots @&#039; t_n&#92;equiv R t_1&#92;ldots t_n&#92;;@&#039;&#92;equiv&#92;varphi&#92;;@&#039; ' class='latex' /></p>
<p>
<p><b> For non-atomic <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi}' title='{&#92;varphi}' class='latex' /> </b></p>
<p> We treat only the case of <img src='http://s0.wp.com/latex.php?latex=%7B%28%5Cexists+x%29%5Cvarphi%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{(&#92;exists x)&#92;varphi}' title='{(&#92;exists x)&#92;varphi}' class='latex' />. There are two cases.</p>
<p>
(i) The variable <img src='http://s0.wp.com/latex.php?latex=%7Bx%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{x}' title='{x}' class='latex' /> is not free in <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi}' title='{&#92;varphi}' class='latex' />. By assumption, for each <img src='http://s0.wp.com/latex.php?latex=%7Ba%5Cin+A%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{a&#92;in A}' title='{a&#92;in A}' class='latex' /> and for each free variable <img src='http://s0.wp.com/latex.php?latex=%7By%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{y}' title='{y}' class='latex' /> of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi}' title='{&#92;varphi}' class='latex' />, <img src='http://s0.wp.com/latex.php?latex=%7B%40+%7C%5Ex_a%5C%3B+y+%3D+%40+y+%3D+%40%27+y+%3D+%7B%40%27%7D%7C%5Ex_a%5C%3B+y%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@ |^x_a&#92;; y = @ y = @&#039; y = {@&#039;}|^x_a&#92;; y}' title='{@ |^x_a&#92;; y = @ y = @&#039; y = {@&#039;}|^x_a&#92;; y}' class='latex' />. By the induction hypothesis,
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cbegin%7Barray%7D%7Bll%7D+%28%5Cexists+x%29%5Cvarphi+%5C%3B%40+%26%5Cequiv+%5Cexists+a%5Cin+A%2C+%5Cvarphi%5C%3B+%40%7C%5Ex_a+%5Cequiv+%5Cexists+a%5Cin+A%2C+%5Cvarphi%5C%3B+%40%5C%5C+%26%5Cequiv+%5Cexists+a%5Cin+A%2C+%5Cvarphi%5C%3B+%40%27+%5Cequiv+%5Cexists+a%5Cin+A%2C+%5Cvarphi%5C%3B+%7B%40%27%7D%7C%5Ex_a%5C%5C+%26%5Cequiv+%28%5Cexists+x%29%5Cvarphi+%5C%3B%40%27.+%5Cend%7Barray%7D+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  &#92;begin{array}{ll} (&#92;exists x)&#92;varphi &#92;;@ &amp;&#92;equiv &#92;exists a&#92;in A, &#92;varphi&#92;; @|^x_a &#92;equiv &#92;exists a&#92;in A, &#92;varphi&#92;; @&#92;&#92; &amp;&#92;equiv &#92;exists a&#92;in A, &#92;varphi&#92;; @&#039; &#92;equiv &#92;exists a&#92;in A, &#92;varphi&#92;; {@&#039;}|^x_a&#92;&#92; &amp;&#92;equiv (&#92;exists x)&#92;varphi &#92;;@&#039;. &#92;end{array} ' title='&#92;displaystyle  &#92;begin{array}{ll} (&#92;exists x)&#92;varphi &#92;;@ &amp;&#92;equiv &#92;exists a&#92;in A, &#92;varphi&#92;; @|^x_a &#92;equiv &#92;exists a&#92;in A, &#92;varphi&#92;; @&#92;&#92; &amp;&#92;equiv &#92;exists a&#92;in A, &#92;varphi&#92;; @&#039; &#92;equiv &#92;exists a&#92;in A, &#92;varphi&#92;; {@&#039;}|^x_a&#92;&#92; &amp;&#92;equiv (&#92;exists x)&#92;varphi &#92;;@&#039;. &#92;end{array} ' class='latex' /></p>
<p>
(ii) The variable <img src='http://s0.wp.com/latex.php?latex=%7Bx%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{x}' title='{x}' class='latex' /> has a free occurrence in <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi}' title='{&#92;varphi}' class='latex' />. By assumption, for each <img src='http://s0.wp.com/latex.php?latex=%7Ba%5Cin+A%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{a&#92;in A}' title='{a&#92;in A}' class='latex' /> and for each free variable <img src='http://s0.wp.com/latex.php?latex=%7By%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{y}' title='{y}' class='latex' /> of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi}' title='{&#92;varphi}' class='latex' />, <img src='http://s0.wp.com/latex.php?latex=%7B%40%7C%5Ex_a+y+%3D+%7B%40%27%7D%7C%5Ex_a+y%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@|^x_a y = {@&#039;}|^x_a y}' title='{@|^x_a y = {@&#039;}|^x_a y}' class='latex' />. Hence by the induction hypothesis, for each <img src='http://s0.wp.com/latex.php?latex=%7Ba%5Cin+A%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{a&#92;in A}' title='{a&#92;in A}' class='latex' />, <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%5C%3B%40%7C%5Ex_a%5Cequiv+%5Cvarphi%5C%3B%7B%40%27%7D%7C%5Ex_a.%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi&#92;;@|^x_a&#92;equiv &#92;varphi&#92;;{@&#039;}|^x_a.}' title='{&#92;varphi&#92;;@|^x_a&#92;equiv &#92;varphi&#92;;{@&#039;}|^x_a.}' class='latex' /> It follows that
