Unphysically large finite computer

An un­phys­i­cally large finite com­puter is one that’s vastly larger than any­thing that could pos­si­bly fit into our uni­verse, if the char­ac­ter of phys­i­cal law is any­thing re­motely like it seems to be.

We might be able to get a googol (\(10^{100}\)) com­pu­ta­tions out of this uni­verse by be­ing clever, but to get \(10^{10^{100}}\) com­pu­ta­tions would re­quire out­run­ning pro­ton de­cay and the sec­ond law of ther­mo­dy­nam­ics, and \(9 \uparrow\uparrow 4\) op­er­a­tions (\(9^{9^{9^9}}\)) would re­quire amounts of com­put­ing sub­strate in con­tigu­ous in­ter­nal com­mu­ni­ca­tion that wouldn’t fit in­side a sin­gle Hub­ble Vol­ume. Even tricks that per­mit the cre­ation of new uni­verses and en­cod­ing com­pu­ta­tions into them prob­a­bly wouldn’t al­low a sin­gle com­pu­ta­tion of size \(9 \uparrow\uparrow 4\) to re­turn an an­swer, if the char­ac­ter of phys­i­cal law is any­thing like what it ap­pears to be.

Thus, in a prac­ti­cal sense, com­pu­ta­tions that would re­quire suffi­ciently large finite amounts of com­pu­ta­tion are prag­mat­i­cally equiv­a­lent to com­pu­ta­tions that re­quire hy­per­com­put­ers, and serve a similar pur­pose in un­bounded anal­y­sis—they let us talk about in­ter­est­ing things and crisply en­code re­la­tions that might take a lot of un­nec­es­sary over­head to de­scribe us­ing small finite com­put­ers. Nonethe­less, since there are some math­e­mat­i­cal pit­falls of con­sid­er­ing in­finite cases, re­duc­ing a prob­lem to one guaran­teed to only re­quire a vast finite com­puter can some­times be an im­prove­ment or yield new in­sights—es­pe­cially when deal­ing with in­ter­est­ing re­cur­sions.

An ex­am­ple of an in­ter­est­ing com­pu­ta­tion re­quiring a vast finite com­puter is AIXI-tl, or An­drew Critch’s para­met­ric bounded analogue of Lob’s The­o­rem.


  • Methodology of unbounded analysis

    What we do and don’t un­der­stand how to do, us­ing un­limited com­put­ing power, is a crit­i­cal dis­tinc­tion and im­por­tant fron­tier.