How are you converting the inductor into a betting decision?
What you should get from the inductor—after sufficient input—is very close to equal probability for head and tails (probability arising from the weighted sum of surviving programs).
edit: namely, you’ll have (among the craziness of that program soup) a family of programs that relay subsequent bits from the program tape to the output. So you have the family of programs that begin with “print 11010111001...” , and within those, there’s equal number of those that print zero after 11010111001 and those that print 1 .
There’s also various craziness which offsets the probabilities—e.g. a family of programs that do that but start outputting zeroes (or ones) after some very large number of bits. Those aren’t eliminated until they fail to predict the future.
edit2: though, yeah, if you keep decreasing your epsilon here, it’ll accept the bet now and then. If you’re utilizing the prior history of bets it made (e.g. to adjust that small punishment down until it’s willing to bet) , you effectively have an environment with hypercomputation.
How are you converting the inductor into a betting decision?
What you should get from the inductor—after sufficient input—is very close to equal probability for head and tails (probability arising from the weighted sum of surviving programs).
edit: namely, you’ll have (among the craziness of that program soup) a family of programs that relay subsequent bits from the program tape to the output. So you have the family of programs that begin with “print 11010111001...” , and within those, there’s equal number of those that print zero after 11010111001 and those that print 1 .
There’s also various craziness which offsets the probabilities—e.g. a family of programs that do that but start outputting zeroes (or ones) after some very large number of bits. Those aren’t eliminated until they fail to predict the future.
edit2: though, yeah, if you keep decreasing your epsilon here, it’ll accept the bet now and then. If you’re utilizing the prior history of bets it made (e.g. to adjust that small punishment down until it’s willing to bet) , you effectively have an environment with hypercomputation.
The obvious way—if EV(bet)>0, take bet.
We could also talk about “the number of times the probability assignment is off by epsilon.”