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cbegin%7Barray%7D%7Bll%7D+%28%5Cexists+x%29%5Cvarphi+%5C%3B%40+%26%5Cequiv+%5Cexists+a%5Cin+A%2C+%5Cvarphi%5C%3B+%40%7C%5Ex_a%5C%5C+%26%5Cequiv+%5Cexists+a%5Cin+A%2C+%5Cvarphi%5C%3B+%7B%40%27%7D%7C%5Ex_a+%5Cequiv+%28%5Cexists+x%29%5Cvarphi+%5C%3B%40%27.+%5Cend%7Barray%7D+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  &#92;begin{array}{ll} (&#92;exists x)&#92;varphi &#92;;@ &amp;&#92;equiv &#92;exists a&#92;in A, &#92;varphi&#92;; @|^x_a&#92;&#92; &amp;&#92;equiv &#92;exists a&#92;in A, &#92;varphi&#92;; {@&#039;}|^x_a &#92;equiv (&#92;exists x)&#92;varphi &#92;;@&#039;. &#92;end{array} ' title='&#92;displaystyle  &#92;begin{array}{ll} (&#92;exists x)&#92;varphi &#92;;@ &amp;&#92;equiv &#92;exists a&#92;in A, &#92;varphi&#92;; @|^x_a&#92;&#92; &amp;&#92;equiv &#92;exists a&#92;in A, &#92;varphi&#92;; {@&#039;}|^x_a &#92;equiv (&#92;exists x)&#92;varphi &#92;;@&#039;. &#92;end{array} ' class='latex' /></p>
<p><b> Substitution </b></p>
<p> If <img src='http://s0.wp.com/latex.php?latex=%7Bx%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{x}' title='{x}' class='latex' /> is a variable and if <img src='http://s0.wp.com/latex.php?latex=%7Bs%2C+t%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{s, t}' title='{s, t}' class='latex' /> are terms <img src='http://s0.wp.com/latex.php?latex=%7B%7C%5Ex_s%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{|^x_s}' title='{|^x_s}' class='latex' /> is a substitution operation on terms, where <img src='http://s0.wp.com/latex.php?latex=%7B%7C%5Ex_s%5C%3B+t%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{|^x_s&#92;; t}' title='{|^x_s&#92;; t}' class='latex' /> is the term arising from <img src='http://s0.wp.com/latex.php?latex=%7Bt%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{t}' title='{t}' class='latex' /> by simultaneous substitution of <img src='http://s0.wp.com/latex.php?latex=%7Bs%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{s}' title='{s}' class='latex' /> for each occurrence of <img src='http://s0.wp.com/latex.php?latex=%7Bx%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{x}' title='{x}' class='latex' /> in <img src='http://s0.wp.com/latex.php?latex=%7Bt%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{t}' title='{t}' class='latex' />. (This can be defined inductively.) This operation acts on the left of terms; the similar notation <img src='http://s0.wp.com/latex.php?latex=%7B%7C%5Ex_a%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{|^x_a}' title='{|^x_a}' class='latex' />, which is differently typed (<img src='http://s0.wp.com/latex.php?latex=%7Ba%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{a}' title='{a}' class='latex' /> is an element of the domain) acts on the right of an @ operator. </p>
<p>
A standard exercise in our notation is more readable than in the usual presentation. Let <img src='http://s0.wp.com/latex.php?latex=%7Bs%2C+t%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{s, t}' title='{s, t}' class='latex' /> be terms and let @ be an <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' />&#8211;assignment, If <img src='http://s0.wp.com/latex.php?latex=%7B%40+s+%3D+a%5Cin+A%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@ s = a&#92;in A}' title='{@ s = a&#92;in A}' class='latex' />, then <a name="EqElegant">
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28%40+%7C%5Ex_a%29+t+%3D+%40+%28%7C%5Ex_s+%5C%3B+t%29+%5C+%5C+%5C+%5C+%5C+%282%29&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  (@ |^x_a) t = @ (|^x_s &#92;; t) &#92; &#92; &#92; &#92; &#92; (2)' title='&#92;displaystyle  (@ |^x_a) t = @ (|^x_s &#92;; t) &#92; &#92; &#92; &#92; &#92; (2)' class='latex' /></p>
<p></a></p>
<p>
This is almost associativity. Could it be made associative? Two possibilities suggest themselves.</p>
<p>
(a) Logicians speak of the expansion of a language by a structure: add the elements of the domain as new, self-evaluating symbols to the language. More precisely, given an <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />&#8211;structure <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%3D%5Clangle+A%2C+%7C_%5Cmathcal%7BA%7D%5Crangle%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}=&#92;langle A, |_&#92;mathcal{A}&#92;rangle}' title='{&#92;mathcal{A}=&#92;langle A, |_&#92;mathcal{A}&#92;rangle}' class='latex' />, define a new language <img src='http://s0.wp.com/latex.php?latex=%7B%28%5Cmathcal%7BL%7D%2C%5Cmathcal%7BA%7D%29%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{(&#92;mathcal{L},&#92;mathcal{A})}' title='{(&#92;mathcal{L},&#92;mathcal{A})}' class='latex' /> extending <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' /> called the <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' />&#8211;expansion of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />. If the signature of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' /> is <img