Lecture 4: Unsolvable Problems

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Lecture 4 Unsolvable Problems

• ???

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Content

• Enumerable and Decidable Sets
• Algorithm Solvability
• Problem Reduction
• Diagonalization Method

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Lecture 4 Unsolvable Problems

• Enumerable and Decidable Sets

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Set Enumerability and Generability
A ? ? is enumerable (generable) if
? computable function g N?? such that
If g is not totally defined, then D(g)0, 1, ,
k ?1.
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The Generation Process
Enumerable (Generable)
A
Generate
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The Enumeration Process
Enumerable (Generable)
A
Enumerate
?
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The Enumeration Process
Enumerable (Generable)
Is such a process always terminable?
A
Enumerate
?
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The Enumeration Process
Enumerable (Generable)
A
Enumerate
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The Labeling Process
Enumerable (Generable)
A
Labeling
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Exercises

• Write two algorithms to enumerate (generate) the
  set of all even numbers of integer in different
  sequences.
• What is difference between countable (studied
  in discrete math) and enumerable?

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The Countability of Real Number
Is R countable?
Is (0,1)?R countable?
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Prove (0,1)?R is uncountable
Assume that (0, 1) is countable.
Suppose that it is counted as
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Prove (0,1)?R is uncountable
Assume that (0, 1) is countable.
Suppose that it is counted as
Consider the following number
d00
d11
d22
d33
d44

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Prove (0,1)?R is uncountable
Assume that (0, 1) is countable.
Suppose that it is counted as
Consider the following number

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Prove (0,1)?R is uncountable
Assume that (0, 1) is countable.
Suppose that it is counted as
Consider the following number

0.
h0
h1
h2
h3
h4
vfk
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Prove (0,1)?R is uncountable
Assume that (0, 1) is countable.
Suppose that it is counted as
Consider the following number

0.
h0
h1
h2
h3
h4
vfk
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Semidecidability
A ? ? is semidecidable if there exists an
algorithm which, when applying to any ? ? A, can
decide that ? is in A, i.e., there exists an
membership algorithm for A.
True
False or Never Halt
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Theorem 1
A is enumerable iff A is semidecidable.
?
Pf)
A is enumerable
? g N?? such that g(N)A.
Hence, given any ??A, we generate strings
. . . . . . . .
g(1)?1
g(0)?0
Then, we will finally reach g(k) ?k ?.
This allows us to concludes that ??A.
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Theorem 1
We have
Want
A is enumerable iff A is semidecidable.
?
Pf)
Thinking first.
What do we know?
There exists an membership algorithm ? for A on
TM such that
A is semidecidable
How to prove?
Find g such that
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Theorem 1
We have
Want
A is enumerable iff A is semidecidable.
?
Pf)
Numbering the following black cells
Preliminaries
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Theorem 1
We have
Want
A is enumerable iff A is semidecidable.
?
Pf)
Numbering the following black cells
Preliminaries
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? 0, 1
0
1
5
? ?, 0, 1, 00, 01, 10, 11, 000,
2
3
4
7
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Theorem 1
We have
Want
A is enumerable iff A is semidecidable.
?
Pf)
Step 1. Set n k 1.
Step 2. List the first n strings
Step 3. Hand simulates n steps of ?
on TM for each
The algorithm to generate g(k)
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Theorem 1
We have
Want
A is enumerable iff A is semidecidable.
?
Pf)
Step 1. Set n k 1.
Step 2. List the first n strings
Step 3. Hand simulates n steps of ?
on TM for each
The algorithm to generate g(k)
Terminating Record
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Theorem 1
We have
Want
A is enumerable iff A is semidecidable.
?
Pf)
Step 1. Set n k 1.
Step 2. List the first n strings
Step 3. Hand simulates n steps of ?
on TM for each
The algorithm to generate g(k)
Step 4. Let m terminating computations.
Step 5. if m lt k 1 n n 1 goto Step
2 else return the kth string.
k
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Theorem 2
A is enumerable iff A is a domain of some
computable functions.
Pf)
Exercise
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Theorem 2
A is enumerable iff A is a domain of some
computable functions.
How about if A is not enumerable?
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Discussion
Program Computability
Enumeration
(Some reformation may be required)
Enumeration
Membership Determination
(Semidecidability)
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Complement Set
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Decidable Set
A ? ? is decidable if there exists an computable
function, say, D such that
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Theorem 3
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Theorem 3
enumerable.
semidecidable.
Pf)
?
Show that A is enumerable as follows

• Fact ? can be enumerated as ?0, ?1, ?2,
• Since A is decidable, ? a computable function D
  st.
• The computable function g to enumerate A can be

g
A is enumerable.
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Theorem 3
enumerable.
semidecidable.
Pf)
?

