<?xml version="1.0" encoding="utf-8"?>
<feed xmlns="http://www.w3.org/2005/Atom">
  <title>Archive</title>
  <link rel="alternate" type="text/html" href="http://kosara.net/topics/Archive"/>
  <link rel="self" type="application/atom+xml" href="http://kosara.net/taxonomy/term/13/atom/feed"/>
  <id>http://kosara.net/taxonomy/term/13/atom/feed</id>
  <updated>2008-04-06T09:57:38-04:00</updated>
  <entry>
    <title>German vs. English, Kuriositaeten, etc.</title>
    <link rel="alternate" type="text/html" href="http://kosara.net/thoughts/german.html" />
    <id>http://kosara.net/thoughts/german.html</id>
    <published>2008-04-07T21:45:37-04:00</published>
    <updated>2008-04-07T22:26:52-04:00</updated>
    <author>
      <name>Robert Kosara</name>
    </author>
    <category term="Archive" />
    <category term="Article" />
    <category term="Meta/Site News" />
    <summary type="html"><![CDATA[<p>The language pages have been replaced by the <a href="http://kosara.net/topics/Language">language category</a> on this website. That category will be updated with postings on German, English, and the relationship between the two on a more or less regular basis.</p><p>Die Seiten zum Thema Sprache wurden durch die <a href="http://kosara.net/topics/Language">Kategorie Language</a> ersetzt. Diese Kategorie enth&auml;lt alle Postings zum Thema Sprache und wird (einigerma&szlig;en) regelm&auml;&szlig;ig mit neuen Artikeln aktualisiert.</p>    ]]></summary>
    <content type="html"><![CDATA[<p>The language pages have been replaced by the <a href="http://kosara.net/topics/Language">language category</a> on this website. That category will be updated with postings on German, English, and the relationship between the two on a more or less regular basis.</p><p>Die Seiten zum Thema Sprache wurden durch die <a href="http://kosara.net/topics/Language">Kategorie Language</a> ersetzt. Diese Kategorie enth&auml;lt alle Postings zum Thema Sprache und wird (einigerma&szlig;en) regelm&auml;&szlig;ig mit neuen Artikeln aktualisiert.</p>    ]]></content>
  </entry>
  <entry>
    <title>The Value of A(4, 2)</title>
    <link rel="alternate" type="text/html" href="http://kosara.net/thoughts/ackermann42.html" />
    <id>http://kosara.net/thoughts/ackermann42.html</id>
    <published>2008-02-23T21:57:45-05:00</published>
    <updated>2008-02-23T22:01:26-05:00</updated>
    <author>
      <name>Robert Kosara</name>
    </author>
    <category term="Archive" />
    <category term="Article" />
    <category term="Math" />
    <summary type="html"><![CDATA[<p>This is the result of the <a href="/thoughts/ackermann.html">Ackermann Function A(m, n)</a> for a specific set of parameters.&nbsp;This number has 19,729 digits.</p>    ]]></summary>
    <content type="html"><![CDATA[<p>This is the result of the <a href="/thoughts/ackermann.html">Ackermann Function A(m, n)</a> for a specific set of parameters.&nbsp;This number has 19,729 digits.</p>   <pre>
A(4, 2) =
20035299304068464649790723515602557504478254755697514192650169737108940595563114
53089506130880933348101038234342907263181822949382118812668869506364761547029165
04187191635158796634721944293092798208430910485599057015931895963952486337236720
30029169695921561087649488892540908059114570376752085002066715637023661263597471
44807111774815880914135742720967190151836282560618091458852699826141425030123391
10827360384376787644904320596037912449090570756031403507616256247603186379312648
47037437829549756137709816046144133086921181024859591523801953310302921628001605
68670105651646750568038741529463842244845292537361442533614373729088303794601274
72495841486491593064725201515569392262818069165079638106413227530726714399815850
88112926289011342377827055674210800700652839633221550778312142885516755540733451
07213112427399562982719769150054883905223804357045848197956393157853510018992000
02414196370681355984046403947219401606951769015611972698233789001764151719005113
34663068981402193834814354263873065395529696913880241581618595611006403621197961
01859534802787167200122604642492385111393400464351623867567078745259464670903886
54774348321789701276445552940909202195958575162297333357615955239488529757995402
84719435299135437637059869289137571537400019863943324648900525431066296691652434
19174691389632476560289415199775477703138064781342309596190960654591300890188887
58808473362595606544488850144733570605881709016210849971452956834406197969056546
98136311620535793697914032363284962330464210661362002201757878518574091620504897
11781820400187282939943446186224328009837323764931814789848119452713007440220765
68091037620399920349202390662626449190916798546151577883906039772075927937885224
12943010174580868622633692847258514030396155585643303854506886522131148136384083
84778263790459607186876728509763471271988890680478243230394718650525660978150729
86114143030581692792497140916105941718535227588750447759221830115878070197553572
22414000195481020056617735897814995323252085897534635470077866904064290167638081
61740550405117670093673202804549339027992491867306539931640720492238474815280619
16690093380573212081635070763435166986962502096902316285935007187419057916124153
68975148082619048479465717366010058924766554458408383347905441448176842553272073
15586349347605137419779525190365032198020108764738368682531025183377533908861426
18480037400808223810407646887847164755294532694766170042446106331123802113458869
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40780879056283663406965030844225855967039271869461158513793386475699748568670079
82396060439347885086164926030494506174341236582835214480672667684180708375486221
14082365798029612000274413244384324023312574035450193524287764308802328508558860
89962774458164680857875115807014743763867976955049991643998284357290415378143438
84730348426190338884149403136613985425763557710533558020662218557706008255128889
33322264362819848386132395706761914096385338323743437588308592337222846442879962
45605476932428998432652677378373173288063210753211238680604674708428051166488709
08477029120816110491255559832236624486855665140268464120969498259056551921618810
43412268389962830716548685255369148502995396755039549383718534059000961874894739
92880432496373165753803673586710175783994818471798498246948060532081996066183434
01247609663951977802144119975254670408060849934417825628509272652370989865153946
21930046073645079262129759176982938923670151709920915315678144397912484757062378
04600009918293321306880570046591458387208088016887445835557926258465124763087148
56631352893416611749061752667149267217612833084527393646924458289257138887783905
63004824837998396920292222154861459023734782226825216399574408017271441461795592
26175083889020074169926238300282286249284182671243405751424188569994272331606998
71298688277182061721445314257494401506613946316919762918150657974552623619122484
80638900336690743659892263495641146655030629659601997206362026035219177767406687
77463549375318899587866282125469797102065747232721372918144666659421872003474508
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48529830023476068454114178139110648560236549754227497231007615131870024053910510
91381784372179142252858743209852495787803468370333781842144401713868812424998441
86181292711985333153825673218704215306311977485352146709553346263366108646673322
92409879849256691109516143618601548909740241913509623043612196128165950518666022
03071561368473236466086890501426391390651506390819937885231836505989729912540447
94434251667742996598118492331515552728832740283526884424087528112832899806259126
73699546247341543333500147231430612750390307397135252069338173843322950701049061
86753943313078479801565513038475815568523621801041965025559618193498631591323303
60964619059902361126811960234418433633345949276319461017166529138237171823942992
16272538461776065694542297877071383198817036964588689811863210976900355735884624
46483570629145305275710127887202796536447972402540544813274839179412882642383517
19491972097971459368875371987291308317380339110161285474153773777159517280841116
27597186384924222802373441925469991983672192131287035585307966942713416391033882
75431861364349010094319740904733101447629986172542442335561223743571582593338280