src='http://s0.wp.com/latex.php?latex=%7B%5Csigma%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;sigma}' title='{&#92;sigma}' class='latex' />, the signature of the <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' />-expansion is <img src='http://s0.wp.com/latex.php?latex=%7B%5Csigma%5Ccup+A%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;sigma&#92;cup A}' title='{&#92;sigma&#92;cup A}' class='latex' />; i.e., the expansion has, in addition to the set <img src='http://s0.wp.com/latex.php?latex=%7B%5Csigma%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;sigma}' title='{&#92;sigma}' class='latex' /> of constant, function and relation symbols of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />, the elements of the domain <img src='http://s0.wp.com/latex.php?latex=%7BA%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{A}' title='{A}' class='latex' /> as new constant symbols. (We assume <img src='http://s0.wp.com/latex.php?latex=%7B%5Csigma%5Ccap+A%3D%5Cemptyset%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;sigma&#92;cap A=&#92;emptyset}' title='{&#92;sigma&#92;cap A=&#92;emptyset}' class='latex' />.) The <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' /> expansion of <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' /> has a natural structure <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%5E%2B%3D%5Clangle+A%2C+%7C_%5Cmathcal%7BA%7D%5E%2B%5Crangle%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}^+=&#92;langle A, |_&#92;mathcal{A}^+&#92;rangle}' title='{&#92;mathcal{A}^+=&#92;langle A, |_&#92;mathcal{A}^+&#92;rangle}' class='latex' />, obtained from the old structure <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%3D%5Clangle+A%2C+%7C_%5Cmathcal%7BA%7D%5Crangle%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}=&#92;langle A, |_&#92;mathcal{A}&#92;rangle}' title='{&#92;mathcal{A}=&#92;langle A, |_&#92;mathcal{A}&#92;rangle}' class='latex' />: the domain <img src='http://s0.wp.com/latex.php?latex=%7BA%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{A}' title='{A}' class='latex' /> is the same, and the map <img src='http://s0.wp.com/latex.php?latex=%7B%7C_%5Cmathcal%7BA%7D%5E%2B%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{|_&#92;mathcal{A}^+}' title='{|_&#92;mathcal{A}^+}' class='latex' /> is given by
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++s%7C_%5Cmathcal%7BA%7D%5E%2B%3D+%5Cbegin%7Bcases%7D+s+%7C_%5Cmathcal%7BA%7D%2C%26s%5Cin%5Csigma%3B%5C%5C+s%2C+%26s%5Cin+A.+%5Cend%7Bcases%7D+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  s|_&#92;mathcal{A}^+= &#92;begin{cases} s |_&#92;mathcal{A},&amp;s&#92;in&#92;sigma;&#92;&#92; s, &amp;s&#92;in A. &#92;end{cases} ' title='&#92;displaystyle  s|_&#92;mathcal{A}^+= &#92;begin{cases} s |_&#92;mathcal{A},&amp;s&#92;in&#92;sigma;&#92;&#92; s, &amp;s&#92;in A. &#92;end{cases} ' class='latex' /></p>
<p>
 In the expanded structure, it makes sense to write
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28%40+%7C%5Ex_a%29+t+%3D+%40+%28%7C%5Ex_a+%5C%3B+t%29.+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  (@ |^x_a) t = @ (|^x_a &#92;; t). ' title='&#92;displaystyle  (@ |^x_a) t = @ (|^x_a &#92;; t). ' class='latex' /></p>
<p>
(b) Alternatively, we could extend the definition of the right operation <img src='http://s0.wp.com/latex.php?latex=%7B%7C%5Ex_a%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{|^x_a}' title='{|^x_a}' class='latex' /> on @ operators for an <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />&#8211;structure <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BA%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{A}}' title='{&#92;mathcal{A}}' class='latex' /> to allow <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />&#8211;term parameters by defining
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%40+%7C%5Ex_s+%3A%3D+%40+%7C%5Ex_%7B%40s%7D+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  @ |^x_s := @ |^x_{@s} ' title='&#92;displaystyle  @ |^x_s := @ |^x_{@s} ' class='latex' /></p>
<p> where <img src='http://s0.wp.com/latex.php?latex=%7Bs%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{s}' title='{s}' class='latex' /> is an <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BL%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{L}}' title='{&#92;mathcal{L}}' class='latex' />&#8211;term. With this inevitable definition, (<a href="#EqElegant">2</a>) becomes</p>
<p>
<a name="EqAss">
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28%40+%7C%5Ex_s%29+t+%3D+%40+%28%7C%5Ex_s+%5C%3B+t%29+%5C+%5C+%5C+%5C+%5C+%283%29&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  (@ |^x_s) t = @ (|^x_s &#92;; t) &#92; &#92; &#92; &#92; &#92; (3)' title='&#92;displaystyle  (@ |^x_s) t = @ (|^x_s &#92;; t) &#92; &#92; &#92; &#92; &#92; (3)' class='latex' /></p>