• Fact ? can be enumerated as ?0, ?1, ?2,
• Since A is decidable, ? a computable function D
  st.
• The computable function h to enumerate can be

h
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Theorem 3
enumerable.
semidecidable.
Pf)
?

• Suppose ?1 semidecides A, ?2 semidecides .
• For any ???, input it to ?1 and ?2
  simultaneously.
• If ?1 terminates, return true.
• If ?2 terminates, return false.

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Lecture 4 Unsolvable Problems

• Algorithm Solvability

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Enumeration, Decidability and Solvability
Enumeration
(Semidecidability)
Decidability
To deal with the existence of algorithm (program)
to solve problems.
Solvability
That is, given a problem, whether there exists a
program to solve (any instance of) the problem.
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Example
Problem (PE Program Equivalence)
?PE
q Are program ?1 and program ?2 equivalent?
Any Algorithm?
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Example
Solvable? Computable?
Problem (PE Program Equivalence)
?PE
q Are program ?1 and program ?2 equivalent?
Decidable?
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Example
Solvable? Computable?
Problem Reformulation
Problem (PE Program Equivalence)
?PE
q Are program ?1 and program ?2 equivalent?
Decidable?
Problem PE is solvable iff A is decidable
(recursive).
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Definitions

• Semisolvable
• Solvable
• Totally Unsolvable

if q is true, it answers yes. if q is false, it
may not give answer (never halt).
? algorithm
if q is true, it answers yes. if q is false, it
answers no.
? algorithm
not semisolvale.
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Problem and Complement Problem
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Problem and Complement Problem
Example
Turing Machine Version
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Problem and Complement Problem
Example
Register Machine Version
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Theorem 4
Pf)
?
?algorithm ?P such that
P is solvable
if q is true, it answers yes. if q is false, it
answers no.
To show that P is semisolvable, we need to find a
algorithm ? such that
?
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Theorem 4
Pf)
?
?algorithm ?P such that
P is solvable
if q is true, it answers yes. if q is false, it
answers no.
To show that P is semisolvable, we need to find a
algorithm ? such that
It can be
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Theorem 4
Pf)
?
?algorithm ?P such that
P is solvable
if q is true, it answers yes. if q is false, it
answers no.
It can be
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Theorem 4
Pf)
?
To run ? and ? alternatively,
if ? answers yes, then yes
if ? answers yes, then no.
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Corollary
P is semisolvable but not solvable
is totally unsolvable.
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The Halting Problem
More simple,

• Register Machine Version
• Turing Machine Version

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Theorem 5
?
The HP (halting problem) is not solvable.
????
We will prove a simpler version.
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Theorem 5
?
The HP (halting problem) is not solvable.
????
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Theorem 5
The HP (halting problem) is not solvable.
Pf)
Assume that HP is solvable.
?algorithm, given ? and lt?, 1gt, can tell
whether ? will halt on lt?, 1gt.
Define
We can have an algorithm, say, ?decide to decide
the elements of A.
That is,
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Theorem 5
The HP (halting problem) is not solvable.
Construct ?strange by making use of ?decide
Pf)
What are done by ?strange?
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Theorem 5
The HP (halting problem) is not solvable.
Construct ?strange by making use of ?decide
Pf)
START
Duplicate ? as ? _at_?
iff
?decide
true
false
?
HALT
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Theorem 5
The input to ?strange is a programs string, say,
??. ?strange will never halt on ?? if and only
if program ? will halt on feeding its code as
input parameter.
The HP (halting problem) is not solvable.
Construct ?strange by making use of ?decide
Pf)
START
Duplicate ? as ? _at_?
iff
?decide
true
false
?
HALT
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Theorem 5
The HP (halting problem) is not solvable.
Pf)
Assume that HP is solvable.
. . . . . . . . . . . . . . . .
iff
iff
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Theorem 5
The HP (halting problem) is not solvable.
Pf)
Assume that HP is solvable.
. . . . . . . . . . . . . . . .
iff
iff
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Theorem 6
The HP (halting problem) is not solvable.
The HP (halting problem) is semisolvable.
Pf)
Trivial (Run it)
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The Non-Halting Problem (NHP)

• Register Machine Version
• Turing Machine Version

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Theorem 7
The NHP is not semisolvable.
The NHP is totally unsolvable.
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Lecture 4 Unsolvable Problems

• Problem Reduction

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Is a Problem Solvable?

