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Decidable-1 - DecidableLanguages Recallthat L...

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Fall 2006 Costas Busch - RPI 1 Decidable Languages  
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Fall 2006 Costas Busch - RPI 2 Recall that: A language     is  Turing-Acceptable if there is a Turing machine  that accepts Also known as:  Turing-Recognizable                         or                           Recursively-enumerable                          languages L M L
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Fall 2006 Costas Busch - RPI 3 For any string      : L w w halts in an accept state  M L w halts in a non-accept state M or   loops forever
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Fall 2006 Costas Busch - RPI 4 Definition: A language      is  decidable if there is a Turing machine ( decider which accepts  and halts on every input string Also known as  recursive  languages L L M
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Fall 2006 Costas Busch - RPI 5 For any string      : L w w halts in an accept state  M L w halts in a non-accept state M Every decidable language is Turing-Acceptable
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Fall 2006 Costas Busch - RPI 6 Sometimes, it is convenient to have Turing  machines with single accept and reject states accept q reject q These are the only halting states That result to possible  halting configurations
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Fall 2006 Costas Busch - RPI 7 We can convert any Turing machine to  have single accept and reject states accept q Old machine New machine R x x , R x x , R x x , Multiple  accept states For each tape symbol x One accept state L x x ,
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Fall 2006 Costas Busch - RPI 8 i q i q R x x , For all tape symbols      not used for read in the  other transitions of x i q Old machine New machine reject q Multiple  reject states One reject state Do the following for each possible  halting state: For each
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Fall 2006 Costas Busch - RPI 9 Input string Accept Reject Decider   for L Decision On Halt: accept q reject q For a decidable language     :  L For each input string, the computation halts in the accept or reject state
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Fall 2006 Costas Busch - RPI 10 Input string Turing Machine for L accept q reject q For a Turing-Acceptable language     :  L It is possible that for some input string  the machine enters an infinite loop
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Fall 2006 Costas Busch - RPI 11 }   7,   5,   3,   2,   {1, = PRIMES Is number      prime?  x Corresponding language: Problem: We will show it is decidable
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Fall 2006 Costas Busch - RPI 12 On input number      : x Divide       with all possible numbers between      and   If any of them divides        Then   reject Else   accept   x 2 x Decider for                :   PRIMES x
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Fall 2006 Costas Busch - RPI 13 (Input string) is      prime?
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