<p></a></p>
<p>
The standard notation for (<a href="#EqElegant">2</a>) is this:
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++t%5E%7B%5Cmathcal%7BA%7D%7D%5B%5Calpha%5Ex_a%5D+%3D+%28t%28s%29%29%5E%7B%5Cmathcal%7BA%7D%7D%5B%5Calpha%5D.+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  t^{&#92;mathcal{A}}[&#92;alpha^x_a] = (t(s))^{&#92;mathcal{A}}[&#92;alpha]. ' title='&#92;displaystyle  t^{&#92;mathcal{A}}[&#92;alpha^x_a] = (t(s))^{&#92;mathcal{A}}[&#92;alpha]. ' class='latex' /></p>
<p>
Here is another beginning exercise. The standard notation is shown first. If <img src='http://s0.wp.com/latex.php?latex=%7B%5Calpha%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;alpha}' title='{&#92;alpha}' class='latex' /> is an assignment, if the term <img src='http://s0.wp.com/latex.php?latex=%7Bs%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{s}' title='{s}' class='latex' /> is free for <img src='http://s0.wp.com/latex.php?latex=%7Bx%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{x}' title='{x}' class='latex' /> in <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi}' title='{&#92;varphi}' class='latex' /> and if <img src='http://s0.wp.com/latex.php?latex=%7B%5Calpha+s+%3D+a%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;alpha s = a}' title='{&#92;alpha s = a}' class='latex' />, then
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathcal%7BA%7D%5CvDash+%5Cvarphi%28s%29+%5B%5Calpha%5D+%5Cequiv+%5Cmathcal%7BA%7D%5CvDash+%5Cvarphi+%5B%5Calpha%5Ex_a%5D.+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  &#92;mathcal{A}&#92;vDash &#92;varphi(s) [&#92;alpha] &#92;equiv &#92;mathcal{A}&#92;vDash &#92;varphi [&#92;alpha^x_a]. ' title='&#92;displaystyle  &#92;mathcal{A}&#92;vDash &#92;varphi(s) [&#92;alpha] &#92;equiv &#92;mathcal{A}&#92;vDash &#92;varphi [&#92;alpha^x_a]. ' class='latex' /></p>
<p>
In the new notation, if the term <img src='http://s0.wp.com/latex.php?latex=%7Bs%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{s}' title='{s}' class='latex' /> is free for <img src='http://s0.wp.com/latex.php?latex=%7Bx%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{x}' title='{x}' class='latex' /> in <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi}' title='{&#92;varphi}' class='latex' /> and if <img src='http://s0.wp.com/latex.php?latex=%7B%40+s+%3D+a%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{@ s = a}' title='{@ s = a}' class='latex' />, then
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cvarphi+%7C%5Ex_s%5C%3B+%40+%5Cequiv+%5Cvarphi%5C%3B+%40+%7C%5Ex_a+.+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  &#92;varphi |^x_s&#92;; @ &#92;equiv &#92;varphi&#92;; @ |^x_a . ' title='&#92;displaystyle  &#92;varphi |^x_s&#92;; @ &#92;equiv &#92;varphi&#92;; @ |^x_a . ' class='latex' /></p>
<p>
If we adopt the convention (b), this reads as follows. If the term <img src='http://s0.wp.com/latex.php?latex=%7Bs%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{s}' title='{s}' class='latex' /> is free for <img src='http://s0.wp.com/latex.php?latex=%7Bx%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{x}' title='{x}' class='latex' /> in <img src='http://s0.wp.com/latex.php?latex=%7B%5Cvarphi%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;varphi}' title='{&#92;varphi}' class='latex' />, then
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cvarphi+%7C%5Ex_s%5C%3B+%40+%5Cequiv+%5Cvarphi%5C%3B+%40+%7C%5Ex_s+.+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  &#92;varphi |^x_s&#92;; @ &#92;equiv &#92;varphi&#92;; @ |^x_s . ' title='&#92;displaystyle  &#92;varphi |^x_s&#92;; @ &#92;equiv &#92;varphi&#92;; @ |^x_s . ' class='latex' /></p>
<p>
(We let the term substitution operator <img src='http://s0.wp.com/latex.php?latex=%7B%7C%5Ex_s%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{|^x_s}' title='{|^x_s}' class='latex' /> act on the left of terms and on the right of formulas and @ operators.)</p>
<p>
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			<media:title type="html">florianlengyel</media:title>
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		<title>Alpha- and beta- characteristic functions</title>
		<link>http://opusminimax.wordpress.com/2012/01/02/alpha-and-beta-characteristic-functions/</link>
		<comments>http://opusminimax.wordpress.com/2012/01/02/alpha-and-beta-characteristic-functions/#comments</comments>
		<pubDate>Mon, 02 Jan 2012 22:12:18 +0000</pubDate>
		<dc:creator>F Lengyel</dc:creator>
				<category><![CDATA[Math]]></category>

		<guid isPermaLink="false">http://opusminimax.wordpress.com/?p=700</guid>
