Summary
Samenvatting Automata and Complexity (X_401049)
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Summary of the course Automata & Complexity given at VU University Amsterdam.
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Automata & ComplexityIntroduction
computers are based on a universal computing mechanism
( ie ,
roughly speaking ,
no other computational model can
compute more ) the basic mathematical principles
,
the operations it can to
execute are
strong enough
compute anything computable
However ,
there are certain limits to what is
computable !
in
Turing machines were invented 1936 by Alan
Turing :
-
abstract computation model 011 R
-
automata with read / write tape
9-0 9-
computation
'
-
universal mechanism
limitation of
we can
study the
012 R
modern computers by studying Turing machines .
different kinds of automata for different applications :
finite automata
give rise to
regular languages :
application :
pattern recognising
•
equivalent to :
regular expressions
-
pushdown automata free languages
give rise to context :
-
-
application :
parsing ( e.g .
.
programming languages )
a
equivalent to : context -
free
grammars
turing machines
yield recursively enumerable languages :
application general computation
-
:
↳ form of
strongest automata
depending on the application in mind , you should choose
the appropriate form of automata !
, what can a computer do ?
some problems undecidable (e. termination )
are g. , program
-
-
some problems ( NP -
hard problems ) are (probably) not
efficiently
solvable by a computer leg .
, travelling salesman problems
aspects of
languages :
the form of words in the ( course)
syntax
the
language ← this
-
:
semantics : the meaning of the words in the
language
-
Words & Languages
Words
word =
finite sequence of symbols from an alphabet E
↳
symbols : A, b, C , . . .
↳ words :
v.v , W , ✗ 2
,
y .
↳ E from the alphabet E
a E means a is a
symbol
I !=
"
write ✗ for ( alphabet
"
the word letter in the
we
empty
Everything stored on a computer is a word (a
sequence of bits ) .
so, in particular ,
a computer program is a word .
From an
abstract view ,
a
program
:
-
takes words as input
-
produces a word as output
operations on words
-
concatenation :
if V =
a, . . -
an and W =
bi . . .
bm ,
then VW =
011 . . -
an bi ' ' '
bm
the concatenation of aab and ba is Gabba
length :
-
if ✓ =
9, . . -
an ,
then I V1 = h .
The
length defined 1×1=0 & Nat I V1 1-
can be
inductively 1
: =
the length of abbba is labbba 1=5
, power
- :
the power vk consists of K concatenationS Of v 'S :
"t '
v0 = ✗ & ✓ = vkv
then W2
'
let W =
aba ,
W0 =
✗ ,
W =
aba ,
=
☐baaba
>
and W =
abaabaaba
-
reverse
the of is Cai an IR
=
reverse a, . . -
an . . .
an - . .
a,
R
the reverse can be inductively defined : ✗ = ✗ & ( ✓a)R =
a CVR )
"
the reverse of abcb is Cabcb) =
bcba
Languages
A formal language is a set of words .
E* is
*
A (format L is of LEE
)
language a subset , that
*
[ is the set of all words over E
E ab ,
aab ,
bbaaabb 3 is a finite
language over E =
{ a ,b3
operations on
languages
-
set operations
a language is a set of words .
So the usual set operations
have meaning for languages :
E, E ,
n ,
U, I . . . .
ba C- { a. aba.ba 3 ab ¢ {a ,
aba.ba 3
{ a. ba3 E E a, aba.ba 3 { a. B3 4EA , aba.ba 3
{ a. aba.ba 3 n { a , ab.ba 3 =
{ a .ba 3
{ a. aba.ba 3 U { a. ab.ba 3 =
{ a. ab aba.ba } ,
E a. aba.ba 3 I [ a. ab.ba 3 =
{ aba3
-
complement
The complement [ is all words that are not in the
language L:
*
[ =
[ 1L
for E =
EA3 and L= [ a. aaa 3 .
Then I =
{ ✗ AA3W [ an 1h24 3
,
-
reverse
the is LR { R
I EL3
reverse of language
=
a ✗ ✗
the reverse of L= EX ab , ,
bbaba 3 is LR =
{ t.ba .
ababb 3
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