49862438924982227807159517627578471094751190334822414120251826887137281931042534
78196128440176479531505057110722974314569915223451643121848657575786528197564843
50895838472292353455946452121583165775147129870822590929265563883665112068194383
69041162526687100445602437042006637090019411855571604720446436969328500600469281
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66636662609020704888743889890749815286544438186291738290105182086993638266186830
39152732645812867828066013375000965933646251460917231803129303478774212346791184
54791311109897794648216922505629399956793483801699157439700537542134485874586856
04728675106542334189383909911058646559511364606105515683854121745980180713316361
25730796111683438637676673073545834947897883163301292408008363568259391571131309
78030516441716682518346573675934198084958947940983292500086389778563494693212473
42610306271374507728615692259662857385790553324064184901845132828463270926975383
08673084091422476594744399733481308109863994173797896570106870267341619671965915
99588537834822988270125605842365589539690306474965584147981310997157542043256395
77607048510088157829140825077773855979012912940730946278594450585941227319481275
32251523248015034665190482289614066468903051025109162377704484862302294889667113
80555607956620732449373374027836767300203011615227008921843515652121379215748206
85935692079021450227713309998772945959695281704458218195608096581170279806266989
12050615607423256868422713062950098644218534708104071289176469065508361299166947
78023822502789667843489199409657361704586786242554006942516693979292624714524945
40885842272615375526007190433632919637577750217600519580069384763578958687848953
68721228985578068265181927036320994801558744555751753127364714212955364940843855
86615208012115079075068553344489258693283859653013272046970694571546959353658571
78889486233329246520273585318853337094845540333656535698817258252891805663548836
37437933484118455801683318276768346462919956055134700391478768086403226296166415
60667508153710646723108461964247537490553744805318226002710216400980584497526023
03564003808347205314994117296573678506642140084269649710324191918212121320693976
91439233683747092282677387081322366800869247034915868409911530983154120635661231
87504305467536983230827966457417620806593177265685841681837966106144963432544111
70694170022265781735835125982108076910196105222926387974504901925431190062056190
65774524161919131875339840493439768233102984658933183730158095925228292068208622
30332585280119266496314441316442773003237792274712330696417149945532261035475145
63129066885434542686978844774298177749371011761465162418361668025481529633530849
08499430067636548061029400946937506098455885580439704859144495844450799784970455
83550685408745163316464118083123079704389849190506587586425810738422420591191941
67418249045270028826398305795005734171148703118714283418449915345670291528010448
51451760553069714417613685823841027876593246626899784183196203122624211773914772
08004883578333569204533935953254564897028558589735505751235129536540502842081022
78524877660357424636667314868027948605244578267362623085297826505711462484659591
42102781227889414481639949738818846227682448516220518170767221698632657016543169
19742651230041757329904473537672536845792754365412826553581858046840069367718605
02007054724754840080553042495185449526724726134731817474218007857469346544713603
69758841180294080396167469462885406791721386012254195038197045384172680063988206
56328792839582708510919958839448297775647152026132871089526163417707151642899487
95356485455355314875497813400996485449863582484769059003311696130376612792346432
31297066284113074270462020320133683503854253603136367635752126047074253112092334
02837482949453104727418969287275572027615272268283376741393425652653283068469997
59709775000556088993268502504921288406827413988163154045649035077587168007405568
57240217586854390532281337707074158307562696283169556874240605277264858530506113
56384851965918968649596335568216975437621430778665934730450164822432964891270709
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28750519442356402644303271737341591281496056168353988188569484045342311424613559
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59621871992791865754908578529500128402290350615149373101070094461510116137124237
61426722541732055959202782129325725947146417224977321316381845326555279604270541
87149623658525245864893325414506264233788565146467060429856478196846159366328895
42997807225422647904006160197519750074605451500602918066382714970161109879513366
33771378434416194053121445291855180136575558667615019373029691932076120009255065
08158327550849934076879725236998702356793102680413674571895664143185267905471716
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90600915472919636381673596673959975710326015571920237348580521128117458610065152
59888384311451189488055212914577569914657753004138471712457796504817585639507289
5337539755822087777506072339445587895905719156733
</pre> <p>&nbsp;</p>    ]]></content>
  </entry>
  <entry>
    <title>Impressions of Istanbul</title>
    <link rel="alternate" type="text/html" href="http://kosara.net/thoughts/istanbul.html" />
    <id>http://kosara.net/thoughts/istanbul.html</id>
    <published>2008-02-22T23:43:44-05:00</published>
    <updated>2008-02-22T23:43:44-05:00</updated>
    <author>
      <name>Robert Kosara</name>
    </author>
    <category term="Archive" />
    <category term="Article" />
    <summary type="html"><![CDATA[<p>This page used to be called &quot;Links to Istanbul&quot;, but somehow the number of links decreased until none of the worked any more. So what is left are a few impressions of this city. I wrote this after a short visit there in 1996 or so. This was before I had been to other big cities like New York, so the numbers might not impress you that much.</p>  <p>Still, this city has a very strong athmosphere to it and it is certainly worth a visit.</p>    ]]></summary>
    <content type="html"><![CDATA[<p>This page used to be called &quot;Links to Istanbul&quot;, but somehow the number of links decreased until none of the worked any more. So what is left are a few impressions of this city. I wrote this after a short visit there in 1996 or so. This was before I had been to other big cities like New York, so the numbers might not impress you that much.</p>  <p>Still, this city has a very strong athmosphere to it and it is certainly worth a visit.</p>  <hr /> <p>This city is built on seven hills, like some other important cities, but, unlike any other city, it is built on two continents: Europe and Asia. Today,  an estimated 12 million people live there, in a city that has a diameter of 120 km. But it is not like a city at all, it looks like many, many small  villages, that have been thrown together, loosely connected by streets and bridges. When driving through this city, you see a lot of green spots, and meadows with sheep, and even barracks!</p>  <p>Osman, our guide, said that in Istanbul, there were only two kinds of pedestrians: fast ones and dead ones. There are 2.5 million cars there, and around 22.000 <i>Taksis</i>, easily recognisable by their yellow color.</p>  <p>I'm not sure if that applies to all Turks, but in Istanbul, at least, driving means one hand on the horn, never stopping to let pedestrians cross the street, and always going as closely to the one before you as by any means possible.  However, I haven't seen one accident or damaged car there, although I witnessed two quite dangerous situations.</p>  <p>Istanbul seems to be a culture in itself, a kind of city-state. Can you believe that there are over 3000 mosques there? The number of ordinary houses must be 50 times as many! But there also more modern things there: I counted 42 radio stations on the USW scale, and there are at least four 'local' tv stations: Channel 6, HBR, KRAL and NumberOne (the latter two being music channels that harldly play any other than Turkish music). Istanbul's citizens seem to feel more like Istanbulians than like Turks, yet they might get lost in certain parts of this huge city, since they can't know every part of it!