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Fall 2003Costas Busch - RPI1 Decidability

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Fall 2003Costas Busch - RPI2 Recall: A language is decidable (recursive), if there is a Turing machine (decider) that accepts the language and halts on every input string Turing Machine Input string Accept Reject Decider for Decision On Halt: Decidable Languages

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Fall 2003Costas Busch - RPI3 Is number prime? Corresponding language: Problem:

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Fall 2003Costas Busch - RPI4 On input number : Divide with all possible numbers between and If any of them divides Then reject Else accept Decider for :

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Fall 2003Costas Busch - RPI5 We also say that the corresponding prime number problem is solvable: we can give an answer (positive or negative) for every input instance of the problem Thus, PRIMES is decidable

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Fall 2003Costas Busch - RPI6 (Input string) Accept Reject is number prime? Decider for PRIMES Input number YES NO the decider for the language solves the corresponding problem (Accept) (Reject)

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Fall 2003Costas Busch - RPI7 Problem: Does DFA accept the empty language ? Corresponding Language: Description of DFA as a string (For example, we can represent as a binary string, as we did for Turing machines)

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Fall 2003Costas Busch - RPI8 Determine whether there is a path from the initial state to any accepting state Decider for : On input : DFA Reject Decision: Accept

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Fall 2003Costas Busch - RPI9 Problem: Does DFA accept a finite language? Corresponding Language: Decidable language

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Fall 2003Costas Busch - RPI10 Decider for : On input : DFA Reject Decision: Accept Check if there is a walk with cycle from the initial state to an accepting state infinitefinite

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Fall 2003Costas Busch - RPI11 Problem: Does DFA accept string ? Corresponding Language: Decidable language

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Fall 2003Costas Busch - RPI12 Decider for : On input string : Run DFA on input string If accepts Then accept (and halt) Else reject (and halt)

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Fall 2003Costas Busch - RPI13 Problem: Do DFAs and accept the same language? Corresponding Language: Decidable language

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Fall 2003Costas Busch - RPI14 Let be the language of DFA Decider for : On input : Construct DFA such that: (combination of DFAs)

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Fall 2003Costas Busch - RPI15 and

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Fall 2003Costas Busch - RPI16 or

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Fall 2003Costas Busch - RPI17 Therefore, we only need to determine whether which is a solvable problem for DFAs:

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Fall 2003Costas Busch - RPI18 Undecidable Languages there is no Turing Machine that reaches a decision (halts) for all input strings of the language undecidable language = language is not decidable There is no decider for the language: (decision may be reached for some strings)

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Fall 2003Costas Busch - RPI19 For an undecidable language, the corresponding problem is unsolvable: there is no Turing Machine (Algorithm) that gives an answer (solves the problem) for all input instances of the problem (answer may be given for some input instances)

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Fall 2003Costas Busch - RPI20 We have shown before that there are undecidable languages: Decidable Turing recognizable is Turing recognizable but not decidable

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Fall 2003Costas Busch - RPI21 We will prove that two particular problems are unsolvable: Membership problem Halting problem

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Fall 2003Costas Busch - RPI22 Membership Problem Input:Turing Machine String Question: Does accept ? Corresponding language:

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Fall 2003Costas Busch - RPI23 Theorem: (The membership problem is unsolvable) Proof: We will assume that is decidable; We will then prove that every decidable language is also Turing recognizable is undecidable Basic idea: A contradiction!

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Fall 2003Costas Busch - RPI24 Suppose that is decidable YES accepts NO rejects Decider for Input string

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Fall 2003Costas Busch - RPI25 Let be a Turing recognizable language Let be the Turing Machine that accepts We will prove that is also decidable: we will build a decider for

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Fall 2003Costas Busch - RPI26 NO YES accept reject Decider for (and halt) Decider for Input string accepts ?

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Fall 2003Costas Busch - RPI27 Therefore,is decidable But there are Turing recognizable languages which are not decidable Contradiction!!!! Since is chosen arbitrarily, every Turing recognizable language is also decidable END OF PROOF

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Fall 2003Costas Busch - RPI28 We have shown: Decidable Undecidable

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Fall 2003Costas Busch - RPI29 We can actually show: Decidable Turing recognizable

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Fall 2003Costas Busch - RPI30 Turing machine that accepts : Run on input If accepts then accept is Turing recognizable

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Fall 2003Costas Busch - RPI31 Halting Problem Input:Turing Machine String Question: Does halt while processing input string ? Corresponding language:

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Fall 2003Costas Busch - RPI32 Theorem: (The halting problem is unsolvable) Proof: Suppose that is decidable; we will prove that every decidable language is also Turing recognizable is undecidable Basic idea: A contradiction!

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Fall 2003Costas Busch - RPI33 YES halts on input doesn’t halt on input NO Suppose that is decidable Decider for Input string

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Fall 2003Costas Busch - RPI34 Let be a Turing recognizable language Let be the Turing Machine that accepts We will prove that is also decidable: we will build a decider for

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Fall 2003Costas Busch - RPI35 halts and accepts halts on ?YES NO Run with input reject accept reject Decider for Input string Decider for and halt halts and rejects and halt

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Fall 2003Costas Busch - RPI36 Thereforeis decidable But there are Turing recognizable languages which are not decidable Contradiction!!!! Since is chosen arbitrarily, every Turing recognizable language is also decidable END OF PROOF

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Fall 2003Costas Busch - RPI37 Theorem: Proof: Assume for contradiction that the halting problem is decidable; (The halting problem is unsolvable) is undecidable we will obtain a contradiction using a diagonilization technique An alternative proof Basic idea:

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Fall 2003Costas Busch - RPI38 YEShalts on doesn’t halt on NO Suppose that is decidable Decider for Input string

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Fall 2003Costas Busch - RPI39 Input string: YES NO Looking inside Decider for halts on ?

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Fall 2003Costas Busch - RPI40 NO Loop forever YES Construct machine : If halts on input Then Loop Forever Else Halt halts on ?

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Fall 2003Costas Busch - RPI41 Construct machine : Copy on tape If halts on input Then loop forever Else halt

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Fall 2003Costas Busch - RPI42 Run with input itself Copy on tape If halts on input Then loops forever on input Else halts on input END OF PROOF CONTRADICTION!!!

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Fall 2003Costas Busch - RPI43 We have shown: Decidable Undecidable

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Fall 2003Costas Busch - RPI44 We can actually show: Decidable Turing recognizable

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Fall 2003Costas Busch - RPI45 Turing machine that accepts : Run on input If halts on then accept is Turing recognizable

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