The Post Correspondence Problem

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The Post Correspondence Problem

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1. We will prove that the MPC problem is
undecidable
2. We will prove that the PC problem is
undecidable
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1. We will prove that the MPC problem is
undecidable
We will reduce the membership problem to the MPC
problem
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Membership problem
Input recursive language string
Question
Undecidable
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Membership problem
Input unrestricted grammar string
Question
Undecidable
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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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Grammar
start variable
special symbol
For every symbol
For every variable
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Grammar
string
special symbol
For every production
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Example
Grammar
String
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Theorem
has an MPC-solution if and only if
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Algorithm for membership problem
Input unrestricted grammar string
Construct the pair
If has an MPC-solution then
else
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Membership machine
solution
construct
MPC algorithm
No-solution
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2. We will prove that the PC problem is
undecidable
We will reduce the MPC problem to the PC problem
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input to the MPC problem
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We construct a new sequences
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We insert a special symbol between any two
symbols
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Special Cases
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Observation
There is a PC-solution for if and only if there
is a MPC-solution for
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PC-solution
MPC-solution
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MPC-algorithm
Input sequences
Construct sequences
Solve the PC problem for
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MPC algorithm
solution
construct
PC algorithm
No-solution