</p>  <p>Another interesting fact is, that for certain shops, there are certain parts of the city. There is, for example, a part where all the newspapers, printing shops and bookbinderies are, or another, that contains all the headquartiers of banks. I also found that funny in the beginning, until I wanted to buy a CD, and couldn't find a CD shop anywhere! I had to go to a certain part of Istanbul (Unkapani), and there they were: one shop next to the other. It is unbelievable to a European (or American, for that matter), to see so many similar shops in one place, while there aren't any of those in other parts of the city! The CD I bought, by the way, is <i>Nefes Keser Asklar</i> by Sibel T&uuml;z&uuml;n, an amazing CD, that contains many different styles of music, really great! The song that made me buy the CD is track number 3, <i>Kacin Kurrasi</i>, which I saw on one of the music channels ...</p>  <p>The Turkish language has a very strong appeal - to me, anyway. I could just sit there and listen to Turks talking for hours. Like Italian, it is perfect for singing, for it contains many vowels (and about twice as many different ones as Italian), and has its own melody and sound. It must be quite difficult to learn, however, for it is related to Hungarian and Finnish.</p>  <p>Istanbul really is a fascinating city, full of exciting sights and friendly people (except for the pickpockets ...), you should really pay it a visit! Only beware of the Turkish sweets, I ate so much of them, I thought I'd die the next day ...</p>    ]]></content>
  </entry>
  <entry>
    <title>The Ackermann Function</title>
    <link rel="alternate" type="text/html" href="http://kosara.net/thoughts/ackermann.html" />
    <id>http://kosara.net/thoughts/ackermann.html</id>
    <published>2008-02-22T23:40:28-05:00</published>
    <updated>2008-02-23T22:13:13-05:00</updated>
    <author>
      <name>Robert Kosara</name>
    </author>
    <category term="Archive" />
    <category term="Article" />
    <category term="Math" />
    <summary type="html"><![CDATA[<p>The <a href="http://en.wikipedia.org/wiki/Ackermann_function">Ackermann Function</a> is a simple recursive function that produces incredibly large values with very simple inputs. Here is a short description, a way to calculate some of its values using simple formulas, and a very large number.</p>     ]]></summary>
    <content type="html"><![CDATA[<p>The <a href="http://en.wikipedia.org/wiki/Ackermann_function">Ackermann Function</a> is a simple recursive function that produces incredibly large values with very simple inputs. Here is a short description, a way to calculate some of its values using simple formulas, and a very large number.</p> <!--break--> <p>This is the definition of the function, A(<i>m</i>, <i>n</i>):</p> <blockquote> <table>     <tbody>         <tr>             <td><i>m</i> = 0</td>             <td>A(0, <i>n</i>) = <i>n</i>+1</td>         </tr>         <tr>             <td><i>m</i> &gt; 0, <i>n</i> = 0 &nbsp;&nbsp;</td>             <td>A(<i>m</i>, 0) = A(<i>m</i>-1, 1)</td>         </tr>         <tr>             <td><i>m</i> &gt; 0, <i>n</i> &gt; 0</td>             <td>A(<i>m</i>, <i>n</i>) = A(<i>m</i>-1, A(<i>m</i>, <i>n</i>-1))</td>         </tr>     </tbody> </table> </blockquote> <p>This can be easily implemented in any programming language.&nbsp;Calculating some values for the first four lines yields the following table:</p> <blockquote> <table>     <tbody>         <tr>             <td>&nbsp;</td>             <td colspan="11" align="center"><i><b>n</b></i></td>         </tr>         <tr>             <td><i><b>m&nbsp;</b></i></td>             <td align="right"><b>0</b></td>             <td align="right"><b>1</b></td>             <td align="right"><b>2</b></td>             <td align="right"><b>3</b></td>             <td align="right"><b>4</b></td>             <td align="right"><b>5</b></td>             <td align="right"><b>6</b></td>             <td align="right"><b>7</b></td>             <td align="right"><b>8</b></td>             <td align="right"><b>9</b></td>             <td align="right"><b>10</b></td>         </tr>         <tr>             <td><b>0</b></td>             <td align="right">1</td>             <td align="right">2</td>             <td align="right">3</td>             <td align="right">4</td>             <td align="right">5</td>             <td align="right">6</td>             <td align="right">7</td>             <td align="right">8</td>             <td align="right">9</td>             <td align="right">10</td>             <td align="right">11</td>         </tr>         <tr>             <td><b>1</b></td>             <td align="right">2</td>             <td align="right">3</td>             <td align="right">4</td>             <td align="right">5</td>             <td align="right">6</td>             <td align="right">7</td>             <td align="right">8</td>             <td align="right">9</td>             <td align="right">10</td>             <td align="right">11</td>             <td align="right">12</td>         </tr>         <tr>             <td><b>2</b></td>             <td align="right">3</td>             <td align="right">5</td>             <td align="right">7</td>             <td align="right">9</td>             <td align="right">11</td>             <td align="right">13</td>             <td align="right">15</td>             <td align="right">17</td>             <td align="right">19</td>             <td align="right">21</td>             <td align="right">23</td>         </tr>         <tr>             <td><b>3</b></td>             <td align="right">&nbsp;&nbsp; 5</td>             <td align="right">&nbsp;&nbsp; 13</td>             <td align="right">&nbsp;&nbsp; 29</td>             <td align="right">&nbsp;&nbsp; 61</td>             <td align="right">&nbsp;&nbsp; 125</td>             <td align="right">&nbsp;&nbsp; 253</td>             <td align="right">&nbsp;&nbsp; 509</td>             <td align="right">&nbsp;&nbsp; 1021</td>             <td align="right">&nbsp;&nbsp; 2045</td>             <td align="right">&nbsp;&nbsp; 4093</td>             <td align="right">&nbsp;&nbsp; 8189</td>         </tr>     </tbody> </table> </blockquote> <p>This is the table that can be found in many books on programming and/or recursion. The Ackermann function is more like an array than a usual function, the results of lower recursion levels are used to access other parts of the array. This is easy to see when writing the function slightly differently, with square brackets instead of parentheses (i.e., using a notation similar to C, Java, etc.):</p> <blockquote> <table>     <tbody>         <tr>             <td><i>m</i> = 0</td>             <td>A[0, n] = <i>n</i>+1</td>         </tr>         <tr>             <td><i>m</i> &gt; 0, n = 0 &nbsp;&nbsp;</td>             <td>A[<i>m</i>, 0] = A[<i>m</i>-1, 1]</td>         </tr>         <tr>             <td><i>m</i> &gt; 0, n &gt; 0</td>             <td>A[<i>m</i>, <i>n</i>] = A[<i>m</i>-1, A[<i>m</i>, <i>n</i>-1]]</td>         </tr>     </tbody> </table> </blockquote> <p>The first line can be thought of as&nbsp;the initialization for line 0 of the array, and it appears that this function contains hardly any calculations of actual values, but mostly of their indices. Coding this into a computer program is pretty straight-forward, but the range of values that can be computes using most programming languages is rather limited. It also takes surprisingly long to calculate the numbers in the third row, because the number of recursive steps is considerable.</p> <p>Looking at the array of numbers above, the first three lines follow simple patterns that could be expressed as simple, non-recursive formulas.&nbsp;The fourth line (<i>m</i>=3) shows a much more interesting behavior, though. Suddenly, the numbers grow very quickly, in fact, they more than double from <i>n</i> to <i>n</i>+1.