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Bob makes a claim that he made a new record and computed $\pi$ to 10 trillion digits (or your favourite number here). How would Alice verify that the newly computed constant is actually a correct approximation of $\pi$?

Given a finite string $x,$ of $n+1$ decimal digits: $(3\ 1\ 4\ 1\ 5\ 9\ 2\ldots),$ is there an efficient algorithm to decide whether $x$ is an approximation of $\pi$ up to the $0.\underbrace{00 \ldots 01}_{n-1}$ decimal places?


Edit: Clarification.

  1. Alice does not have access to Bob's method (so she can't prove that his method is correct).
  2. Alice only receives $x$ from Bob (in any number bases), and wants to verify that $x$ is indeed a correct approximation. No further communications between them.
  3. Alice could look at all digits of $x$ but should be able to verify in time $\ll$ than what it takes to compute $x$.
  4. Motivation: Assume Alexander J. Yee did not publish his code nor his method. He only publish $x$ in many number bases. He said it took him 3 months to compute $x.$ How could we verify his claims that $x$ is correct in a day or week or two without access to his code and formulas? Is there such a verification formula or algorithm?
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See Approximations of $\pi$. – arjafi Mar 14 '12 at 22:12
    
Not sure if this link helps or not. But you may refer this as well numberworld.org/misc_runs/pi-5t/details.html – Kirthi Raman Mar 14 '12 at 22:28
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Can you ask Bob to give you the digits in base 16? – Aryabhata Mar 15 '12 at 0:12
    
@Aryabhata yes. In any base. – user2468 Mar 15 '12 at 0:27
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My earlier link was just silly, but perhaps this one might be useful in the context you are looking for tinyurl.com/84396h3 – Kirthi Raman Mar 15 '12 at 0:38
up vote 13 down vote accepted

Since you allow any base, (16 in particular) and randomized algorithm, you can use the Bailey-Borwein-Plouffe formula which allows you to compute the $n^{th}$ digit of $\pi$, without having to compute the earlier $n-1$ digits! (Alas, such a algorithm seems to have been discovered only for base-16.)

All Alice needs to do is pick "some" random digits and compare.

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Fascinating! Classic method discovered in 1995. – Kirthi Raman Mar 15 '12 at 1:05
    
How efficient is it to convert between bases? – PyRulez Aug 19 '13 at 1:40
    
@PyRulez: Not sure what you mean by efficient, but given an n digit number in base b, there will be a linear time algorithm ($\mathcal{O}(n)$) to convert between bases. People are still trying to find such formulae in other bases... – Aryabhata Aug 19 '13 at 1:51

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