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Fall 2004COMP 3351 Undecidable problems for Recursively enumerable languages continued…

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Fall 2004COMP 3352 is empty? is finite? contains two different strings of the same length? Take a recursively enumerable language Decision problems: All these problems are undecidable

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Fall 2004COMP 3353 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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Fall 2004COMP 3354 finite language problem decider YES NO Suppose we have a decider for the finite language problem: Let be the TM with finite not finite

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Fall 2004COMP 3355 Halting problem decider YES NO halts on We will build a decider for the halting problem: doesn’t halt on

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Fall 2004COMP 3356 YES NO YES Halting problem decider finite language problem decider We want to reduce the halting problem to the finite language problem

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Fall 2004COMP 3357 YES NO YES Halting problem decider finite language problem decider We need to convert one problem instance to the other problem instance convert input ?

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Fall 2004COMP 3358 Construct machine : If enters a halt state, accept ( inifinite language) Initially, simulates on input Otherwise, reject ( finite language) On arbitrary input string

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Fall 2004COMP 3359 halts on is infinite if and only if

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Fall 2004COMP 33510 construct YES NO YES halting problem decider finite language problem decider

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Fall 2004COMP 33511 is empty? is finite? contains two different strings of the same length? Take a recursively enumerable language Decision problems: All these problems are undecidable

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Fall 2004COMP 33512 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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Fall 2004COMP 33513 Two-strings problem decider YES NO Suppose we have the decider for the two-strings problem: Let be the TM with contains Doesn’t contain two equal length strings

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Fall 2004COMP 33514 Halting problem decider YES NO halts on We will build a decider for the halting problem: doesn’t halt on

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Fall 2004COMP 33515 YES NO YES NO Halting problem decider Two-strings problem decider We want to reduce the halting problem to the empty language problem

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Fall 2004COMP 33516 YES NO YES NO Halting problem decider Two-strings problem decider We need to convert one problem instance to the other problem instance convert inputs ?

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Fall 2004COMP 33517 Construct machine : When enters a halt state, accept if or Initially, simulate on input (two equal length strings ) On arbitrary input string Otherwise, reject ( )

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Fall 2004COMP 33518 halts on if and only if accepts two equal length strings accepts and

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Fall 2004COMP 33519 construct YES NO YES NO Halting problem decider Two-strings problem decider

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Fall 2004COMP 33520 Rice’s Theorem

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Fall 2004COMP 33521 Non-trivial properties of recursively enumerable languages: any property possessed by some (not all) recursively enumerable languages Definition:

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Fall 2004COMP 33522 Some non-trivial properties of recursively enumerable languages: is empty is finite contains two different strings of the same length

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Fall 2004COMP 33523 Rice’s Theorem: Any non-trivial property of a recursively enumerable language is undecidable

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Fall 2004COMP 33524 The Post Correspondence Problem

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Fall 2004COMP 33525 Some undecidable problems for context-free languages: Is context-free grammar ambiguous? Is ? are context-free grammars

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Fall 2004COMP 33526 We need a tool to prove that the previous problems for context-free languages are undecidable: The Post Correspondence Problem

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Fall 2004COMP 33527 The Post Correspondence Problem Input: Two sequences of strings

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Fall 2004COMP 33528 There is a Post Correspondence Solution if there is a sequence such that: PC-solution: Indices may be repeated or omitted

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Fall 2004COMP 33529 Example: PC-solution:

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Fall 2004COMP 33530 Example: There is no solution Because total length of strings from is smaller than total length of strings from

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Fall 2004COMP 33531 1. The MPC problem is undecidable 2. The PC problem is undecidable (by reducing MPC to PC) (by reducing the membership to MPC) We will show:

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Fall 2004COMP 33532 Theorem: The PC problem is undecidable Proof: We will reduce the MPC problem to the PC problem

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Fall 2004COMP 33533 Some undecidable problems for context-free languages: Is context-free grammar ambiguous? Is ? are context-free grammars We reduce the PC problem to these problems

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Fall 2004COMP 33534 Theorem: Proof: Let be context-free grammars. It is undecidable to determine if Rdeduce the PC problem to this problem

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Fall 2004COMP 33535 Suppose we have a decider for the empty-intersection problem Empty- interection problem decider YES NO Context-free grammars

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Fall 2004COMP 33536 For a context-free grammar, Theorem: it is undecidable to determine if G is ambiguous Proof: Reduce the PC problem to this problem

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