What Would You Ask Vina?
What would you ask an all powerful alien?
Carl Sagan was one of the greatest popularizer of science in general, and astronomy and astrophysics in particular. He became world famous for his many books, his award winning TV series, and his novel “Contact,” which was the basis for the film starring Jodie Foster.
Today I want to talk about a lunch or dinner time conversation that I have had many times—often at conferences. The question is what would you ask an all powerful alien?
Sagan often talked about contact with aliens from advanced civilizations. He once wrote:
It’s a stimulating exercise to think of questions to which no human today knows the answers, but where a correct answer would immediately be recognized as such. It’s even more challenging to formulate such questions in fields other than mathematics. Perhaps we should hold a contest and collect the best responses in “Ten Questions to Ask an Alien.”
I thought it would be interesting to discuss asking question to an alien. I want to mention the work of Brendan Juba and Madhu Sudan, where they discuss a different issue—how would you communicate with an alien who does not speak the same language as you do. Perhaps I will discuss their interesting work another time.
Suppose that an all knowing being, perhaps from another world, arrives on Earth. Let’s agree to call her Vina—a name from Star Trek. She knows everything and is always truthful, a pretty neat combination in anyone. She tells you that you will get to ask her exactly one question. What would you ask?
For those of you who are Simpsons fans, you may recall the episode where Apu and Homer meet the all knowing head and master of Kwik-E-Mart. Apu and Homer are told they will get to ask three questions, not just one. The conversation goes like this:
Master: Approach, my sons. [they do] You may ask me three questions.
Apu: That’s great, because all I need is one –
Homer: Are you really the head of the Kwik-E-Mart?
Master: Yes. I hope this has been enlightening for you.
Apu: But I must –
Master: Thank you, come again.
Apu: But –
Master: Thank you, come again.
Our job, which should not be hard, is to make better use of our one question, than Homer did of his three questions.
So what should we ask Vina? Since we are mathematically oriented when we have discussed this question we often hear people say they would ask: Is P not equal to NP? Usually someone quickly points out that they are pretty sure that is true, so why not ask: Is P NP true and is the Riemann Hypothesis true? All agree this is a pretty clever idea, until someone points out there is a danger that she could answer no. In which case we have some information, but we do not know anything for sure. So is this a good idea or not?
Solutions With Randomization
Suppose we are allowed to ask Vina questions that are probabilistic. What about a question like:
Vina, please flip a fair coin. If it is heads answer yes/no for P=NP? If it is tails then answer yes/no for the Riemann Hypothesis.
My first thought was that this question was really the same as the previous question, but that is not quite right. Let be the statement “PNP is true” and let be the statement that the “Riemann Hypothesis is true.” The question we asked before was
The new question is: flip a fair coin, if it is heads answer ; if it is tails answer .
In the non-probabilistic question a yes means that both and are true; a no means that at least one is false. In the probabilistic question a yes means that or is true; a no means that at least one is false. These are clearly different.
What happens if we allow more complex probabilistic questions. Are there reasons, advantages, to ask questions that involve Vina flipping coins? The point is we can only get one bit, so how can we use that bit to get the most information possible? Do we want to be sure about things, or do we want to simply increase our understand a bit—bad pun.
I think the formal question is what is the best use of the one question given our prior beliefs in the probability that each statement is true. Thus, let be statements, and let us believe that is true with some probability . What is our best question to ask Vina to maximize our increase in information?
I am not sure the question is completely specified. Do we also need to know the joint distribution of our beliefs about all the statements?
What would you ask? Clearly that is not a question you can ask Vina—she only gives one bit. Suppose you thought PNP and Riemann and other questions were 90% likely to be true. You might ask her a question like:
Are (PNP & Riemann & Goldbach & Collatz & Jacobian & Twin Prime) all true?
Is this a good idea? Perhaps the questions are all 90%, but are they independent?
Also what happens if several questions are allowed? What is the best way to use the extra bits? Do we use an adaptive method or can we ask all in one question at once? Also how much does having Vina be able to use randomness help us?