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Theory of Computation | Complexity

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O
Chapter
Complexity Theory
Central
·
classifying
·
1
:
question
which
aspect
. Settle
2
for
less than
3 Some
are
hard
.
4
ie)
computational
their
the
of
perfect
a
only
alternative
find
to
in the
computationaly
problems
some
programs according
understand
.
makes
What
problem is
solution
ro o f
closely
Complexity
*
Computability
*
objective
=
objective is
:
Mathematical Notation
*
Sets
in
-
set
sets
a
classify
to
of
I
ones
solable/not solvable
as
deals with definition & properties
-
finite automaton
*
elements
called
i
members
or
Z
As
↳
takes
-
-
-
b
=
and
=
The
set of
The
outputs
A
↳
property
K-any
↳
repetition
possible inputs
of
.
2
.
3
feature
.
R
is
R
is
a
!
function
set of
K-
f
: D
function
the
from
with
captures
A
x
x
every
is
for
range
ies k
=
undirected
of
2
and
y
binary
:
objects being equal
when
If and
Ry
x
and 2 ,
XRy
and
:
E
only if yRx ;
yRz
191
2
3 43
of modes
.
of edges in
=
particular
-cali
edge
graph
,
x
,
for
...
20
,
,
Ca . , 92
=
element
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903 913
Al
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(21
,
;
some times
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j)
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j
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EN
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(2
5)
is set of
i
② i
(2, 4) (3
,
(5
,
i)
P:
,
Boston
,
4)3)
repetition
K- tuples
=
some
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133
sets
a:
function
K-ary
.
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=
=
A ....
Al
arguments
,
of the
arity
f
to
functio
[True Falses
De
(D S)
,
,
S
where
[a ED/P(a)
=
,
,
-
-
i
i =7
that
=
0
,
.
j is multiple
i-j
if
multiple of
98
1
.
<
1
-strego
*
New York
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represented [5 5)
=
57)
(Stone ,
=
True
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Graph
3
Scissor)
It :
multiple of
G
,
of 7
is
a
multiple of
1
=
enter
7 <reflexive)
if
j
i
is
-
multiple of 7
.
<Symmetric)
&In
109
edges
7
Paper-Scissor
,
says
b/C &
XRI
e
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,
sequence
(V E)
,
Labeled
Ithaca
Graph
G is
378
rancisco
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transitive)
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of
Graph
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subgraph
a re
of
subset
0
-0
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/W I
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graph
H
if t h e
of the nodes H
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Modes of
of edges
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30
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subsets of A
all
of
,
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but
set
0
[C1 2) (1 5) (1 4)
(set
of A
.
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.
of
set
21 ,
,
"mot ni
,
,
,
,
,
,
,
,
,
,
,
-
Set
Rock
predicate
unimportant
the Set
Al
&
(7 21
sequence
,
Kargument
**
*
If V is
design
hard warew
,
binary function
;
relation
for
,
:
in
fis K-tuple
to
with
2
set
function
ex)
XRX
x,
,
is ; pair (i , j)
in
compilers
,
Falses
,
equivalence relation
:
every y y
odesventies
mode
matter
order
when
Motion
tuples
=
(7
ie
matter
allowed
power
input
relation
=
not
of
function
relation
2-any
the
symmetric if for every X
G
processing
languages
in some order
order does
,
the
=
-
30 13
domain
True
range
A.:
on
=
=
domain
R
+
:
set
a
tuples Ax. . .
relation
if for
if
text
used in
power set of A is the
A
mapping)
called
*
come
Binare relation
R is transitive
graph
↑order
.
reflexive
*
In
is
(also
.
.
to
function
K-ary
,
Graphs
*
a
with domain
relation
in some
1
b
to
a
equivalence relation
-
output
produces
maps
in
& Tuples
finite sequence
less than "<"
ill
*
input-output relationship
an
Predicate/property
-
-
sets up
input
an
f(a)
-
:
used
grammar
objects
list of
Sequence
*
& Relations
functions
-
context-free
↳
↳
Functions
Theory
↓
Sequences
unit
a
#s N
intergens
Automata
Terminology
are
natural
of
Set
-
&
classify easy/hard
group of object represented
:
objects
-
to
answer
computation
randomized
is
No
situation
complexity
related to
?
difficulty
the
of
Computability Theory
*
easy
others
problem
to the
case
worse
and
difficulty
the
a
hard
0
/
Subgraph
.
of G
path
:
sequence
simple path path
:
#
A
graph
Cycle
:
Tree
:
connected
that doesn't
connected if
is
and
start
simple cycle
nodes
of
doesn't
:
if it is
repeat
connected
repeat any
every
node
end
by edges
two
is
same
any
mode
and has
nodes have
a
path
between them
↳ strongly
relation
:
all
cucle
except
no
-②
①
1st and last
simple cycles
1
⑥
↓
- >
.
degree
③ indegree
out
-
the ar rows
:
if a
pointin
directed
itit *)be
of
*
Scissors
dao
I
O
Paper
the same
graph
directed
direction
connects
graph
->
every
two
handy way
modes
of
nee
-
directed graph
e
&⑥
connected
graph
-
0
Path
Paths
Directed path
·
*
-46b
0
-
nodes
depicting binary
relations
:
:
# of
#
of
arrows
pointing
from particular mode
arow
pointing
to
particular
mode
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