patternModerate
Show that there are infinitely more problems than we will ever be able to compute
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Problem
I was looking at this reading of MIT on computational complexity and on minute 15:00 Erik Demaine embarks on a demonstration to show what is stated in the title of this question. However I cannot follow his reasoning, in practice what he says is this:
we can state a decision problem as string of $0$ and $1$ which in practice is the truth table of the function.
He goes on to say that a decision problem is an infinite string of bits while a program is a finite string of bits and up to here no problem. What I don't understand is the continuation of the proof from this point on: Decision problems are in $\mathbb{R}$ cause you can put a decimal point before the string representing the problem, thus obtaining the decimal part of a real
A program is "just" an integer in $\mathbb{N}$ cause it is a finite string of bits. The point that I fail to understand is how it is possible that a decision problem is comparable to a real number instead of an integer ... I mean, if we use the argument of "put a dot in front of the number" could not the same reasoning also be applied to the number of possible algorithms that can ever be produced?
we can state a decision problem as string of $0$ and $1$ which in practice is the truth table of the function.
He goes on to say that a decision problem is an infinite string of bits while a program is a finite string of bits and up to here no problem. What I don't understand is the continuation of the proof from this point on: Decision problems are in $\mathbb{R}$ cause you can put a decimal point before the string representing the problem, thus obtaining the decimal part of a real
for example if you have 0111011010101010101... it could become x.0111011010101010101...A program is "just" an integer in $\mathbb{N}$ cause it is a finite string of bits. The point that I fail to understand is how it is possible that a decision problem is comparable to a real number instead of an integer ... I mean, if we use the argument of "put a dot in front of the number" could not the same reasoning also be applied to the number of possible algorithms that can ever be produced?
Solution
Reformulating in a more mathematically precise way, what the lecturer is trying to say is this: Any algorithm can be (uniquely) encoded as a finite string of bits, and any finite string of bits (uniquely) encodes a program; hence, there is a bijection between $\mathbb{N}$ and the set of algorithms, so both are countable sets. Conversely, having fixed an ordering of strings, any decision problem $P$ can be (uniquely) encoded as an infinite string of bits, where the $i$-th bit represents whether the $i$-th string is in $P$ or not, and any infinite string of bits (uniquely) encodes a decision problem (in the same fashion); hence, $\mathbb{R}$ and the set of decision problems are both uncountable sets.
Context
StackExchange Computer Science Q#111360, answer score: 13
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