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lect15-tm1 - Undecidability TuringMachines...

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1 Undecidability Everything is an Integer Countable and Uncountable Sets Turing Machines Recursive and Recursively  Enumerable Languages
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2 Integers, Strings, and Other Things Data types have become very important  as a programming tool. But at another level, there is only one  type, which you may think of as integers  or strings. Key point : Strings that are programs are  just another way to think about the  same one data type.
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3 Example : Text Strings of ASCII or Unicode characters  can be thought of as binary strings, with  8 or 16 bits/character. Binary strings can be thought of as  integers. It makes sense to talk about “the i-th  string.”
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4 Binary Strings to Integers There’s a small glitch: If you think simply of binary integers, then  strings like 101, 0101, 00101,… all appear  to be “the fifth string.” Fix by prepending a “1” to the string  before converting to an integer. Thus, 101, 0101, and 00101 are the 13 th 21 st , and 37 th  strings, respectively.
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5 Example : Images Represent an image in (say) GIF. The GIF file is an ASCII string. Convert string to binary. Convert binary string to integer. Now we have a notion of “the i-th  image.”
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6 Example : Proofs A formal proof is a sequence of logical  expressions, each of which follows from  the ones before it. Encode mathematical expressions of  any kind in Unicode. Convert expression to a binary string  and then an integer.
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7 Proofs – (2) But a proof is a sequence of  expressions, so we need a way to  separate them. Also, we need to indicate which  expressions are given.
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8 Proofs – (3) Quick-and-dirty way to introduce new  symbols into binary strings: 1. Given a binary string, precede each bit by 0. Example : 101 becomes 010001. 1. Use strings of two or more 1’s as the  special symbols. Example : 111 = “the following expression is  given”; 11 = “end of expression.”
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9 Example : Encoding Proofs 1110100011111100000101110101 A given expression follows An ex- pression End of expression Notice this 1 could not be part of the “end” A given expression follows Expression End
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10 Example : Programs Programs are just another kind of data. Represent a program in ASCII. Convert to a binary string, then to an  integer. Thus, it makes sense to talk about “the  i-th program.” Hmm…There aren’t all that many  programs.
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11 Finite Sets Intuitively, a  finite set   is a set for which  there is a particular integer that is the  count of the number of members.
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