AUBER F1

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CD5560 FABER Formal Languages, Automata and
Models of Computation Lecture 2 Mälardalen
University 2010
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• Content
• Languages, Alphabets and Strings
• Strings String Operations
• Languages Language Operations
• Regular Expressions
• Finite Automata, FA
• Deterministic Finite Automata, DFA

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Languages, Alphabets and Strings
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Languages
A language is a set of strings
A String is a sequence of letters

• defined over an alphabet

An alphabet is a set of symbols
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Alphabets and Strings

• We will use small alphabets

Strings
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Operations on Strings
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String Operations
Concatenation (sammanfogning)
xy ? abbabbbaaa
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Reverse (reversering)
Example Longest odd length palindrome in a
natural language saippuakauppias (Finnish soap
salesman)
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String Length
Examples
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Recursive Definition of Length

• For any letter
• For any string
• Example

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Length of Concatenation
Example
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Proof of Concatenation Length

• Claim
• Proof By induction on the length
• Induction basis
• From definition of length

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• Inductive hypothesis
• 
  

for
Inductive step we will prove
for
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Inductive Step

• Write , where
• From definition of length
• From inductive hypothesis
• Thus

END OF PROOF
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Empty String

• A string with no letters
• (Also denoted as ?)
• Observations

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Substring (delsträng)

• Substring of a string
• a subsequence of consecutive characters
• String
  Substring

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Prefix and Suffix

• Suffixes

Prefixes
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Repetition
n

w
ww...
w

n

• Example
• Definition

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The (Kleene star) Operation

• the set of all possible strings from
  alphabet

Kleene is pronounced "clay-knee
http//en.wikipedia.org/wiki/Kleene_star
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The (Kleene plus) Operation
the set of all possible strings from the
alphabet except



S
,
b
a
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Example
l
-
S




oj, fy, usch, ojoj, fyfy,uschusch, ojfy, ojusch

K
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Operations on Languages
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Language

• A language is any subset of
• Example
• Languages

S
,
b
a


l

S
,
,
,
,
,
,
,
,

aaa
bb
ba
ab
aa
b
a
K


l


,
,
aab
aa
a
l

,
,
,
,
,

aaaaaa
ab
aa
baba
abba
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Example

• An infinite language

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Operations on Languages

• The usual set operations

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Reverse
Definition
Examples
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Concatenation

• Definition

Example
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Repeat

• Definition
• Special case

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Example

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Star-Closure (Kleene )

• Definition
• Example

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Positive Closure

• Definition

1
2

L
L
L
L
U
U


l
-


L
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Regular Expressions

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Regular Expressions Recursive Definition
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Examples
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Building Regular Expressions

• Zero or more.
• a means "zero or more a's."
• To say "zero or more ab's," that is,
• , ab, abab, ababab, ..., you need to say
  (ab).
• ab denotes a, ab, abb, abbb, abbbb, ....

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Building Regular Expressions

• One or more.
• Since a means "zero or more a's", you can use
  aa (or equivalently, aa) to mean "one or more
  a's.
• Similarly, to describe "one or more ab's,"
  that is,
• ab, abab, ababab, ..., you can use ab(ab).

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Building Regular Expressions

• Any string at all.
• To describe any string at all (with a, b,
  c), you can use (abc).
• Any nonempty string.
• This can be written as any character from
  followed by any string at all (abc)(abc).

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Building Regular Expressions

• Any string not containing....
• To describe any string at all that doesn't
  contain an a (with a, b, c), you can use
  (bc).
• Any string containing exactly one...
• To describe any string that contains exactly one
  a, put "any string not containing an a," on
  either side of the a, like this (bc)a(bc).

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Languages of Regular Expressions
language of regular expression
Example
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Definition

• For primitive regular expressions

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Definition (continued)

• For regular expressions and

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Example
Regular expression
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Example

• Regular expression

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Example

• Regular expression

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Example

• Regular expression

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Example

• Regular expression
• (consists of repeating 1s and 01s).

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Example
all strings without two consecutive 0

Equivalent solution
(In order not to get 00 in a string, after each 0
there must be an 1, which means that strings of
the form 1....101....1 are repeated. That is the
first parenthesis. To take into account strings
that end with 0, and those consisting of 1s
solely, the rest of the expression is added.)
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Equivalent Regular Expressions
Definition

• Regular expressions and

are equivalent if
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Example

and
are equivalent regular expressions.
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Additional Sources

• http//www.math.uu.se/salling/ Lennart Salling
• http//www.math.uu.se/salling/AUTOMATA_DV/index.h
  tml
• Introduktion movie .mov
• Program, strings, integers and integer functions
  .mov
• Different infinities and integer functions that
  can not be calculated by a program .mov
• Strings and languages .mov Regular languages and
  regular expressions .mov