</p> <p>Using the table above, it is possible to derive functions that calculate the values of A(<i>m</i>, <i>n</i>) for fixed values of <i>m</i> from 0 to 3 (where 'x^y' stands for 'x to the power of y'):</p> <blockquote> <table>     <tbody>         <tr>             <td><b>m &nbsp;&nbsp;</b></td>             <td class="rteleft"><b>A(<i>m</i>, <i>n</i>) = f(<i>n</i>)</b></td>         </tr>         <tr>             <td>0</td>             <td>A(0, <i>n</i>) = <i>n</i>+1</td>         </tr>         <tr>             <td>1</td>             <td>A(1, <i>n</i>) = <i>n</i>+2</td>         </tr>         <tr>             <td>2</td>             <td>A(2, <i>n</i>) = 3+2*<i>n</i></td>         </tr>         <tr>             <td>3</td>             <td>A(3, <i>n</i>) = 5+8*(2^<i>n</i>-1)</td>         </tr>     </tbody> </table> </blockquote> <p>Using these formulas, these values can all be calculated in constant time, rather than with a complexity that is growing rapidly with <i>m</i> and <i>n</i>. In practice, the values quickly exceed&nbsp;2^32-1, which is the limit for integers in many programming languages. Using the formulas above, we can easily find out the largest <i>n</i> for any largest number that can be represented, <i>L</i>&nbsp;('ld' is the 'logarithmus dualis', the base 2 logarithm). The right column in the following table lists the values for L=2^32-1 and L=2^64-1 (i.e., 32-bit and 64-bit integers):</p> <blockquote> <table>     <tbody>         <tr>             <td><b>m</b></td>             <td><b>A(<i>m</i>, <i>n</i>) &lt; L &lt;=&gt;&nbsp;</b></td>             <td><b>A(<i>m</i>, <i>n</i>) &lt; 2^32-1 &lt;=&gt;</b></td>             <td><b>A(<i>m</i>, <i>n</i>) &lt; 2^64-1 &lt;=&gt;</b></td>         </tr>         <tr>             <td>0</td>             <td><i>n</i> &lt;= <i>L</i>-1</td>             <td><i>n</i> &lt;= 2^32-2</td>             <td><i>n</i> &lt;= 2^64-2</td>         </tr>         <tr>             <td>1</td>             <td><i>n</i> &lt;= <i>L</i>-2</td>             <td><i>n</i> &lt;= 2^32-3</td>             <td><i>n</i> &lt;= 2^64-3</td>         </tr>         <tr>             <td>2</td>             <td><i>n</i> &lt;= (<i>L</i>-3)/2</td>             <td><i>n</i> &lt;= 2147483646 = 2^31-2 &nbsp; &nbsp; &nbsp;&nbsp;</td>             <td><i>n</i> &lt;= (2^64-3)/2 ~ 2^63</td>         </tr>         <tr>             <td>3</td>             <td><i>n</i> &lt;= ld((<i>L</i>-5)/8+1) &nbsp; &nbsp; &nbsp;&nbsp;</td>             <td><i>n</i> &lt;= 29</td>             <td><i>n</i> &lt;= 61</td>         </tr>     </tbody> </table> </blockquote> <p>Not surprisingly, using 64 bits does not buy much room for larger values of <i>n</i> on the fourth line. Also, we still only have formulas for values of <i>m</i> up to three.&nbsp;Using the table from the very top of this article, we see that A(4, <i>n</i>) =  A(3, A(4, <i>n</i>-1)), and if <i>n</i>-1 = 0, we can use A(<i>m</i>, 0) = A(<i>m</i>-1, 1), so A(4, 0) = A(3, 1)<i>&nbsp;<span class="Apple-style-span" style="font-style: normal; ">&ndash;&nbsp;which we can look up in the second table or calculate easily.</span></i></p> <p>Using this results, we find that <i>A</i>(4, 0) = 13. <i>A</i>(4, 1) is a lot larger, A(4, 1) = A(3, A(4, 0)) = A(3, 13) = 65533. It gets interesting with A(4, 2) = A(3, A(4, 1)) = A(3, 65533) = 5+8*(2^65533-1). The result is a number with 19729 digits! We can go one line further, to A(5, 0) = A(4, 1) = 65533, but A(5, 2) = A(4, A(5, 1)) = A(4, 65533) which is far beyond our reach, we can't even calculate A(4, 3) = A(3, A(4, 2))!<br /> This is also true for A(6, 0) = A(5, 1) - which means, that we can't calculate a <i>single</i> value from the seventh line, or those beyond! So this is the final table (A(4, 2) links to the actual number):</p> <blockquote> <table>     <tbody>         <tr>             <td>&nbsp;</td>             <td colspan="10" align="center"><b>n</b></td>         </tr>         <tr>             <td><b>m</b></td>             <td align="right"><b>0</b></td>             <td align="right"><b>1</b></td>             <td align="right"><b>2</b></td>             <td align="right"><b>3</b></td>             <td align="right"><b>4</b></td>             <td align="right"><b>5</b></td>             <td align="right"><b>6</b></td>             <td align="right"><b>7</b></td>             <td align="right"><b>8</b></td>             <td align="right"><b>9</b></td>         </tr>         <tr>             <td><b>0</b></td>             <td align="right">1</td>             <td align="right">2</td>             <td align="right">3</td>             <td align="right">4</td>             <td align="right">5</td>             <td align="right">6</td>             <td align="right">7</td>             <td align="right">8</td>             <td align="right">9</td>             <td align="right">10</td>         </tr>         <tr>             <td><b>1</b></td>             <td align="right">2</td>             <td align="right">3</td>             <td align="right">4</td>             <td align="right">5</td>             <td align="right">6</td>             <td align="right">7</td>             <td align="right">8</td>             <td align="right">9</td>             <td align="right">10</td>             <td align="right">11</td>         </tr>         <tr>             <td><b>2</b></td>             <td align="right">3</td>             <td align="right">5</td>             <td align="right">7</td>             <td align="right">9</td>             <td align="right">11</td>             <td align="right">13</td>             <td align="right">15</td>             <td align="right">17</td>             <td align="right">19</td>             <td align="right">21</td>         </tr>         <tr>             <td><b>3</b></td>             <td align="right">5</td>             <td align="right">13</td>             <td align="right">29</td>             <td align="right">&nbsp;&nbsp; 61</td>             <td align="right">&nbsp;&nbsp; 125</td>             <td align="right">&nbsp;&nbsp; 253</td>             <td align="right">&nbsp;&nbsp; 509</td>             <td align="right">&nbsp;&nbsp; 1021</td>             <td align="right">&nbsp;&nbsp; 2045</td>             <td align="right">&nbsp;&nbsp; 4093</td>         </tr>         <tr>             <td><b>4</b></td>             <td align="right">13</td>             <td align="right">&nbsp;&nbsp; 65533</td>             <td align="right">&nbsp;&nbsp; <a href="/thoughts/ackermann42.html">A(4, 2)</a></td>         </tr>         <tr>             <td><b>5</b></td>             <td align="right">&nbsp;&nbsp; 65533</td></tr></tbody></table></blockquote>  <h1>Programs</h1> <p>The following programs were used to calculate the numbers above, including A(4, 2). They were written in LISP, for its ability to work with arbitrarily large numbers; the LISP interpreter I used was <a href="http://clisp.cons.org/">CLISP</a>, which is freeware, and runs on many operating systems. Please note that Emacs LISP is not able to handle large numbers, so anything above A(3, 29) will not work.</p><p>Here is the basic program using the simple formulas for m &lt;= 3:</p> <blockquote> <pre>
(defun ackermann (m n) &quot;The Ackermann Function&quot;
(cond ((= m 0) (+ n 1))
      ((= m 1) (+ n 2))
      ((= m 2) (+ 3 (* n 2)))
      ((= m 3) (+ 5 (* 8 (- (expt 2 n) 1))))
      (t (cond ((= n 0) (ackermann (- m 1) 1))
               (t (ackermann (- m 1) (ackermann m (- n 1))))
         ))
))
</pre> </blockquote> <p>It is run using the following line, with <i>m</i> and <i>n</i> replaced with numbers:</p> <blockquote> <pre>
(ackermann <i>m</i> <i>n</i>)
</pre> </blockquote> <p>The following function is used for counting the number of digits in a number, and is very inefficient and also horrible LISP style. It gets the job done, though:</p> <blockquote> <pre>
(defun digits (num) &quot;number of digits of num&quot;
(setq a 0)
(loop
 (setq a (+ a 1))
 (setq num (floor (/ num 10)))
 (when (= num 0) (return a))
))
</pre> </blockquote> <p>Invoking it with</p> <blockquote> <pre>
(digits (ackermann <i>m</i> <i>n</i>))
</pre> </blockquote> <p>will return the number of digits of A(m, n).</p>    ]]></content>
  </entry>
  <entry>
    <title>Arsat 2.8/35mm Shift Lens</title>
    <link rel="alternate" type="text/html" href="http://kosara.net/photo/arsatshift.html" />
    <id>http://kosara.net/photo/arsatshift.html</id>
    <published>2008-02-22T23:37:39-05:00</published>
    <updated>2008-02-22T23:48:38-05:00</updated>
    <author>
      <name>Robert Kosara</name>
