Undecidable problems for Recursively enumerable languages

1
Undecidable problemsfor Recursively enumerable
languages

• continued

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Take a recursively enumerable language
Decision problems

• is empty?

• is finite?

• contains two different strings
• of the same length?

All these problems are undecidable
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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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Let be the TM with
Suppose we have a decider for the finite
language problem
YES
finite
finite language problem decider
NO
not finite
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We will build a decider for the halting problem
YES
halts on
Halting problem decider
doesnt halt on
NO
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We want to reduce the halting problem to the
finite language problem
Halting problem decider
NO
YES
finite language problem decider
YES
NO
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We need to convert one problem instance to the
other problem instance
Halting problem decider
NO
YES
convert input ?
finite language problem decider
YES
NO
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Construct machine
On arbitrary input string
Initially, simulates on input
If enters a halt state, accept (
inifinite language)
Otherwise, reject ( finite language)
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halts on
if and only if
is infinite
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halting problem decider
NO
YES
finite language problem decider
construct
YES
NO
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Take a recursively enumerable language
Decision problems

• is empty?

• is finite?

• contains two different strings
• of the same length?

All these problems are undecidable
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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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Let be the TM with
Suppose we have the decider for the two-strings
problem
YES
contains
Two-strings problem decider
Doesnt contain
NO
two equal length strings
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We will build a decider for the halting problem
YES
halts on
Halting problem decider
doesnt halt on
NO
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We want to reduce the halting problem to the
empty language problem
Halting problem decider
YES
YES
Two-strings problem decider
NO
NO
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We need to convert one problem instance to the
other problem instance
Halting problem decider
YES
YES
Two-strings problem decider
convert inputs ?
NO
NO
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Construct machine
On arbitrary input string
Initially, simulate on input
When enters a halt state, accept if
or
(two equal length strings
)
Otherwise, reject (
)
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halts on
if and only if
accepts two equal length strings
accepts and
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Halting problem decider
YES
YES
Two-strings problem decider
construct
NO
NO
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Rices Theorem

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Definition
Non-trivial properties 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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The Post Correspondence Problem

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Some undecidable problems for context-free
languages

• Is ?

are context-free grammars

• Is context-free grammar ambiguous?

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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
Indices may be repeated or omitted
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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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We will show
1. The MPC problem is undecidable
(by reducing the membership to MPC)
2. The PC problem is undecidable
(by reducing MPC to PC)
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Theorem The PC problem is undecidable
Proof We will reduce the MPC problem
to the PC problem
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Some undecidable problems for context-free
languages

• Is ?

are context-free grammars

• Is context-free grammar
• ambiguous?

We reduce the PC problem to these problems
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Theorem
Let be context-free grammars. It
is undecidable to determine if
Rdeduce the PC problem to this problem
Proof
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Suppose we have a decider for the empty-intersecti
on problem
Context-free grammars
Empty- interection problem decider
YES
NO
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For a context-free grammar ,
Theorem
it is undecidable to determine if G is ambiguous
Reduce the PC problem to this problem
Proof