It’s interesting that K-complexity has this property that there is always a single explanation which dominates the space of possible explanations. Intuitively it seems like there are often cases where there are many qualitatively different yet equally valid explanations for a given phenomenon. I wonder if there are natural complexity measures with this property as well—I think that this might be the case for Levin complexity[1], and this could provide an interesting way of defining ‘emergence’ quantitatively.
This isn’t what is proved. What is proved is that there is some constant K, dependent on choice programming language, such that Komolgorov complexity is dominated by at most K explanations. The proof provides an upper bound on K, as the exponential of the length of program needed to do various bits of fiddling around with encoding schemes. In other words, K may well be >10^100 for sensible choices of programming language.
It’s interesting that K-complexity has this property that there is always a single explanation which dominates the space of possible explanations. Intuitively it seems like there are often cases where there are many qualitatively different yet equally valid explanations for a given phenomenon. I wonder if there are natural complexity measures with this property as well—I think that this might be the case for Levin complexity[1], and this could provide an interesting way of defining ‘emergence’ quantitatively.
Or not?? I haven’t read this paper closely yet but sounds like it might prove a similar “one hypothesis dominates” theorem for Levin complexity
This isn’t what is proved. What is proved is that there is some constant K, dependent on choice programming language, such that Komolgorov complexity is dominated by at most K explanations. The proof provides an upper bound on K, as the exponential of the length of program needed to do various bits of fiddling around with encoding schemes. In other words, K may well be >10^100 for sensible choices of programming language.
Fair. I was summarizing “dominates within a constant factor” as just “dominates”, which admittedly is not totally accurate ;)