		<description><![CDATA[The definition of - and -characteristic functions is paraphrased from The Cooperative Game Theory of Networks and Hierarchies by Robert Gilles. A cooperative game is pair consisting of finite set of players, and a map , called the characteristic function, such that . The and -characteristic functions of a normal form game assign values to [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=opusminimax.wordpress.com&amp;blog=26204618&amp;post=700&amp;subd=opusminimax&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<p><p>
The definition of <img src='http://s0.wp.com/latex.php?latex=%7B%5Calpha%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;alpha}' title='{&#92;alpha}' class='latex' />- and <img src='http://s0.wp.com/latex.php?latex=%7B%5Cbeta%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;beta}' title='{&#92;beta}' class='latex' />-characteristic functions is paraphrased from <a href="http://books.google.com/books?id=g2dqS42F7skC&amp;printsec=frontcover&amp;source=gbs_atb#v=onepage&amp;q&amp;f=false">The Cooperative Game Theory of Networks and Hierarchies</a> by Robert Gilles.</p>
<p>
A cooperative game is pair consisting of finite set <img src='http://s0.wp.com/latex.php?latex=%7BN%3D%5C%7B1%2C%5Cldots%2Cn%5C%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{N=&#92;{1,&#92;ldots,n&#92;}}' title='{N=&#92;{1,&#92;ldots,n&#92;}}' class='latex' /> of players, and a map <img src='http://s0.wp.com/latex.php?latex=%7Bv%3A2%5EN%5Crightarrow%5Cmathbb%7BR%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{v:2^N&#92;rightarrow&#92;mathbb{R}}' title='{v:2^N&#92;rightarrow&#92;mathbb{R}}' class='latex' />, called the characteristic function, such that <img src='http://s0.wp.com/latex.php?latex=%7Bv%28%5Cemptyset%29%3D0%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{v(&#92;emptyset)=0}' title='{v(&#92;emptyset)=0}' class='latex' />.</p>
<p>
The <img src='http://s0.wp.com/latex.php?latex=%7B%5Calpha%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;alpha}' title='{&#92;alpha}' class='latex' /> and <img src='http://s0.wp.com/latex.php?latex=%7B%5Cbeta%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;beta}' title='{&#92;beta}' class='latex' />-characteristic functions of a normal form game
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathcal%7BG%7D%3D+%5Clangle+N%3D%5C%7B1%2C%5Cldots%2Cn%5C%7D%2C+A%3DA_1%5Ctimes%5Ccdots%5Ctimes+A_n%2C+%5Cpi%3AA%5Crightarrow%5Cmathbb%7BR%7D%5EN+%5Crangle+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  &#92;mathcal{G}= &#92;langle N=&#92;{1,&#92;ldots,n&#92;}, A=A_1&#92;times&#92;cdots&#92;times A_n, &#92;pi:A&#92;rightarrow&#92;mathbb{R}^N &#92;rangle ' title='&#92;displaystyle  &#92;mathcal{G}= &#92;langle N=&#92;{1,&#92;ldots,n&#92;}, A=A_1&#92;times&#92;cdots&#92;times A_n, &#92;pi:A&#92;rightarrow&#92;mathbb{R}^N &#92;rangle ' class='latex' /></p>
<p> assign values to coalitions <img src='http://s0.wp.com/latex.php?latex=%7BS%5Csubseteq+N%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{S&#92;subseteq N}' title='{S&#92;subseteq N}' class='latex' /> of the <img src='http://s0.wp.com/latex.php?latex=%7BN%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{N}' title='{N}' class='latex' /> players. An aggregation function is a functional of type
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++F%3A2%5EN%5Ctimes+A%5E%7B%5Cmathbb%7BR%7D%5EN%7D%5Ctimes+A+%5Crightarrow%5Cmathbb%7BR%7D+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  F:2^N&#92;times A^{&#92;mathbb{R}^N}&#92;times A &#92;rightarrow&#92;mathbb{R} ' title='&#92;displaystyle  F:2^N&#92;times A^{&#92;mathbb{R}^N}&#92;times A &#92;rightarrow&#92;mathbb{R} ' class='latex' /></p>
<p> which associates to a subset <img src='http://s0.wp.com/latex.php?latex=%7BS%5Csubseteq+N%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{S&#92;subseteq N}' title='{S&#92;subseteq N}' class='latex' /> and a strategy profile <img src='http://s0.wp.com/latex.php?latex=%7Ba%5Cin+A%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{a&#92;in A}' title='{a&#92;in A}' class='latex' /> the value <img src='http://s0.wp.com/latex.php?latex=%7BF%28S%2C%5Cpi%2C+a%29%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{F(S,&#92;pi, a)}' title='{F(S,&#92;pi, a)}' class='latex' /> in <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathbb%7BR%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathbb{R}}' title='{&#92;mathbb{R}}' class='latex' /> depending on the payoff functions <img src='http://s0.wp.com/latex.php?latex=%7B%5Cpi_i%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;pi_i}' title='{&#92;pi_i}' class='latex' /> of the players of the coalition <img