    </author>
    <category term="Archive" />
    <category term="Article" />
    <category term="Photography" />
    <summary type="html"><![CDATA[<p>Some time ago, I thought I needed a shift (perspective correction)       lens. But the original Nikon PC lenses are extremely expensive (at       least for me, doing photography only as a hobby), so I bought       a Russian Arsat 2.8/35 mm shift lens, which is much cheaper.       Here is some data about that lens, including a table of how far       you have to stop down to avoid vignetting.</p>    ]]></summary>
    <content type="html"><![CDATA[<p>Some time ago, I thought I needed a shift (perspective correction)       lens. But the original Nikon PC lenses are extremely expensive (at       least for me, doing photography only as a hobby), so I bought       a Russian Arsat 2.8/35 mm shift lens, which is much cheaper.       Here is some data about that lens, including a table of how far       you have to stop down to avoid vignetting.</p>      <h2>What is a Shift Lens?</h2>      <p>Buildings are usually taller than people, and so if you want to take       a picture of one, you will be tempted to tilt you camera up so that       you get the whole building on the picture. But that leads to converging       lines, and the buildings seem to tilt back. This is illustrated quite       nicely at       <a href="http://www.photo.net/photo/canon/tilt-shift.html">photo.net</a>.</p>      <p>The right way to correct this problem would be to move the camera       up so that it is at half the height of the building. Now that is       often easier said than done, so an optical trick would be the preferred       way.</p>            <p>A shift lens can be moved perpendicular to the film plane so that        a different part of the picture is projected onto the negative (or       slide) without having to tilt the lens. It is important to understand       that the image is shifted, because       the movement of the lens is negligible compared to the objects usually       depicted (houses). So this way, you can leave the camera in a       position where the lens is horizontal, but move the lens up, and thus       select a part of the image that contains the whole building, but not       the foreground. This way, you get a picture of your house upright       and still fill most of the available negative or slide space.</p>      <p>The perspective is of course changed by this, so that pictures       taken with shift lenses always look a bit unnatural. But that is the       smaller evil compared to houses that seem to tilt and crumble ...</p>      <h2>The Arsat 2.8/35</h2>      <p>The Arsat 2.8/35 is available for different lens mounts, the one       I have is for Nikon. It does not have Auto Focus, of course, neither       does it have the AI feature that tells the camera the relation       between the selected and the maximum f-stop. Thus, you can only measure       exposure when stopped down. And with the F80 (called N80 in the US),       you can't do any       exposure metering at all, because there is no electronic circuitry,       without which the F80 won't activate the exposure meter.</p>      <p>The aperture can be changed continuously between 2.8 and 22, the       aperture preselection locks at whole f-stops (see below). The lens can       be shiftet 11 mm       from its &quot;normal&quot; position, the direction of the shift can be       changed 360&deg; and locks at eight angles.</p>      <p>The procedure when taking a picture is as follows:</p>      <ol>       <li>Compose picture with unshiftet lens</li>       <li>focus</li>       <li>stop down to measure exposure</li>       <li>set time manually at camera</li>       <li>shift lens to get the desired frame</li>       <li>take picture</li>     </ol>      <p>The important part is not to forget stopping down again if you change the     composition or refocus (which you have to do at maximum aperture, of     course). But this is made easier with a nice feature of the Arsat lens:     you can preselect the f-stop, then you cannot stop down more than the     selected f-stop, which makes it a lot easier (i.e., you don't have     to look) to go back and forth from 2.8 to the desired f-stop. In addition,     you will can often just set the lens to infinity, so focusing isn't such     an important issue.</p>      <h2>Vignetting, Sharpness, Distortion</h2>      <p>The lens does not cover enough area to be shiftet the whole 11 mm       without vignetting (i.e., dark areas in the corners). You can       get rid of the vignetting by stopping down. Just how far you have       to stop down is shown in the following table that I came up with       by taking pictures of the same building 72 times ...</p>      <table border="1">       <tbody><tr><th>&nbsp;</th><th>0 mm</th><th>7 mm</th><th>8 mm</th><th>10 mm</th><th>11 mm</th></tr>       <tr><td>Vertical Shift</td><td>2.8</td><td>2.8</td><td>5.6</td><td>5.6</td><td>8</td></tr>       <tr><td>Horizontal Shift</td><td>2.8</td><td>5.6</td><td>8</td><td>8</td><td>11</td></tr>     </tbody></table>      <p>&quot;Vertical&quot; and &quot;horizontal&quot; are meant relative to a landscape picture,       so &quot;vertical&quot; means shifting along the shorter side of the negative,       and &quot;horizontal&quot; along the longer side.</p>      <p>I used Kodak EliteChrome 100 slide film for the test, and my tripod       of course.</p>      <p>Using my 50 mm lens as a loupe, I could see no distortions at the       edges. The test also showed that the sharpness of the lens is quite       good, but I don't have a comparison with the Nikon PC lenses.</p>      <h2>Overall Impression</h2>      <p>The Arsat shift lens seems to be a decent shift lens that is much       cheaper than the Nikon PC lenses but has quite good quality. Especially       if you don't do a lot of architecture shots, you probably don't want       to spend so much money on a lens you only use occasionally.<br />       One thing you should really think of getting for your camera if       you buy a shift lens is a focusing screen with a grid so you can       really get the buildings upright. But that type of focussing screen       is very useful even without a shift lens, because it makes aligning       the horizon and other features a lot easier.</p>    ]]></content>
  </entry>
  <entry>
    <title>Agfa APX 25 vs. Kodak Technical Pan</title>
    <link rel="alternate" type="text/html" href="http://kosara.net/photo/apx25vstp.html" />
    <id>http://kosara.net/photo/apx25vstp.html</id>
    <published>2008-02-22T23:12:54-05:00</published>
    <updated>2008-02-22T23:28:29-05:00</updated>
    <author>
      <name>Robert Kosara</name>
    </author>
    <category term="Archive" />
    <category term="Article" />
    <category term="Photography" />
    <summary type="html"><![CDATA[<p>What's the point of comparing two black-and-white films that are not available anymore? Somebody emailed me with that question several years ago. It was still interesting to do this comparison, and who knows when you might get into an argument between b&amp;w photo-geeks and need a source to back you up?</p>    ]]></summary>
