On Fri, Feb 18, 2011 at 03:46:45PM -0800, Travis Garrett wrote: > Hi Stephen, > > Sorry for the slow reply, I have been working on various things and > also catching up on the many conversations (and naming conventions) on > this board. And thanks for your interest! -- I think I have > discovered a giant "low hanging fruit", which had previously gone > unnoticed since it is rather nonintuitive in nature (in addition to > being a subject that many smart people shy away from thinking > about...). > > Ok, let me address the Faddeev-Popov, "gauge-invariant information" > issue first. I'll start with the final conclusion reduced to its most > basic essence, and give more concrete examples later. First, note > that any one "structure" can have many different "descriptions". When > counting among different structures thus it is crucial to choose only > one description per structure, as including redundant descriptions > will spoil the calculation. In other words, one only counts over the > gauge-invariant information structures.
This is essentially what one does in the derivation of the Solomonoff-Levin distribution, aka "Universal Prior". That is, fix a universal prefix Turing machine, which halts on all input. Then all input programs generating the same output are considered equivalent. The universal prior for a given output is given by summing over the equivalence class of inputs giving that output, weighted exponentially by the length of the unique prefix. This result (which dates from the early 70s) gives rise to the various Occams razor theorems that have been published since. My own modest contribution was to note that any classifier function taking bit strings as input and mapping them to a discrete set (whether integers, or meanings, matters not) in a prefix way (the meaning of the string, once decided, does not change on reading more bits) will work. Turing machines are not strictly needed, and one expects observers to behave this way, so an Occams razor theorem will apply to each and every observer, even if the observers do not agree on the relative complexities of their worlds. However, this only suffices to eliminate what Bruno would call "3rd person white rabbits". There are still 1st person white rabbits that arise through the failure of induction problem. I will explain my solution to that issue further down. > > A very important lemma to this is that all of the random noise is > also removed when the redundant descriptions are cut, as the random > noise doesn't encode any invariant structure. Thus, for instance, I > agree with COMP, but I disagree that white rabbits are therefore a > problem... The vast majority of the output of a universal dovetailer > (which I call A in my paper) is random noise which doesn't actually > describe anything (despite "optical illusions" to the contrary...) and > can therefore be zapped, leaving the union of nontrivial, invariant > structures in U (which I then argue is dominated by the observer class > O due to combinatorics). It is important to remember that random noise events are not white rabbits. A nice physicsy example of the distinction is to consider a room full of air. The random motion of the molecules are not white rabbits, that is just normal thermal noise. All of the molecules being situated in one small corner of the room, however, so that an observer sitting in the room ends up suffocating is a white rabbit. One could say that white rabbits are extremely low entropy states that happen by chance, which is the key to udnerstanding why they're never observed. To be low entropy, the state must have significance to the observer, as well as being of low probability. Otherwise, any arbitrary configuration will have low entropy. When observing data, it is important that observers are relatively insensitive to error. It does not help to not recognise a lion in the African savannah, just because it is partically obscured by a tree. Computers used to be terrible at just this sort of problem - you needed the exact key to extract a record from a database - now various sorts of fuzzy techniques, particularly ones inspired by the neural structure in the brain - mean computers are much better at dealing wiuth noisy data. With this observation, it becomes clear that the myriad of nearby histories that differ only in a few bits are not recognised as different from the original observation. These are not white rabbits. It requires many bits to make a white rabbit, and this, as you eloquently point out, is doubly exponentially suppressed. Bruno will probably still comment that this does not dispose of all the 1st person white rabbits, but I fail to see what other ones could exist. Cheers -- ---------------------------------------------------------------------------- Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 hpco...@hpcoders.com.au Australia http://www.hpcoders.com.au ---------------------------------------------------------------------------- -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to firstname.lastname@example.org. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.