The full laws are uncomputable due to the inclusion of omega, yet you could compute a finite prefix of omega
You can’t compute a prefix of Chaitin’s omega of any arbitrary length. You can compute prefixes only up to some finite length, and this length is itself uncomputable.
From our perspective, the length which it is computable is going to be arbitrary, and until we hit it, at each digit we will confront the same epistemic problem: “is the prefix of omega that seems to be embedded in our particular physics a finite computable prefix of omega and so perfectly computable and so our physics is perfectly computable, or is this a genuine uncomputable constant buried in our physics?”
This has been discussed in the past as the ‘oracle hypercomputation’ problem: suppose you found a device which claimed to be an oracle for Turing machines halting, and you test it out and it seems to be accurate on the n Turing machines you are able to run to the point where they either halt or loop their state. How much, if at all, do you credit its claim to be an oracle doing hypercomputation? Since, after all, it could just be a random number generator, and in 1 out of the 2^n possible outputs its predictions will be completely correct ‘by chance’, or it could be using heuristics or something. What is one’s prior for hypercomputation/oracles being possible and how much does one update?
This was discussed on the advanced decision theory ML a while back, IIRC, and I don’t think they came to any solid conclusion either way.
You can’t compute a prefix of Chaitin’s omega of any arbitrary length. You can compute prefixes only up to some finite length, and this length is itself uncomputable.
From our perspective, the length which it is computable is going to be arbitrary, and until we hit it, at each digit we will confront the same epistemic problem: “is the prefix of omega that seems to be embedded in our particular physics a finite computable prefix of omega and so perfectly computable and so our physics is perfectly computable, or is this a genuine uncomputable constant buried in our physics?”
This has been discussed in the past as the ‘oracle hypercomputation’ problem: suppose you found a device which claimed to be an oracle for Turing machines halting, and you test it out and it seems to be accurate on the n Turing machines you are able to run to the point where they either halt or loop their state. How much, if at all, do you credit its claim to be an oracle doing hypercomputation? Since, after all, it could just be a random number generator, and in 1 out of the 2^n possible outputs its predictions will be completely correct ‘by chance’, or it could be using heuristics or something. What is one’s prior for hypercomputation/oracles being possible and how much does one update?
This was discussed on the advanced decision theory ML a while back, IIRC, and I don’t think they came to any solid conclusion either way.