Church-Turing Thesis: Every Function F: {0,1}* -> {0,1}* Which Is "Effectively Computable" Is Computable by a Turing Machine
Church-Turing Thesis: Every function f: {0,1}* -> {0,1}* which is "effectively computable" is computable by a Turing Machine. Observation: Every Turing Machine (equivalently, every python program, or evey program in your favorite language) M can be encoded by a finite binary string #M. Moreover, our encoding guarantees #M != #M' whenever M and M' are different Turing Machines (different programs). Recall: Every finite binary string can be thought of as a natural number written in binary, and vice versa. Therefore, we can think of #M as a natural number. E.g. If #M = 1057, then we are talking about the 1057th Turing Machine, or 1057th python program. Intuitive takeaway: At most, there are as many different Turing Machines (Programs) as there are natural numbers. We will now show that some (in fact most) decision problems f: {0,1}* -> {0,1} are not computable by a TM (python program). If a decision problem is not comptuable a TM/program then we say it is "undecidable". Theorem: There are functions from {0,1}* to {0,1} which are not computable by a Turing Machine (We say, undecidable). Corollary: Assuming the CT thesis, there are functions which are not "effectively computable". Proof of Theorem: - Recall that we can think of instead of of {0,1}* - "How many" functions f: -> {0,1} are there? - We can think of each each such function f as an infinite binary string - For each x in (0,1), we can write its binary expansion as an infinite binary string to the right of a decimal point - The string to the right of the decimal point can be interpreted as a function f_x from natural numbers to {0,1} -- f_x(i) = ith bit of x to the right of the decimal point - In other words: for each x in (0,1) we get a different function f_x from natural numbers to {0,1}, so there are at least as many decision problems as there are numbers in (0,1).
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