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1 Uncountable Sets continued...
2 Theorem: Let be an infinite countable set. The powerset of is uncountable
3 Application: Languages Example Alphabet : Set of Strings: infinite and countable Powerset: all languages uncountable
4 Languages: Uncountable Turing machines: Countable There are infinitely many more languages than Turing Machines
5 There are some languages not accepted by Turing Machines These languages cannot be described by algorithms
6 Recursively Enumerable Languages and Recursive Languages
7 Definition: A language is recursively enumerable if some Turing machine accepts it
8 For string : Let be a recursively enumerable language and be the Turing Machine that accepts it ifthen halts in a final state ifthen halts in some state or loops forever
9 Definition: A language is recursive if some Turing machine accepts it and halts on any input string In other words: A language is recursive if there is a membership algorithm for it
10 For string : Let be a recursive language and be the Turing Machine that accepts it ifthen halts in a final state ifthen halts in a non-final state
11 We will prove: 1. There is a specific language which is not recursively enumerable 2. There is a specific language which is recursively enumerable but not recursive
12 Recursive Recursively Enumerable Non Recursively Enumerable
13 First we prove: If a language is recursive then there is an enumeration procedure for it A language is recursively enumerable if and only if there is an enumeration procedure for it
14 Theorem: if a language is recursive then there is an enumeration procedure for it
15 Proof: Enumeration Machine Accepts Enumerates all strings of input alphabet
16 Enumeration procedure Repeat: generates a string checks if YES: print to output NO: ignore End of proof
17 Theorem: if language is recursively enumerable then there is an enumeration procedure for it
18 Proof: Enumeration Machine Accepts Enumerates all strings of input alphabet
19 Enumeration procedure Repeat:generates a string checks if YES: print to output NO: ignore NAIVE APPROACH Problem: If machine may loop forever
20 executes first step on BETTER APPROACH Generates second string executes first step on second step on Generates first string
21 Generates third string executes first step on second step on third step on And so on............
22 1 Move 2 3
23 If for string machine halts in a final state then it prints on the output End of proof
24 Theorem: If for language there is an enumeration procedure then is recursively enumerable
25 Proof: Input Tape Enumerator for Compare Machine that accepts
26 Turing machine that accepts Repeat: Using the enumerator, generate the next string of For input string Compare generated string with If same, accept and exit loop End of proof
27 This is not a membership algorithm. Why? Question: Answer: The enumeration procedure may not produce strings in proper order
28 We have proven: A language is recursively enumerable if and only if there is an enumeration procedure for it
29 A Language which is not Recursively Enumerable
30 We want to find a language that is not Recursively Enumerable This language is not accepted by any Turing Machine
31 Consider alphabet Strings:
32 Consider Turing Machines that accept languages over alphabet They are countable:
33 Example language accepted by Alternative representation
35 Consider the language consists from the 1’s in the diagonal
37 Consider the language consists from the 0’s in the diagonal
39 Theorem: Language is not recursively enumerable
40 Proof: is recursively enumerable Assume for contradiction that There must exist some machine that accepts
47 Similarly: for any Because either: or
48 Therefore the machine cannot exist CONTRADICTION!!! Therefore the language is not recursively enumerable End of proof
49 Observation: There is no algorithm that describes (otherwise it would be accepted by a Turing Machine)
50 A Language which is Recursively Enumerable and not Recursive
51 We want to find a language which There is a Turing Machine that accepts the language The machine doesn’t necessarily halt on any input Is recursively enumerable But not recursive
52 We will prove that the language Is recursively enumerable but not recursive
53 The language Theorem: is recursively enumerable
54 Proof: We will give a Turing Machine that accepts
55 Turing Machine that accepts For any input string Write Find Turing machine (using the enumeration procedure for Turing Machines) Simulate on input If accepts, then accept End of proof
56 Observation: Recursively enumerable Not recursively enumerable (Thus, not recursive)
57 Theorem: The language is not recursive
58 Proof: Assume for contradiction that is recursive Then is recursive: Take the Turing Machine that accepts halts on any input If accepts then reject If rejects then accept
59 Therefore: recursive But we know: not recursively enumerable thus, not recursive CONTRADICTION!!!!
60 Therefore, is not recursive End of proof
61 Recursive Recursively Enumerable Non Recursively Enumerable
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