3 ms·

I emailed back and forth with Leslie a lot about this, when I first read his paper. There is an earlier analysis of the same phenomenon by different people (the

by EGreg 4y ago

I emailed back and forth with Leslie a lot about this, when I first read his paper. There is an earlier analysis of the same phenomenon by different people (the arbiter problem) which I discovered later. At first, I had trouble to believe it, and tried to find counterexamples, until I realized that the composition of continuous functions is continuous. I still wonder sometimes whether an infinite composition of continuous functions has to be continuous (limit of a sequence) if it represents some real world evolving process. Can it produce a discontinuity in the limit? Anyway, Buridan’s principle informs our consensus process in Intercoin Protocol[1] [2] — it is why the consensus groups are always of a bounded finite size, so we don’t approach the errors in Buridan’s priciple (I think this is also called unstable problems). 1. Beyond blockchains https://community.intercoin.org/t/intercoin-technology-consensus/ https://community.intercoin.org/t/intercoin-technology-conse... 2. Experimental improvements https://community.intercoin.org/t/experimental-improvements-for-intercoin-consensus/ https://community.intercoin.org/t/experimental-improvements-... Anyone who reads the above … I would be very interested in your feedback on our forum.

deleted 4y ago
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raegis 4y ago
> I still wonder sometimes whether an infinite composition of continuous functions has to be continuous (limit of a sequence) ... No, as you expected. Take f(x) = x^2 on [0,1]. Then f(f(f(...f(x))) converges to g(x) = { 0, if 0 <= x < 1; 1, if x=1 }
- EGreg 4y ago
  Right. So now the question is -- are you able to give a physical example of physical (continuous) processes which can create a discontinuous output from a continuous initial state, in a finite amount of time? I think this is a sort of Zeno's paradox...
  - juhanima 4y ago
    Tossing a coin?
    - EGreg 4y ago
      No. The coin could land on its side, for instance, and stay there. Consider a pencil that is placed almost vertically, and then later drops to a horizontal position. There is a period where it is "unsure" where to go, and the more perfectly balanced it starts out, the longer this period is (assuming no wind etc.) That's what is being discussed.

by EGreg 4y ago

deleted 4y ago

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raegis 4y ago

> I still wonder sometimes whether an infinite composition of continuous functions has to be continuous (limit of a sequence) ... No, as you expected. Take f(x) = x^2 on [0,1]. Then f(f(f(...f(x))) converges to g(x) = { 0, if 0 <= x < 1; 1, if x=1 }

EGreg 4y ago
Right. So now the question is -- are you able to give a physical example of physical (continuous) processes which can create a discontinuous output from a continuous initial state, in a finite amount of time? I think this is a sort of Zeno's paradox...
- juhanima 4y ago
  Tossing a coin?
  - EGreg 4y ago
    No. The coin could land on its side, for instance, and stay there. Consider a pencil that is placed almost vertically, and then later drops to a horizontal position. There is a period where it is "unsure" where to go, and the more perfectly balanced it starts out, the longer this period is (assuming no wind etc.) That's what is being discussed.