• Methods to identify the problems solvability
• Problem Reduction
• Diagonalization method
• Rices Theorem

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Problem Reduction
r
Let P and P be two problems. If there is an
algorithm (r) which will map any q?P into q?P st.
q is true iff q is true.
Then, we say that problem P reduces to P.
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Problem Reduction
Conceptually, P is larger.
r
Let P and P be two problems. If there is an
algorithm (r) which will map any q?P into q?P
st.
q is true iff q is true.
Then, we say that problem P reduces to P.
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Discussion
r
q is true iff q is true.
semisolvable
How about P?
If P is
solvable
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Discussion
r
q is true iff q is true.
semisolvable
semisolvable
If P is
P is
solvable
solvable
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Discussion
r
q is true iff q is true.
How about P?
semisolvable
If P is
solvable
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Discussion
r
q is true iff q is true.
semisolvable
If P is
P is ?????????
solvable
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Discussion
r
q is true iff q is true.
not solvable
How about P?
If P is
totally unsolvable
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Discussion
r
q is true iff q is true.
not solvable
If P is
P is ????????
totally unsolvable
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Discussion
r
q is true iff q is true.
not solvable
How about P?
If P is
totally unsolvable
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Discussion
r
q is true iff q is true.
not solvable
not solvable
If P is
P is
totally unsolvable
totally unsolvable
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Discussion
r
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Theorem 8
The halting-problems for TM(?), R, SR, SR2, are
semisolvable but not solvable.
r
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Theorem 9
The problem of TM equivalence (PE) is totally
unsolvable.
Any Algorithm?
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Theorem 9
The problem of TM equivalence (PE) is totally
unsolvable.
Pf)
Fact NHP is totally unsolvable.
Method of problem reduction
Find r st.
iff
How?
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Theorem 9

• doesnt
• halt on lt?, kgt

The problem of TM equivalence (PE) is totally
unsolvable.
Pf)
Fact NHP is totally unsolvable.
Input lt?, kgt?
?
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Theorem 9

• doesnt
• halt on lt?, kgt

The problem of TM equivalence (PE) is totally
unsolvable.
Pf)
Fact NHP is totally unsolvable.
Input lt?, kgt?
?
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Lecture 4 Unsolvable Problems

• Diagonalization Method

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Example
Show that HP is not solvable using
diagonalization method.

• Fact The set of all programs is enumerable.
  (why?)
• Let be the string
  representation of all programs.
• Suppose that HP is solvable. Then, we can have
  the following termination record
• Construct ?strange such that

Can you write such a program?
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Example
Show that HP is not solvable using
diagonalization method.

• Fact The set of all programs is enumerable.
  (why?)
• Let be the string
  representation of all programs.
• Suppose that HP is solvable. Then, we can have
  the following termination record
• Construct ?strange such that

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Example
Show that HP is not solvable using
diagonalization method.

• Fact The set of all programs is enumerable.
  (why?)
• Let be the string
  representation of all programs.
• Suppose that HP is solvable. Then, we can have
  the following termination record
• Construct ?strange such that

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Theorem 10
totally unsolvable ? not semisolvable
The total function problem is totally unsolvable.
Pf)

• Assume it is semisolvable.

• Then, we can enumerate the all total function as

• Let x1 be the input/output register.

Totally defined functions

• Then,

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Theorem 10
totally unsolvable ? not semisolvable
The total function problem is totally unsolvable.
Pf)

• Assume it is semisolvable.

. . . . . . . . . . . . . . . . . . . . .

• Define

• Clearly, g is total.

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Theorem 10
totally unsolvable ? not semisolvable
The total function problem is totally unsolvable.
Pf)

• Assume it is semisolvable.

. . . . . . . . . . . . . . . . . . . . .

• Define

• Clearly, g is total.

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Theorem 10
totally unsolvable ? not semisolvable
The total function problem is totally unsolvable.
Pf)

• Assume it is semisolvable.

. . . . . . . . . . . . . . . . . . . . .

• Define

• Clearly, g is total.

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Theorem 11
There is a total function which is not the
associate function of any count program.
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Exercises

• Special halting problem (SHP) and general halting
  problem (GHP) are defined as follows
• Prove
• If GHP is not solvable, then SHP is not solvable.
• If SHP is not solvable, then GHP is not solvable.

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Exercises

• Special equivalence problem (SE) is defined as
• Show that SE is not solvable.
• General equivalence problem (GE) is defined as
• Show that GE is not solvable.

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Exercises

• Define
• Show that MEMBER is not solvable.