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1 Decidability continued…

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2 Theorem: For a recursively enumerable language it is undecidable to determine whether is finite Proof: We will reduce the halting problem to this problem

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3 Algorithm for finite language problem YES NO Assume we have the finite language algorithm: Let be the machine that accepts finite not finite

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4 Algorithm for Halting problem YES NO halts on We will design the halting problem algorithm: doesn’t halt on

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5 First construct machine : When enters a halt state, accept any input (inifinite language) Initially, simulates on input Otherwise accept nothing (finite language)

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6 halts on is not finite if and only if

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7 Algorithm for halting problem: Inputs: machine and string 1. Construct 2. Determine if is finite YES: then doesn’t halt on NO: then halts on

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8 construct Check if is finite YES NO YES Machine for halting problem

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9 Theorem: For a recursively enumerable language it is undecidable to determine whether contains two different strings of same length Proof: We will reduce the halting problem to this problem

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10 Algorithm for two-strings problem YES NO Assume we have the two-strings algorithm: Let be the machine that accepts contains Doesn’t contain two equal length strings

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11 Algorithm for Halting problem YES NO halts on We will design the halting problem algorithm: doesn’t halt on

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12 First construct machine : When enters a halt state, accept symbols or Initially, simulates on input (two equal length strings)

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13 halts on if and only if accepts and (two equal length strings)

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14 Algorithm for halting problem: Inputs: machine and string 1. Construct 2. Determine if accepts two strings of equal length YES: then halts on NO: then doesn’t halt on

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15 construct Check if has two equal length strings YES NO YES NO Machine for halting problem

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16 The Post Correspondence Problem

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17 Some undecidable problems for context-free languages: Is context-free grammar ambiguous? Is ?

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18 We need a tool to prove that the previous problems for context-free languages are undecidable: The Post Correspondence Problem

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19 The Post Correspondence Problem Input: Two sequences of strings

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20 There is a Post Correspondence Solution if there is a sequence such that: PC-solution

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21 Example: PC-solution:

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22 Example: There is no solution Because total length of strings from is smaller than total length of strings from

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23 The Modified Post Correspondence Problem Inputs: MPC-solution:

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24 Example: MPC-solution:

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25 1. We will prove that the MPC problem is undecidable 2. We will prove that the PC problem is undecidable

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26 1. We will prove that the MPC problem is undecidable We will reduce the membership problem to the MPC problem

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27 Membership problem Input: recursive language string Question: Undecidable

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28 Membership problem Input: unrestricted grammar string Question: Undecidable

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29 The reduction of the membership problem to the MPC problem: For unrestricted grammar and string we construct a pair such that has an MPC-solution if and only if

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30 : special symbol For every symbol Grammar : start variable For every variable

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31 Grammar For every production : special symbol string

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32 Example: Grammar : String

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35 Grammar :

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39 Theorem: has an MPC-solution if and only if

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40 Algorithm for membership problem: Input: unrestricted grammar string Construct the pair If has an MPC-solution then else

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41 construct MPC algorithm solution No-solution Membership machine

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42 2. We will prove that the PC problem is undecidable We will reduce the MPC problem to the PC problem

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43 : input to the MPC problem

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44 We construct new sequences

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45 We insert a special symbol between any two symbols

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47 Special Cases

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48 Observation: There is a PC-solution for if and only if there is a MPC-solution for

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49 PC-solution MPC-solution

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50 MPC-algorithm Input: sequences Construct sequences Solve the PC problem for

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51 construct PC algorithm solution No-solution MPC algorithm

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