Reductions

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Reductions

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Problem is reduced to problem
If we can solve problem then we can solve
problem
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Definition
Language is reduced
to language There is a computable function
(reduction) such that
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Recall
Computable function
There is a deterministic Turing machine
which for any string computes
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Theorem
If a Language is reduced to b
Language is decidable Then is
decidable
Proof
Basic idea
Build the decider for using the decider for
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Decider for
Reduction
YES
Input string
YES
accept
accept
compute
Decider for
(halt)
(halt)
NO
NO
reject
reject
(halt)
(halt)
END OF PROOF
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Example
is reduced to
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We only need to construct
Turing Machine for reduction
DFA
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Let be the language of DFA Let be
the language of DFA
Turing Machine for reduction
DFA
construct DFA by combining and
so that
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(No Transcript)
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Decider for
Reduction
Input string
YES
compute
YES
Decider
NO
NO
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Theorem (version 1)
If a Language is reduced to b
Language is undecidable Then is
undecidable
(this is the negation of the previous theorem)
Proof
Suppose is decidable
Using the decider for build the decider for
Contradiction!
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If is decidable then we can build
Decider for
Reduction
YES
Input string
YES
accept
accept
compute
Decider for
(halt)
(halt)
NO
NO
reject
reject
(halt)
(halt)
CONTRADICTION!
END OF PROOF
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Observation In order to prove that some
language is undecidable we only need to
reduce a known undecidable language to
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State-entry problem
Input

• Turing Machine

• State

• String

Question
Does
enter state
while processing input string ?
Corresponding language
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Theorem
is undecidable
(state-entry problem is unsolvable)
Proof
Reduce (halting problem) to
(state-entry problem)
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Halting Problem Decider
Decider for
state-entry problem decider
Reduction
YES
YES
Decider
Compute
NO
NO
Given the reduction, if is
decidable, then is decidable
A contradiction! since is undecidable
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We only need to build the reduction
Reduction
Compute
So that
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Construct from
special halt state
halting states
A transition for every unused tape symbol
of
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special halt state
halting states
halts on state
halts
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halts on input
Therefore
halts on state on input
Equivalently
END OF PROOF
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Blank-tape halting problem
Input
Turing Machine
Question
Does
halt when started with
a blank tape?
Corresponding language
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Theorem
is undecidable
(blank-tape halting problem is unsolvable)
Proof
Reduce (halting problem) to
(blank-tape problem)
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Halting Problem Decider
Decider for
blank-tape problem decider
Reduction
YES
YES
Decider
Compute
NO
NO
Given the reduction, If is
decidable, then is decidable
A contradiction! since is undecidable
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We only need to build the reduction
Reduction
Compute
So that
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Construct from
no
Tape is blank?
yes
Run
Write on tape
with input
If halts then halt
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no
Tape is blank?
yes
Run
Write on tape
with input
halts on input
halts when started on blank tape
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halts on input
halts when started on blank tape
Equivalently
END OF PROOF
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Theorem (version 2)
If a Language is reduced to b
Language is undecidable Then is
undecidable
Proof
Suppose is decidable
Then is decidable
Using the decider for build the decider for
Contradiction!
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Suppose is decidable
reject
Decider for
(halt)
accept
(halt)
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Suppose is decidable
Then is decidable
(we have proven this in previous class)
Decider for
NO
YES
reject
accept
Decider for
(halt)
(halt)
YES
NO
accept
reject
(halt)
(halt)
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If is decidable then we can build
Decider for
Reduction
YES
Input string
YES
accept
accept
compute
Decider for
(halt)
(halt)
NO
NO
reject
reject
(halt)
(halt)
CONTRADICTION!
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Alternatively
Decider for
Reduction
NO
Input string
YES
reject
accept
compute
Decider for
(halt)
(halt)
YES
NO
accept
reject
(halt)
(halt)
CONTRADICTION!
END OF PROOF
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Observation In order to prove that some
language is undecidable we only need to
reduce some known undecidable language to or to
(theorem version 1)
(theorem version 2)
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Undecidable Problems for Turing Recognizable
languages
Let be a Turing-acceptable language

• is empty?

• is regular?

• has size 2?

All these are undecidable problems
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Let be a Turing-acceptable language

• is empty?

• is regular?

• has size 2?

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Empty language problem
Input
Turing Machine
Question
Is
empty?
Corresponding language
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Theorem
is undecidable
(empty-language problem is unsolvable)
Proof
Reduce (membership problem) to
(empty language problem)
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membership problem decider
Decider for
empty problem decider
Reduction
YES
YES
Decider
Compute
NO
NO
Given the reduction, if is
decidable, then is decidable
A contradiction! since is undecidable
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We only need to build the reduction
Reduction
Compute
So that
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Construct from
Tape of
input string
write on tape
skip input string
run on input
yes
If
accepts ?
then accept
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Tape of
write on tape
skip input string
run on input
yes
If
accepts ?
then accept
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Tape of
write on tape
skip input string
run on input
yes
If
accepts ?
then accept
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During this phase this area is not touched
working area of
write on tape
skip input string
run on input
yes
If
accepts ?
then accept
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Simply check if entered an accept state
altered
write on tape
skip input string
run on input
yes
If
accepts ?
then accept
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Now check input string
write on tape
skip input string
run on input
yes
If
accepts ?
then accept
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The only possible accepted string
t r o y
write on tape
skip input string
run on input
yes
If
accepts ?
then accept
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accepts
does not accept
write on tape
skip input string
run on input
yes
If
accepts ?
then accept
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Therefore
accepts
Equivalently
END OF PROOF
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Let be a Turing-acceptable language

• is empty?