    <content type="html"><![CDATA[<p>What's the point of comparing two black-and-white films that are not available anymore? Somebody emailed me with that question several years ago. It was still interesting to do this comparison, and who knows when you might get into an argument between b&amp;w photo-geeks and need a source to back you up?</p><p>These are the results of a comparison I did between the Agfa APX 25       and the Kodak Technical Pan. Both are black and white films with       extremely fine grain, so I wanted to know which is better.       This page reports the results of this comparison and also includes       some points that came out of a discussion after I posted my       results to the german-speaking newsgroup de.rec.fotografie.</p>      <p>The Techical Pan is usually used for reproductions of drawings and       other applications that require a very fine grain and extremely high       contrast. But it can be developed to be used for pictorial photography as       well. It is said to be the finest grain film available today. Well,       we'll see ...</p>      <p>This is a practical test to see which film has finer grain under       the circumstances I will use it in. This is clearly not a scientific       study, but you should get similar results when using reasonable       amateur equipment. If your equipment is much better, your results       might be different.</p>            <h2>Methodology</h2>      <p>The comparison was done with the 35mm versions of the films to be       compared. They were exposed with the same time/f-stop combination       using the same lens mounted to different bodies. Here is a summary       of the exposure data:</p>      <table border="1">       <tbody><tr><td><b>Lens:</b></td><td>Nikkor AI 2.8/24</td></tr>       <tr><td><b>F-Stop:</b></td><td>8</td></tr>       <tr><td><b>Exposure Time:</b></td><td>1/30s (I think)</td></tr>       <tr><td><b>Camera:</b></td><td>Nikon FE-2 and FT-2</td></tr>       <tr><td><b>Tripod:</b></td><td>Manfrotto 055CB with head 141RC</td></tr>     </tbody></table>      <p>The films were developed in a small tank according to the       recommendations in the datasheets (which are available for the APX 25       (<a href="http://www.agfaphoto.com/library/datasheet/pdf/film/pfe.pdf">PDF</a>)       and the Technical Pan (<a href="http://www.kodak.com/global/en/professional/support/techPubs/p255/p255.shtml">HTML</a> and       <a href="http://www.kodak.com/global/en/professional/support/techPubs/p255/p255.pdf">PDF</a>)). Here is the data (all at 20&deg;C):</p>        <table border="1"> 	<tbody><tr><th>&nbsp;</th><th>Agfa APX 25</th><th>Kodak Technical Pan</th></tr> 	<tr><td><b>Developer:</b></td><td>Agfa Rodinal 1+50</td><td>Kodak Technidol</td></tr> 	<tr><td><b>Duration:</b></td><td>10 min</td><td>9 min</td></tr> 	<tr><td><b>Agitation:</b></td><td>continously first minute,<br /> then four inversions every 30s</td><td>10-12 times up and down in 2s every 30s</td></tr>       </tbody></table>      <p>The negatives were then enlarged to the same magnification (about       16 times the size of the negative, 59x39 cm (about 23x15 in)) and the       prints were scanned in (with an Agfa SnapScan at 300 dpi).</p>            <h2>Results</h2>      <p>You can look at the detail images <a href="#images">below</a>.</p>      <p>In these images, you can see that the APX 25 is slightly sharper       than the Technical Pan. There is no visible grain in either of the       images. But due to the limitations of the process, I would call it       a draw ...</p>      <p>What you cannot see, is that the APX 25 needs about &frac12; f-stop       less to reach the same grey level on the paper (grade 3) than       the Technical Pan. I can never figure out what that means in       terms of densitometry, so I leave that as an exercise to the reader ...</p>      <p>I also had to refocus the enlarger after putting in the other film.       I don't quite understand why this is necessary, since the emulsion should       be at the same position no matter how thick the film carrier is (the       Technical Pan is supposedly thinner).</p>      <p>Using the grain focuser, you can see the grain, but only as a       pattern, not really the single grains. It looks like a 100 ASA film       enlarged to 13x18 cm (5x7 in). I will make a comparison with       a microscope when I get a chance.</p>          <h2 id="images">Images</h2>      <p>These are scans from the prints that were not sharpened or unsharp       masked in       order not to introduce any artifacts (but they probably appear       to be a bit blurrier than they really are). The bottom images show       a detail that is about 0.74mm big on the negative (or about 3% of       the shorter side of the negative). Each pixel has a size of about       2.5 &micro;m x 2.5 &micro;m on the negative. This is the maximum       I could do with the equipment I have access to.</p>      <table border="1">       <tbody><tr><th>Agfa APX 25</th><th>Kodak Technical Pan</th></tr>       <tr><td><img alt="APX 25 big detail" width="386" height="284" src="/files/images/apx25more.png" /></td> 	<td><img alt="Technical Pan big detail" width="386" height="284" src="/files/images/tpmore.png" /></td></tr>       <tr><td align="center"><img alt="APX 25 small detail" width="308" height="298" src="/files/images/apx25detail.png" /></td> 	<td align="center"><img alt="Technical Pan small detail" width="308" height="297" src="/files/images/tpdetail.png" /></td></tr>       </tbody></table>        <h2>Discussion</h2>      <p>The two films are almost identical, safe for the slight difference       in density (which can easily be corrected by adjusting development).       Up to an enlargement of about 16-fold there is no visible difference,       and even small details are very precisely depicted. The APX 25 seems to       be slightly sharper, but that is difficult to say.</p>      <p>But if you take the cost of the films into account, the APX 25 is the       clear winner: It costs only about one third of the Technical Pan, and       Rodinal is also much cheaper than Technidol (and Rodinal can also be used       for many other films). With Technidol, you also have to develop two       films at the same time, because it comes in small bottles for two       films and cannot be stored.</p>      <p>&nbsp;</p>    ]]></content>
  </entry>
  <entry>
    <title>The Turing Test</title>
    <link rel="alternate" type="text/html" href="http://kosara.net/thoughts/turing.html" />
    <id>http://kosara.net/thoughts/turing.html</id>
    <published>2008-02-22T22:42:36-05:00</published>
    <updated>2008-04-05T23:52:16-04:00</updated>
    <author>
      <name>Robert Kosara</name>
    </author>
    <category term="Archive" />
    <category term="Article" />
    <summary type="html"><![CDATA[<p>This test was invented by Alan M. Turing (1912-1954) and first described in his 1950 article <i>Computing machinery and intelligence</i> (<i>Mind</i>, Vol. 59, No. 236, pp. 433-460).</p>    ]]></summary>
    <content type="html"><![CDATA[<p>This test was invented by Alan M. Turing (1912-1954) and first described in his 1950 article <i>Computing machinery and intelligence</i> (<i>Mind</i>, Vol. 59, No. 236, pp. 433-460).</p>  <h2>How It Works</h2><p>An interrogator is connected to one person and one machine via a terminal, and therefore can't see her counterparts. Her task is to find out which of the two candidates is the machine, and which is human only by asking them questions. If the interrogator cannot make a decision within a certain time (Turing proposed five minutes, but the exact amount of time is generally  considered irrelevant), the machine is considered to be intelligent.</p>  <p><span class="Apple-style-span" style="font-family: 'Century Gothic'; font-size: 16px; font-weight: bold; ">Criticism</span></p>  <p>This test has been subject to many different kinds of criticism, but it is the only one known - and as long as there is no definition for (human) intelligence, it will most probably remain so. The Turing Test is quite amazing in that it provides a test for something that today's science doesn't even have the remotest idea of!</p>  <p>The most important argument against the Turing Test, in my opinion, is that it only provides a test for <i>human</i> intelligence (see French, Robert M.: <a href="http://www.ulg.ac.be/cogsci/rfrench/turing.pdf"><i>Subcognition and the Limits of the Turing Test</i></a>). Even a person from a different culture might be considered a 'machine' (i.e., not intelligent) because of certain questions she wouldn't be able to answer, or would answer in an unexpected way. For example, asking about the side of the road you drive on would be answered in different ways, and there are more subtle differences people might not be aware of (a minor detail in everyday life you take for granted to be one way, while it is different in a different part of the world). Thus, the Turing Test shares its fate with early IQ tests the US Army used, and that immigrants usually failed because of their lack of knowledge of American culture.</p>      <p><p style="margin-top: 0.5em; margin-bottom: 0.9em; ">For a funny take (and a nice illustration, at the same time) on the Turing Test, see&nbsp;<a style="text-decoration: none; color: rgb(68, 34, 68); " href="http://modernhumorist.com/mh/0107/turing/">Are You A Computer?</a>.</p></p>    ]]></content>
