Reducibility

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Reducibility

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Problem is reduced to problem
If we can solve problem then we can solve
problem
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Problem is reduced to problem
If is decidable then is decidable
If is undecidable then is undecidable
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Example
the halting problem
is reduced to
the state-entry problem
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The state-entry problem
Inputs

• Turing Machine

• State

• String

Question
Does
enter state
on input ?
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Theorem
The state-entry problem is undecidable
Proof
Reduce the halting problem to
the state-entry problem
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Suppose we have a Decider for the state-entry
algorithm
YES
enters
state-entry problem decider
NO
doesnt enter
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We want to build a decider for the halting
problem
YES
halts on
Halting problem decider
NO
doesnt halt on
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We want to reduce the halting problem to the
state-entry problem
Halting problem decider
YES
YES
State-entry problem decider
NO
NO
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We need to convert one problem instance to the
other problem instance
Halting problem decider
YES
Convert Inputs ?
YES
State-entry problem decider
NO
NO
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Convert to

• Add new state

• From any halting state of add transitions
  to

Single halt state
halting states
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halts on input
if and only if
halts on state on input
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Halting problem decider
YES
YES
State-entry problem decider
Generate
NO
NO
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We reduced the halting problem to the state-entry
problem
Since the halting problem is undecidable, the
state-entry problem is undecidable
END OF PROOF
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Another example
the halting problem
is reduced to
the blank-tape halting problem
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The blank-tape halting problem
Input
Turing Machine
Question
Does
halt when started with
a blank tape?
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Theorem
The blank-tape halting problem is undecidable
Proof
Reduce the halting problem to the
blank-tape halting problem
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Suppose we have a decider for the blank-tape
halting problem
halts on blank tape
YES
blank-tape halting problem decider
NO
doesnt halt on blank tape
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We want to build a decider for the halting
problem
YES
halts on
halting problem decider
NO
doesnt halt on
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We want to reduce the halting problem to the
blank-tape halting problem
Halting problem decider
YES
YES
Blank-tape halting 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
Blank-tape halting problem decider
Convert Inputs ?
NO
NO
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Construct a new machine

• When started on blank tape, writes

• Then continues execution like

step 1
step2
if blank tape
execute
then write
with input
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halts on input string
if and only if
halts when started with blank tape
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Halting problem decider
YES
YES
blank-tape halting problem decider
Generate
NO
NO
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We reduced the halting problem to the blank-tape
halting problem
Since the halting problem is undecidable, the
blank-tape halting problem is undecidable
END OF PROOF
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Summary of Undecidable Problems
Halting Problem
Does machine halt on input ?
Membership problem
Does machine accept string ?
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Blank-tape halting problem
Does machine halt when starting on blank
tape?
State-entry Problem
Does machine enter state on input
?
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Uncomputable Functions

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Uncomputable Functions
Values region
Domain
A function is uncomputable if it cannot be
computed for all of its domain
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An uncomputable 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
Assume for contradiction that is
computable
Then the blank-tape halting problem is decidable

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Decider 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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Therefore, the blank-tape halting problem is
decidable
However, the blank-tape halting problem is
undecidable
Contradiction!!!
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Therefore, function in uncomputable
END OF PROOF
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Undecidable Problems for Recursively Enumerable
Languages
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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 any 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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Let be the TM with
Suppose we have a decider for the empty language
problem
YES
empty
empty language problem decider
NO
not empty
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We will build the decider for the membership
problem
YES
accepts
membership problem decider
NO
rejects
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We want to reduce the membership problem to the
empty language problem
Membership problem decider
NO
YES
empty language problem decider
YES
NO
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We need to convert one problem instance to the
other problem instance
Membership problem decider
NO
YES
Convert inputs ?
empty language problem decider
YES
NO
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Construct machine
On arbitrary input string
executes the same as with
When enters a final state, compare
with
Accept only if
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if and only if
is not empty
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Membership problem decider
NO
YES
empty language problem decider
construct
YES
NO
END OF PROOF