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CSE 2001, Winter 2006
1
CSE 2001:
Introduction to Theory of Computation
Winter 2006
Suprakash Datta
[email protected]
Office: CSEB 3043
Phone: 4167362100 ext 77875
Course page: http://www.cs.yorku.ca/course/2001
Some of these slides are adapted from Wim van Dam’s slides
(
www.cs.berkeley.edu/~vandam/CS172/
)
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CSE 2001, Winter 2006
2
Next
Towards undecidability:
•
The Halting Problem
•
Countable and uncountable infinities
•
Diagonalization arguments
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CSE 2001, Winter 2006
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The Halting Problem
The existence of the universal TM U shows that
A
TM
= {
<
M,w
>
 M is a TM that accepts w }
is TMrecognizable, but can we also
decide
it?
The problem lies with the cases when M does
not halt on w.
In short: the halting problem
.
We will see that this is an insurmountable
problem: in general one cannot decide if a TM
will halt on w or not, hence A
TM
is undecidable.
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CSE 2001, Winter 2006
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Counting arguments
•
We need tools to reason about
undecidability.
•
The basic argument is that there are
more languages than Turing machines
and so there are languages than Turing
machines. Thus some languages
cannot be decidable
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CSE 2001, Winter 2006
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Baby steps
•
What is counting?
– Labeling
with integers
– Correspondence with integers
•
Let us review basic properties of functions
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CSE 2001, Winter 2006
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Mappings and Functions
The function F:A
→
B
maps one set A to
another set B:
A
B
F
F is onetoone
(injective) if every x
∈
A has a
unique image F(x): If F(x)=F(y) then x=y.
F is onto
(surjective) if every z
∈
B is ‘hit’ by F:
If z
∈
B then there is an x
∈
A such that F(x)=z.
F is a correspondence
(bijection) between A
and B if it is both onetoone and onto.
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3/28/2006
CSE 2001, Winter 2006
7
Cardinality
A set S has k elements if and only if there exists
a bijection between S and {1,2,…,k}.
S and {1,…,k} have the same cardinality
.
If there is a surjection possible from {1,…,n}
to S, then n
≥
S.
We can generalize this way of comparing the
sizes of sets to infinite ones.
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CSE 2001, Winter 2006
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How Many Languages?
For
Σ
={0,1}, there are 2
k
words of length k.
Hence, there are
languages L
⊆ Σ
k
.
Proof
: L has two options for every word
∈Σ
k
;
L can be represented by a string
.
That’s a lot, but finite.
There are infinitely many languages
⊆ Σ
*
.
But we can say more than that…
Georg Cantor defined a way of comparing infinities.
)
2
(
k
2
)
2
(
k
}
1
,
0
{
∈
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CSE 2001, Winter 2006
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Countably Infinite Sets
A set S is countable
if there exists a surjective
function F:
Ν →
S
“The set S has not more elements than
1
.”
A set S is infinite
if there exists a surjective
function F:S
→Ν
.
“The set
Ν
has no more elements than S.”
A set S is countably infinite
if there exists a
bijective function F:
Ν →
S.
“The sets
Ν
and S are of equal size.”
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CSE 2001, Winter 2006
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Counterintuitive facts
•
Cardinality of even integers
– Bijection i
↔
2i
– A proper subset of N has the same
cardinality as N !
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 Halting problem, Countable set, Georg Cantor

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