• is regular?

• has size 2?

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Regular language problem
Input
Turing Machine
Question
Is
a regular language?
Corresponding language
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Theorem
is undecidable
(regular language problem is unsolvable)
Proof
Reduce (membership problem) to
(regular language problem)
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membership problem decider
Decider for
regular problem decider
Reduction
YES
YES
Decider
Compute
NO
NO
Given the reduction, If is
decidable, then is decidable
A contradiction! since is undecidable
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We only need to build the reduction
Reduction
Compute
So that
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Construct from
Tape of
input string
write on tape
skip input string
run on input
yes
If has the form
accepts ?
then accept
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not regular
accepts
does not accept
regular
write on tape
skip input string
run on input
yes
If has the form
accepts ?
then accept
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Therefore
accepts
is not regular
Equivalently
END OF PROOF
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Let be a Turing-acceptable language

• is empty?

• is regular?

• has size 2?

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Size2 language problem
Input
Turing Machine
Does have size 2?
Question
(accepts exactly two strings?)
Corresponding language
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Theorem
is undecidable
(regular language problem is unsolvable)
Proof
Reduce (membership problem) to
(size 2 language problem)
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membership problem decider
Decider for
size2 problem decider
Reduction
YES
YES
Decider
Compute
NO
NO
Given the reduction, If is
decidable, then is decidable
A contradiction! since is undecidable
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We only need to build the reduction
Reduction
Compute
So that
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Construct from
Tape of
input string
write on tape
skip input string
run on input
yes
If
accepts ?
then accept
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2 strings
accepts
does not accept
0 strings
write on tape
skip input string
run on input
yes
If
accepts ?
then accept
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Therefore
accepts
has size 2
Equivalently
END OF PROOF
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RICEs Theorem
Undecidable problems

• is empty?

• is regular?

• has size 2?

This can be generalized to all non-trivial propert
ies of Turing-acceptable languages
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Non-trivial property
A property possessed by some
Turing-acceptable languages but not all
is empty?
Example
YES
NO
NO
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More examples of non-trivial properties
is regular?
YES
YES
NO
has size 2?
NO
NO
YES
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Trivial property
A property possessed by ALL
Turing-acceptable languages
has size at least 0?
Examples
True for all languages
is accepted by some Turing machine?
True for all Turing-acceptable languages
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We can describe a property as the set of
languages that possess the property
If language has property then
is empty?
Example
YES
NO
NO
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Example
Suppose alphabet is
has size 1?
NO
YES
NO
NO
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Non-trivial property problem
Input
Turing Machine
Does have the non-trivial property ?
Question
Corresponding language
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Rices Theorem
is undecidable
(the non-trivial property problem is unsolvable)
Proof
Reduce (membership problem) to

or
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We examine two cases
Case 1
Examples
is empty?
is regular?
Case 2
Example
has size 2?
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Case 1
Since is non-trivial, there is a
Turing-acceptable language such that
Let be the Turing machine that accepts
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Reduce (membership problem) to

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membership problem decider
Decider for
Non-trivial property problem decider
Reduction
YES
YES
Decider
Compute
NO
NO
Given the reduction, if is
decidable, then is decidable
A contradiction! since is undecidable
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We only need to build the reduction
Reduction
Compute
So that
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Construct from
Tape of
input string
write on tape
skip input string
run on input
yes
If
accepts ?
then accept
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For this phase we can run machine that accepts
, with input string
write on tape
skip input string
run on input
yes
If
accepts ?
then accept
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accepts
does not accept
write on tape
skip input string
run on input
yes
If
accepts ?
then accept
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Therefore
accepts
Equivalently
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Case 2
Since is non-trivial, there is a
Turing-acceptable language such that
Let be the Turing machine that accepts
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Reduce (membership problem) to

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membership problem decider
Decider for
Non-trivial property problem decider
Reduction
YES
YES
Decider
Compute
NO
NO
Given the reduction, if is
decidable, then is decidable
A contradiction! since is undecidable
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We only need to build the reduction
Reduction
Compute
So that
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Construct from
Tape of
input string
write on tape
skip input string
run on input
yes
If
accepts ?
then accept
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accepts
does not accept
write on tape
skip input string
run on input
yes
If
accepts ?
then accept
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Therefore
accepts
Equivalently
END OF PROOF