Undecidable problems for Recursively enumerable languages - PowerPoint PPT Presentation

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Undecidable problems for Recursively enumerable languages

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Rice's Theorem: Any non-trivial property of. a recursively ... are context-free grammars. We reduce the PC problem to these problems. Costas Busch - RPI ... – PowerPoint PPT presentation

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Title: Undecidable problems for Recursively enumerable languages


1
Undecidable problemsfor Recursively enumerable
languages
  • continued

2
Take a recursively enumerable language
Decision problems
  • is empty?
  • is finite?
  • contains two different strings
  • of the same length?

All these problems are undecidable
3
Theorem
For a recursively enumerable language
it is undecidable to determine whether is
finite
Proof
We will reduce the halting problem to this problem
4
Let be the TM with
Suppose we have a decider for the finite
language problem
YES
finite
finite language problem decider
NO
not finite
5
We will build a decider for the halting problem
YES
halts on
Halting problem decider
doesnt halt on
NO
6
We want to reduce the halting problem to the
finite language problem
Halting problem decider
NO
YES
finite language problem decider
YES
NO
7
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
8
Construct machine
On arbitrary input string
Initially, simulates on input
If enters a halt state, accept (
inifinite language)
Otherwise, reject ( finite language)
9
halts on
if and only if
is infinite
10
halting problem decider
NO
YES
finite language problem decider
construct
YES
NO
11
Take a recursively enumerable language
Decision problems
  • is empty?
  • is finite?
  • contains two different strings
  • of the same length?

All these problems are undecidable
12
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
13
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
14
We will build a decider for the halting problem
YES
halts on
Halting problem decider
doesnt halt on
NO
15
We want to reduce the halting problem to the
empty language problem
Halting problem decider
YES
YES
Two-strings problem decider
NO
NO
16
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
17
Construct machine
On arbitrary input string
Initially, simulate on input
When enters a halt state, accept if
or
(two equal length strings
)
Otherwise, reject (
)
18
halts on
if and only if
accepts two equal length strings
accepts and
19
Halting problem decider
YES
YES
Two-strings problem decider
construct
NO
NO
20
Rices Theorem

21
Definition
Non-trivial properties of recursively enumerable
languages
any property possessed by some (not
all) recursively enumerable languages
22
Some non-trivial properties of recursively
enumerable languages
  • is empty
  • is finite
  • contains two different strings
  • of the same length

23
Rices Theorem
Any non-trivial property of a recursively
enumerable language is undecidable
24
The Post Correspondence Problem

25
Some undecidable problems for context-free
languages
  • Is ?

are context-free grammars
  • Is context-free grammar ambiguous?

26
We need a tool to prove that the
previous problems for context-free languages are
undecidable
The Post Correspondence Problem
27
The Post Correspondence Problem
Input
Two sequences of strings
28
There is a Post Correspondence Solution if there
is a sequence such that
PC-solution
Indeces may be repeated or ommited
29
Example
PC-solution
30
Example
There is no solution
Because total length of strings from is smaller
than total length of strings from
31
The Modified Post Correspondence Problem
Inputs
MPC-solution
32
Example
MPC-solution
33
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)
34
Theorem The MPC problem is undecidable
Proof We will reduce the membership
problem to the MPC problem
35
Membership problem
Input recursive language string
Question
Undecidable
36
Membership problem
Input unrestricted grammar string
Question
Undecidable
37
Suppose we have a decider for the MPC problem
MPC solution?
String Sequences
YES
MPC problem decider
NO
38
We will build a decider for the membership
problem
YES
Membership problem decider
NO
39
The reduction of the membership problem to the
MPC problem
Membership problem decider
yes
yes
MPC problem decider
no
no
40
We need to convert the input instance of one
problem to the other
Membership problem decider
yes
yes
convert inputs ?
MPC problem decider
no
no
41
Grammar
start variable
special symbol
For every symbol
For every variable
42
Grammar
string
special symbol
For every production
43
Example
Grammar
String
44
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45
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46
Grammar
47
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48
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49
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50
has an MPC-solution
if and only if
51
Membership problem decider
yes
yes
Construct
MPC problem decider
no
no
52
Since the membership problem is undecidable, The
MPC problem is uncedecidable
END OF PROOF
53
Theorem The PC problem is undecidable
Proof We will reduce the MPC problem
to the PC problem
54
Suppose we have a decider for the PC problem
PC solution?
String Sequences
YES
PC problem decider
NO
55
We will build a decider for the MPC problem
MPC solution?
String Sequences
YES
MPC problem decider
NO
56
The reduction of the MPC problem to the PC
problem
MPC problem decider
yes
yes
PC problem decider
no
no
57
We need to convert the input instance of one
problem to the other
MPC problem decider
yes
yes
convert inputs ?
PC problem decider
no
no
58
input to the MPC problem
59
We construct new sequences
60
We insert a special symbol between any two
symbols
61
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62
Special Cases
63
has a PC solution
if and only if
has an MPC solution
64
PC-solution
MPC-solution
65
MPC problem decider
yes
yes
Construct
PC problem decider
no
no
66
Since the MPC problem is undecidable, The PC
problem is undecidable
END OF PROOF
67
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
68
Theorem
Let be context-free grammars. It
is undecidable to determine if
Rdeduce the PC problem to this problem
Proof
69
Suppose we have a decider for the empty-intersecti
on problem
Context-free grammars
Empty- interection problem decider
YES
NO
70
We will build a decider for the PC problem
PC solution?
String Sequences
YES
PC problem decider
NO
71
The reduction of the PC problem to the
empty-intersection problem
PC problem decider
yes
yes
Empty- interection problem decider
no
no
72
We need to convert the input instance of one
problem to the other
PC problem decider
no
yes
Empty- interection problem decider
convert inputs ?
no
yes
73
input to the PC problem
74
Introduce new unique symbols
75
Context-free grammar
76
Context-free grammar
77
has a PC solution
if and only if
78
Because are unique
There is a PC solution
79
PC problem decider
no
yes
Empty- interection problem decider
Construct Context-Free Grammars
no
yes
80
Since PC is undecidable, the empty-intersection
problem is undecidable
END OF PROOF
81
For a context-free grammar ,
Theorem
it is undecidable to determine if G is ambiguous
Reduce the PC problem to this problem
Proof
82
PC problem decider
no
yes
Ambiguous- grammar problem decider
Construct Context-Free Grammar
no
yes
83
start variable of
start variable of
start variable of
84
has a PC solution
if and only if
if and only if
is ambiguous
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