Decidability

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Decidability

• continued

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Undecidable Problems
Halting Problem
Does machine halt on input ?
State-entry Problem
Does machine enter state halt on input
?
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Blank-tape halting problem
Does machine halt when starting on blank
tape?
Membership problem
Is a string member of a recursively
enumerable language ?
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Uncomputable Functions
Values region
Domain
A function is uncomputable if it cannot be
computed for all the domain
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Function
maximum number of moves until any Turing machine
with states halts when started with the
blank tape
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Theorem
Function is uncomputable
Proof
If was computable then the
blank-tape halting problem would be decidable
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Algorithm for blank-tape halting problem
Input machine
1. Count states of
2. Compute
3. Simulate for steps
starting with empty tape
If halts then return YES
otherwise return NO
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Rices Theorem

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Non-trivial property of recursively enumerable
languages
any property possessed by some (not
all) recursively enumerable languages
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Some non-trivial properties of recursively
enumerable languages

• is empty

• is finite

• contains two different strings
• of the same length

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Rices Theorem
Any non-trivial property of a recursively
enumerable language is undecidable
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Theorem
For a recursively enumerable language
it is undecidable to determine whether is
empty
Proof
We will reduce the membership problem to this
problem
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Membership problem
Inputs machine and string
Question ?
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Construct machine
When enters a final state, compare input
with
Observations
if and only if is empty
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Algorithm for membership problem
Inputs machine and string
1. Construct
2. Determine if is empty
Yes then
No then
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Membership machine
yes
no
Check if
construct
no
is empty
yes
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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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Halting problem
Inputs machine and string
Question does halt on input ?
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Construct machine
When enters a halt state, accept any
input
Initially, simulates on input
(virtual input)
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Observations
If is finite then
halts on if and only if is infinite
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Algorithm for halting problem
Inputs machine and string
1. Construct
2. Determine if is finite
Yes then doesnt halt on
No then halts on
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Machine for halting problem
yes
no
Check if
construct
no
is finite
yes
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Theorem
For a recursively enumerable language
it is undecidable to determine whether
contains two different string of same length
Proof
We will reduce the halting problem to this problem
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Halting problem
Inputs machine and string
Question does halt on input ?
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Construct machine
When enters a halt state, accept symbols
or
Initially, simulates on input
(virtual input)
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Observation
halts on if and only if accepts
and
(strings of equal length)
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Algorithm for halting problem
Inputs machine and string
1. Construct
2. Determine if accepts
strings of equal length
Yes then halts on
No then doesnt halt on
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Machine for halting problem
yes
yes
Check if
construct
no
Has equal length strings
no
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The Post Correspondence Problem

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Some undecidable problems for context-free
languages

• Is context-free grammar ambiguous?

• Is ?

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We need a tool to prove that the
previous problems for context-free languages are
undecidable
The Post Correspondence Problem
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The Post Correspondence Problem
Input
Two sequences of strings
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There is a Post Correspondence Solution if there
is a sequence such that
PC-solution
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Example
PC-solution
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Example
There is no solution
Because total length of strings from is smaller
than total length of strings from
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The Modified Post Correspondence Problem
Inputs
MPC-solution
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Example
MPC-solution
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We will show that The Modified Post
Correspondence Problem is undecidable
In other words There is not MPC-solution
for any pair
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Proof Technique
We will reduce the membership problem to the MPC
problem
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Membership problem
Does Turing machine accept string
Equivalent Problem
Does unrestricted Grammar generate string
?