src='http://s0.wp.com/latex.php?latex=%7BS%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{S}' title='{S}' class='latex' />. We emphasize the dependence of the aggregator on the payoff functions <img src='http://s0.wp.com/latex.php?latex=%7B%5Cpi_i%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;pi_i}' title='{&#92;pi_i}' class='latex' /> of the players <img src='http://s0.wp.com/latex.php?latex=%7Bi%5Cin+S%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{i&#92;in S}' title='{i&#92;in S}' class='latex' /> by writing <img src='http://s0.wp.com/latex.php?latex=%7BF%28S%2C+%5Cpi_%7Bi%5Cin+S%7D%2C+a%29%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{F(S, &#92;pi_{i&#92;in S}, a)}' title='{F(S, &#92;pi_{i&#92;in S}, a)}' class='latex' />. We write <img src='http://s0.wp.com/latex.php?latex=%7BA_S%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{A_S}' title='{A_S}' class='latex' /> for the <img src='http://s0.wp.com/latex.php?latex=%7BS%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{S}' title='{S}' class='latex' />-subcartesian product of <img src='http://s0.wp.com/latex.php?latex=%7BA%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{A}' title='{A}' class='latex' />, which pedantically is the image of the map which takes a strategy profile <img src='http://s0.wp.com/latex.php?latex=%7Ba%5Cin+A%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{a&#92;in A}' title='{a&#92;in A}' class='latex' />, viewed as a choice function <img src='http://s0.wp.com/latex.php?latex=%7Ba%3AN%5Crightarrow+%5Ccup_%7Bi%5Cin+N%7D+S_i%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{a:N&#92;rightarrow &#92;cup_{i&#92;in N} S_i}' title='{a:N&#92;rightarrow &#92;cup_{i&#92;in N} S_i}' class='latex' /> with <img src='http://s0.wp.com/latex.php?latex=%7Ba%28i%29%5Cin+S_i%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{a(i)&#92;in S_i}' title='{a(i)&#92;in S_i}' class='latex' />, and restricts the domain of <img src='http://s0.wp.com/latex.php?latex=%7Ba%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{a}' title='{a}' class='latex' /> to the subset <img src='http://s0.wp.com/latex.php?latex=%7BS%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{S}' title='{S}' class='latex' /> and corestricts the codomain to <img src='http://s0.wp.com/latex.php?latex=%7B%5Ccup_%7Bi%5Cin+S%7D+S_i%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;cup_{i&#92;in S} S_i}' title='{&#92;cup_{i&#92;in S} S_i}' class='latex' />. For <img src='http://s0.wp.com/latex.php?latex=%7Ba%5Cin+A%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{a&#92;in A}' title='{a&#92;in A}' class='latex' /> we write <img src='http://s0.wp.com/latex.php?latex=%7Ba%3Dx%5Ccup+y%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{a=x&#92;cup y}' title='{a=x&#92;cup y}' class='latex' /> where <img src='http://s0.wp.com/latex.php?latex=%7Bx%5Cin+A_S%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{x&#92;in A_S}' title='{x&#92;in A_S}' class='latex' />, <img src='http://s0.wp.com/latex.php?latex=%7By%5Cin+A_%7BN%5Csetminus+S%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{y&#92;in A_{N&#92;setminus S}}' title='{y&#92;in A_{N&#92;setminus S}}' class='latex' /> and where <img src='http://s0.wp.com/latex.php?latex=%7B%5Ccup%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;cup}' title='{&#92;cup}' class='latex' /> is the union (coproduct, if you like) of functions.</p>
<p>
The <img src='http://s0.wp.com/latex.php?latex=%7B%5Calpha%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;alpha}' title='{&#92;alpha}' class='latex' />-characteristic function for the game <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BG%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{G}}' title='{&#92;mathcal{G}}' class='latex' /> and aggregator <img src='http://s0.wp.com/latex.php?latex=%7BF%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{F}' title='{F}' class='latex' /> is defined by
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++v%5E%5Calpha%28S%29%3D%5Cmax_%7Bx%5Cin+A_S%7D+%5Cmin_%7By%5Cin+A_%7BN%5Csetminus+S%7D%7D+F%28S%2C+%5Cpi_%7Bi%5Cin+S%7D%2C+x%5Ccup+y%29.+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  v^&#92;alpha(S)=&#92;max_{x&#92;in A_S} &#92;min_{y&#92;in A_{N&#92;setminus S}} F(S, &#92;pi_{i&#92;in S}, x&#92;cup y). ' title='&#92;displaystyle  v^&#92;alpha(S)=&#92;max_{x&#92;in A_S} &#92;min_{y&#92;in A_{N&#92;setminus S}} F(S, &#92;pi_{i&#92;in S}, x&#92;cup y). ' class='latex' /></p>