  </entry>
  <entry>
    <title>Thoughts on Artificial Intelligence</title>
    <link rel="alternate" type="text/html" href="http://kosara.net/thoughts/ai.html" />
    <id>http://kosara.net/thoughts/ai.html</id>
    <published>2008-02-22T22:32:43-05:00</published>
    <updated>2008-04-06T09:57:38-04:00</updated>
    <author>
      <name>Robert Kosara</name>
    </author>
    <category term="Archive" />
    <category term="Article" />
    <summary type="html"><![CDATA[<p><b>Please note:</b> This was written many years ago, after reading the article <a href="http://www.ulg.ac.be/cogsci/rfrench/turing.pdf"><i>Subcognition and the Limits of the Turing Test</i></a> by Robert M. French. AI is not my field of research, so take this as a bit of speculation, nothing more. I am keeping the article here because of the number of links that are pointing to it, and for historical reasons (this was the first thing I ever published on the Internet, back in 1996 or so).&nbsp;</p>    ]]></summary>
    <content type="html"><![CDATA[<p><b>Please note:</b> This was written many years ago, after reading the article <a href="http://www.ulg.ac.be/cogsci/rfrench/turing.pdf"><i>Subcognition and the Limits of the Turing Test</i></a> by Robert M. French. AI is not my field of research, so take this as a bit of speculation, nothing more. I am keeping the article here because of the number of links that are pointing to it, and for historical reasons (this was the first thing I ever published on the Internet, back in 1996 or so).&nbsp;</p>     <h2 id="intro">1. Introduction</h2>  <p>Although there is no clear definition of AI (or even of intelligence), it  can be described as the attempt to build machines that think and act like  humans, that are able to learn and to use their knowledge to solve  problems on their own.</p> <p>A 'by-product' of the intensive studies of the human brain by AI  researchers is a far better understanding of how it works.&nbsp;The <a href="http://en.wikipedia.org/wiki/Brain">human brain</a> consists of 10 to 100 billion neurons, each of which is  connected to between 10 and 10,000 others through synapses. The  single brain cell is comparatively slow (compared to a microprocessor)  and has a very simple function: calculating the weighted sum of its inputs and  issuing an output if that sum exceeds a certain value. Through its highly parallel way of operation, however, the human brain achieves a performance that has not been reached by computers yet; and while we might see single computers with performance comparable to the human brain by 2020, it will take a lot more understanding of how the brain works to even just copy its function in a rudimentary way.&nbsp;</p>  <p>The&nbsp;<a href="turing.html">Turing Test</a>&nbsp;is currently the only accepted way to test a machine for exhibiting intelligent behavior. Such a thinking machine has yet to be built.</p>   <h2 id="why">2. Why the current approach of AI is wrong</h2>  <p>These following thoughts do not deal with technical problems of AI, nor  am I going to try to prove that humans are the only intelligent species (exactly the opposite, see <a href="#intell">section 5</a>). What I  want to show is that the whole idea of AI needs to be changed in order  to lead to more than just partial results.</p>  <h3 id="ref">2.1. Reference Points</h3>  <p>Today's AI concentrates entirely on the brain. If you look at the human  body, however, it is not clear where to draw the line between which parts  of the nervous system belong to the brain, and which don't. A number of  functions are performed by the spinal cord, for example, like  withdrawing the hand quickly when touching something hot. It can be  vitally important that this action is taken as fast as possible, in order  to limit the damage. The only way of doing so is through reflexes,  without the intervention of the brain. This is not an example of  intelligent decisions outside the brain, but it provides an entry point  to the following.</p>  <p>Whenever you talk to somebody, you use a huge amount of assumptions about  the background of your counterpart. You usually start with assuming that  other person is almost identical to yourself, and by small  misunderstandings and questions of him you correct that picture  you have. When you know somebody, you don't have a list of all his  features in your mind, but you know the differences between you and that  person (or the difference between him and a third party, be it a single  person or a group). This explains why it often is  so difficult to describe somebody to another person, since that other  person's reference point is different from your own.</p> <p>We build relations between all the things we know, and we build classes  of things by putting objects with certain similar features into one such  class. But the most basic difference we see is the difference between  ourselves and the objects, the not-ourselves. Man's first reference point  is himself, which is obvious when looking around oneself: Isn't that cat  looking very nosy at you? Doesn't that monitor's face look at you?  Don't that car's headlights look into a certain direction?</p> <p>People's categories are based on people, first and foremost. This is not an  intellectual decision, but a natural necessity. How could you ever find  out anything without a first reference point that you could relate it to? This also is the reason why in children's minds everything 'lives': They  live themselves, so why shouldn't other things? Why shouldn't that  teddybear feel hunger, exactly like I do?</p> <p>The point is clear: No knowledge can be accumulated without a reference  point. AI doesn't obey this. Most 'intelligent' programs are equipped  with knowledge, but none has ever had a clear picture of itself that it  could relate everything else to. This is the first deficiency.</p>  <h3 id="body">2.2. The Role of the Body</h3>  <p>The most obvious difference between man and other animals is his mind, his ability to accumulate knowledge and pass it on to his descendants.  Yet many of man's highly developed abilities can be completely switched off by the sheer terror of a single aching tooth. This also applies for other strong feelings like hatred, grief or pain in general. They can make people act against their better knowledge and their principles - these being higher developed parts of the mind. This leads to a conclusion that is obvious from looking at the ancestry of man: the vital functions rule over everything else. Man has not been built to wear digital watches (as Douglas Addams states), but he is the winner of a game that is as old as life itself: Evolution. If people were able to simply ignore hunger, they would starve to death; if they would have to control their lungs consciously, they would sooner or later suffocate. Vital functions must have priority over everything else. Considering this, it is not surprising that many of our expressions involve basic needs, like 'being hungry for love', 'being tired of something', 'being fed up', 'having a bleeding heart', 'saving someone's skin'. This is also an example for what was said in the last paragraph: The main reference point is oneself, this is of course also true for strong emotions that are on a less 'basic' level.</p> <p>Additionally, most (if not all) emotions are accompanied by physical symptoms,  such as the production of hormones, shivering, gnashing of teeth or  goose-pimples.</p> <p>The point I want to make here is that the human body plays an  important role in all intellectual processes, since they are mere  subordinates of its needs in order to stay alive. It is therefore  short-sighted (another example!) to try to build artificial minds  not only without any body, but also without even the <i>concept</i> of a  body. How should an artificial mind ever be able to understand tiredness,  excitment, happiness or fear without ever having <i>felt</i> it? And by  feeling, I mean the physical symptoms, and the intellectual processes  that accompany the fear of injury or death, for example. A body-less mind  can never understand that, und thus will never be able to understand  humans, let alone act like one. This is the second deficiency.