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Comment by throwawaymath | Hacker News Reader

> For example, when you define probabilistic computation, if you declare that a probabilistic gate is a stochastic matrix with real number entries, you will incorrectly find that there exists probabilistic programs that solve the halting problem.

Great example! Do you have an outline of this proof? I can see the error in defining a unitary matrix over R instead of C, but I'm not immediately seeing how you can exploit that error to bypass the halting problem.

I'm guessing the concrete error would be introduced by overlooking that the real-valued matrix won't preserve the correct probability amplitude? If so, what's the next step to (falsely) deciding that a given algorithm will complete?

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> Do you have an outline of this proof?

Let p be the real number where, iff the k'th program halts, the k'th bit of p (after the decimal point, in binary) is 1.

Let M be the operation "toggle a bit with probability p", which has matrix [[1-p, p], [p, 1-p]].

When given a program k to solve, keep applying M while counting up how many toggles you see. Do this until the chance that the k'th bit of the sample mean equals the k'th bit of p is greater than 2/3 or whatever other threshold you want, which will take a finite number of samples. Return the k'th bit of the sample mean. This is a probabilistic algorithm which solves the halting problem with arbitrarily high probability.

Maybe the problem is that you can use uncomputable numbers in the matrix? (e.g. https://en.wikipedia.org/wiki/Chaitin%27s_constant)

Unless I’m misunderstanding the parent commenter’s definition of a probabilistic gate - I’m assuming a unitary matrix - then that shouldn’t be an issue. Chaitin’s constant is in both R and C. More generally, all uncomputable real numbers are also in C, because C is complete over R.

In other words that shouldn't be the issue, because the correct "setting" for the stochastic matrix still allows for that possibility. It's not something you introduce by using real entries.

I don't think the issue here is requiring the matrix coefficients to be real (I don't think that complex values make sense there at all), but allowing arbitrary real numbers. In such case you can show algorithm, for which there exist matrix with real coefficients such that it solves halting problem - the trick is encoding infinite amount of information in the real constant.

Complex values (a unitary matrix) make sense if the stochastic matrix is representing a probability amplitude.

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