<p> This is the pessimistic view: the opposing coalition <img src='http://s0.wp.com/latex.php?latex=%7BN%5Csetminus+S%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{N&#92;setminus S}' title='{N&#92;setminus S}' class='latex' /> fixes a strategy <img src='http://s0.wp.com/latex.php?latex=%7Bx%5Cin+A_S%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{x&#92;in A_S}' title='{x&#92;in A_S}' class='latex' /> and chooses the strategy <img src='http://s0.wp.com/latex.php?latex=%7By%28x%29%5Cin+A_%7BN%5Csetminus+S%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{y(x)&#92;in A_{N&#92;setminus S}}' title='{y(x)&#92;in A_{N&#92;setminus S}}' class='latex' /> that gives <img src='http://s0.wp.com/latex.php?latex=%7BS%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{S}' title='{S}' class='latex' /> the worst payoff; <img src='http://s0.wp.com/latex.php?latex=%7BS%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{S}' title='{S}' class='latex' /> chooses the best <img src='http://s0.wp.com/latex.php?latex=%7Bx%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{x}' title='{x}' class='latex' /> of the worst.</p>
<p>
The <img src='http://s0.wp.com/latex.php?latex=%7B%5Cbeta%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;beta}' title='{&#92;beta}' class='latex' />-characteristic function for the game <img src='http://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BG%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;mathcal{G}}' title='{&#92;mathcal{G}}' class='latex' /> and aggregator <img src='http://s0.wp.com/latex.php?latex=%7BF%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{F}' title='{F}' class='latex' /> is defined by
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++v%5E%5Cbeta%28S%29%3D%5Cmin_%7By%5Cin+A_%7BN%5Csetminus+S%7D%7D+%5Cmax_%7Bx%5Cin+A_S%7D+F%28S%2C+%5Cpi_%7Bi%5Cin+S%7D%2C+x%5Ccup+y%29.+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  v^&#92;beta(S)=&#92;min_{y&#92;in A_{N&#92;setminus S}} &#92;max_{x&#92;in A_S} F(S, &#92;pi_{i&#92;in S}, x&#92;cup y). ' title='&#92;displaystyle  v^&#92;beta(S)=&#92;min_{y&#92;in A_{N&#92;setminus S}} &#92;max_{x&#92;in A_S} F(S, &#92;pi_{i&#92;in S}, x&#92;cup y). ' class='latex' /></p>
<p> This is the optimistic view: given a fixed strategy <img src='http://s0.wp.com/latex.php?latex=%7By%5Cin+A_%7BN%5Csetminus+S%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{y&#92;in A_{N&#92;setminus S}}' title='{y&#92;in A_{N&#92;setminus S}}' class='latex' /> the coalition <img src='http://s0.wp.com/latex.php?latex=%7BS%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{S}' title='{S}' class='latex' /> chooses the strategy <img src='http://s0.wp.com/latex.php?latex=%7Bx%28y%29+%5Cin+A_S%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{x(y) &#92;in A_S}' title='{x(y) &#92;in A_S}' class='latex' /> with the best payoff for itself; the opposing coalition <img src='http://s0.wp.com/latex.php?latex=%7BN%5Csetminus+S%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{N&#92;setminus S}' title='{N&#92;setminus S}' class='latex' /> chooses the most hostile strategy profile <img src='http://s0.wp.com/latex.php?latex=%7By%5Cin+A_%7BN%5Csetminus+S%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{y&#92;in A_{N&#92;setminus S}}' title='{y&#92;in A_{N&#92;setminus S}}' class='latex' /> in response. The optimism of <img src='http://s0.wp.com/latex.php?latex=%7BS%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{S}' title='{S}' class='latex' /> in the face of the hostile coalition of <a href="http://opusminimax.wordpress.com/glossary/#GL_mansplainer">mansplainers</a> <img src='http://s0.wp.com/latex.php?latex=%7BN%5Csetminus+S%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{N&#92;setminus S}' title='{N&#92;setminus S}' class='latex' /> motivates the title of this blog. </p>
<p>
<span id="more-700"></span> Perhaps my optimism for the future of the mathematics of the social sciences is unjustified. Herbert Gintis has pointed out that strategic interaction distinguishes the behavioral sciences from the physical sciences. But mathematics lacks a characterization of interaction that distinguishes strategic interaction from non-strategic physical interaction. Gintis&#8217; distinction isn&#8217;t mathematically definable&#8211;yet. </p>
<p>
We have that <img src='http://s0.wp.com/latex.php?latex=%7Bv%5E%5Calpha%5Cle+v%5E%5Cbeta%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{v^&#92;alpha&#92;le v^&#92;beta}' title='{v^&#92;alpha&#92;le v^&#92;beta}' class='latex' />; this is a consequence of the following easy exercise.</p>
<blockquote><p><b>Proposition 1</b> <em> Let <img src='http://s0.wp.com/latex.php?latex=%7Bf%3A+A%5Ctimes+B%5Crightarrow%5Cmathbb%7BR%7D%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{f: A&#92;times B&#92;rightarrow&#92;mathbb{R}}' title='{f: A&#92;times B&#92;rightarrow&#92;mathbb{R}}' class='latex' /> be a map.