</p>  <p><font size="-1">Two additional points here: AI, being a science, acts very  unscientific  here. What is being done (although not on purpose) is, that while denying  the existence of an immaterial soul that is independent of the body, only  the soul is taken into account, but not the effects of the body.<br /> The second point is an observation: The reason why men and women often  have such difficulties understanding one another, might be their  different reference points, and also their different bodies, that affect  the way they think. As shown above, these two points can easily lead to  misunderstandings, when one assumes the counterpart to be too similar to  oneself. This does not, however, mean that one of the sexes is superior,  but that they are different, and that these differences have to be worked  out and brought into people's minds.</font></p>  <h3 id="evol">2.3. Evolution</h3>  <p>This little interlude also brought another point up: the desire to  replicate. This being the most basic of all basic desires, it cannot be  ignored. No experiment in the field of AI has yet specified which gender the mind would be. In a world, where there is no need for a partner to  replicate (because there is a simple 'copy'-mechanism, for example),  social structures would be completely different from man's. It is even  the question, if different genders were necessary at all. Thus, an  artificial mind would also need the 'ability' to die, as well as that to  mate and replicate, otherwise the resulting being would be beyond any  recognition by a human mind.</p> <p>A mind that cannot die and that doesn't feel the need to replicate in a  manner similar to humans, would be very different from man. Such an  environment would have to be created artificially, but in a different  sense: The conditions would have to be made more difficult than they  needed to be, only to force the beings to act human-like. And only an  evolutionary process would lead to a mind similar to a human one.  This is the third deficiency.</p>  <h2 id="learn">3. The Art of Learning</h2>  <p>Learning isn't a 'static' ability, but one which continuously changes: You must learn how to learn, and the way you learn changes. The new-born child can only learn by first-hand experience, and hardly generalize. But the older the child gets, the more he/she can learn without having experienced a corresponding situation. Indeed, most of what you know is what you were taught by others, read in books, etc. This probably is the main advantage of man over all other animals: that we can pass on knowledge, so that the next generation doesn't have to make the same mistakes again (it does anyway, but that's a different problem ...).  Your knowledge includes the first-hand experiences of hundreds of thousands of people, whose knowledge and experiences were collected and put into a structured form, in order to make learning these facts easier. People nowadays aren't more intelligent than 1000 years ago, but we have more knowledge, and thus can achieve far higher goals. Like Newton said: &quot;I am a dwarf, but I can see very far for I am standing on the shoulders of giants&quot;</p>  <p>But what effects does this have on AI? The brain changes, and not only does its knowledge change, but also the way it accumulates knowledge and makes use of it. An artificial intelligence must be able to change its own programme.</p>  <h2 id="outside">4. Thinking Outside the Brain</h2>  <p>The previous point contains another interesting thought: Whatever you do, whatever you think or say - it hasn't been thought up entirely by you, it always contains parts from other people. This is the key to developing beyond what a single generation can reach (see previous section). But it also leads us to a somewhat discomforting question: How much of that brain is actually mine? What percentage of what I think has been thought by others already? How different am I?</p>  <p>Imagine a group of five people working on the solution of a complicated problem. They're sitting around a table, discussing ideas and problems, taking notes, and thinking about whether or not that last proposal was good. In the end, they will come up with a solution that will be far better than one that any of them had worked out in his/her own! The result will even be better than if each of them was assigned a fifth of the problem, and solved it alone. This team is able to achieve more than five single persons can do independently.</p>  <p>But what is the difference between five isolated persons and a team of five? If the team develops an idea that the single persons don't, which of the members created it? It's not a person that created the idea, but the interaction process, the discussion. An act of thinking has been done by an immaterial process, not a single person.</p>  <h2 id="intell">5. The idea of a General Concept of Intelligence</h2>  <p>All the points made above make one thing clear: In order to build an artificial intelligence, it must be built as human-like as possible.  Without basic human 'ingredients', the resulting mind might not even be recognized as such. This boils down to the feeling that the goal is to build a mere copy of the human mind.</p>  <p>Why on earth, one might wonder, would anybody want to build a copy of the human mind? Isn't the original working fine? Isn't it superior to everything known? Isn't one's mind the most difficult thing to be examined by itself? What would be the use of such an artificial mind, that would need even more artificial means, only to stay human-like?</p>  <p>The only logical solution to this is to completely separate human from artificial intelligence, in order to build something entirely new.</p>  <p>This naturally leads to the idea of a higher principle of Intelligence, that human intelligence is only one manifestation of (in order to distinguish between the traditional human intelligence and this new idea of a more general concept, I want to spell the latter with a capital I:  Intelligence). Another one would be artificial intelligence, and another one still the intelligence developed on a planet many lightyears from here. Again, I remind you of the points just made. Considering these, how should a mind that is the result of evolution on an entirely different planet be similar to ours in any way? There must be similarities, but on a higher level: on the level of Intelligence (note the capital I).</p>  <p>In that hierarchy, AI is on the same level as human intelligence,  together with animal intelligence and any other kind of intelligence that  one might encounter. The following figure illustrates this:</p>  <center><img width="441" height="241" alt="" src="/files/images/aitree.png" /><br /></center>  <h2 id="see">6. See also ...</h2>  <dl> <dt><a href="http://www.intelligent-systems.com.ar/intsyst/index.html">Intelligent Systems and  their Societies</a> by Walter Fritz</dt><dd> An excellent (albeit unfinished) book that puts many facinating ideas into a whole building of thoughts.</dd> <dt><a href="http://www.yahoo.com/Science/Computer_Science/Artificial_Intelligence/">Yahoo's AI page</a></dt><dd> Contains links to a lot of interesting sites, including this one ;-)</dd> <dt><a href="http://www.perceptualcontroltheory.org/">The Control Systems Group</a></dt><dd> A page on Perceptual Control Theory (PCT), which is a very promising approach to how the brain works. Includes a number of great demos you should really try out!</dd> <dt><a href="http://www.ai.mit.edu/projects/humanoid-robotics-group/">Cog, the Robot</a></dt><dd>A project at the MIT to build a human-like robot. That's probably the most hands-on approach to what is called 'embodied AI'.</dd> </dl><p>&nbsp;</p>    ]]></content>
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