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csup_%7Ba%5Cin+A%7D+%5Cinf_%7Bb%5Cin+B%7D+f%28a%2Cb%29%5Cle+%5Cinf_%7Bb%5Cin+B%7D+%5Csup_%7Ba%5Cin+A%7D+f%28a%2Cb%29.+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  &#92;sup_{a&#92;in A} &#92;inf_{b&#92;in B} f(a,b)&#92;le &#92;inf_{b&#92;in B} &#92;sup_{a&#92;in A} f(a,b). ' title='&#92;displaystyle  &#92;sup_{a&#92;in A} &#92;inf_{b&#92;in B} f(a,b)&#92;le &#92;inf_{b&#92;in B} &#92;sup_{a&#92;in A} f(a,b). ' class='latex' /></p>
<p> </em></p></blockquote>
<p> <em>Proof:</em>  Fix <img src='http://s0.wp.com/latex.php?latex=%7Ba%5Cin+A%2C%5C%2Cb%5Cin+B%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{a&#92;in A,&#92;,b&#92;in B}' title='{a&#92;in A,&#92;,b&#92;in B}' class='latex' />. Then
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cinf_%7Bb%5Cin+B%7D+f%28a%2Cb%29%5Cle+f%28a%2Cb%29%5Cle+%5Csup_%7Ba%5Cin+A%7D+f%28a%2Cb%29.+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  &#92;inf_{b&#92;in B} f(a,b)&#92;le f(a,b)&#92;le &#92;sup_{a&#92;in A} f(a,b). ' title='&#92;displaystyle  &#92;inf_{b&#92;in B} f(a,b)&#92;le f(a,b)&#92;le &#92;sup_{a&#92;in A} f(a,b). ' class='latex' /></p>
<p> Hence for any <img src='http://s0.wp.com/latex.php?latex=%7Ba%5Cin+A%2C%5C%2Cb%5Cin+B%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{a&#92;in A,&#92;,b&#92;in B}' title='{a&#92;in A,&#92;,b&#92;in B}' class='latex' />,
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cinf_%7Bb%5Cin+B%7D+f%28a%2Cb%29%5Cle+%5Csup_%7Ba%5Cin+A%7D+f%28a%2Cb%29.+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  &#92;inf_{b&#92;in B} f(a,b)&#92;le &#92;sup_{a&#92;in A} f(a,b). ' title='&#92;displaystyle  &#92;inf_{b&#92;in B} f(a,b)&#92;le &#92;sup_{a&#92;in A} f(a,b). ' class='latex' /></p>
<p> Fix <img src='http://s0.wp.com/latex.php?latex=%7Bb%5Cin+B%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{b&#92;in B}' title='{b&#92;in B}' class='latex' /> on the right. The right is an upper bound for the left (over <img src='http://s0.wp.com/latex.php?latex=%7Ba%5Cin+A%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{a&#92;in A}' title='{a&#92;in A}' class='latex' />). Taking the <img src='http://s0.wp.com/latex.php?latex=%7B%5Csup%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;sup}' title='{&#92;sup}' class='latex' /> of the left over <img src='http://s0.wp.com/latex.php?latex=%7Ba%5Cin+A%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{a&#92;in A}' title='{a&#92;in A}' class='latex' /> yields
<p align="center"><img src='http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csup_%7Ba%5Cin+A%7D+%5Cinf_%7Bb%5Cin+B%7D+f%28a%2Cb%29%5Cle+%5Csup_%7Ba%5Cin+A%7D+f%28a%2Cb%29.+&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;displaystyle  &#92;sup_{a&#92;in A} &#92;inf_{b&#92;in B} f(a,b)&#92;le &#92;sup_{a&#92;in A} f(a,b). ' title='&#92;displaystyle  &#92;sup_{a&#92;in A} &#92;inf_{b&#92;in B} f(a,b)&#92;le &#92;sup_{a&#92;in A} f(a,b). ' class='latex' /></p>
<p>
The left is now a constant (extended real number) lower bound of the right. Take the <img src='http://s0.wp.com/latex.php?latex=%7B%5Cinf%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{&#92;inf}' title='{&#92;inf}' class='latex' /> over <img src='http://s0.wp.com/latex.php?latex=%7Bb%5Cin+B%7D&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='{b&#92;in B}' title='{b&#92;in B}' class='latex' />. <img src='http://s0.wp.com/latex.php?latex=%5CBox&amp;bg=ffffff&amp;fg=000000&amp;s=0' alt='&#92;Box' title='&#92;Box' class='latex' /></p>
<p>
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