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Languages and Finite Automata

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Recursive Definition of Length. For any letter: For any string : Example: 7 ... Definition: A substring of a string is any sequence of consecutive characters ... – PowerPoint PPT presentation

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Title: Languages and Finite Automata


1
Languages and Finite Automata
  • or
  • how to talk to machines...

2
Languages
  • A language is a set of strings
  • String
  • A sequence of letters (a word)
  • Examples cat, dog, house,
  • Defined over an alphabet
  • set of symbols (letters)

3
Alphabets and Strings
  • We will use small alphabets
  • Strings

4
String Operations
  • Concatenation
  • Reverse

5
String Length
  • Length
  • Examples

6
Recursive Definition of Length
  • For any letter
  • For any string
  • Example

7
Length of Concatenation
  • Example

8
Proof of Concatenation Length
  • Claim
  • Proof
  • By induction on the length
  • Induction basis
  • is only one symbol
  • From definition of length

9
  • Inductive hypothesis
  • for all
    with
  • Inductive step
  • we will prove
  • for

10
Inductive Step
  • Write , where
  • From definition of length
  • From inductive hypothesis
  • Thus

11
Empty String
  • A string with no letters
  • Observations

12
Substring
  • Definition
  • A substring of a string is any sequence of
    consecutive characters
  • Example
    Substrings

13
Prefix and Suffix
  • Prefixes Suffixes

prefix
suffix
14
Another Operation
  • Example
  • Definition
  • for any

15
The Operation
  • the set of all possible strings from
  • alphabet
  • Example

16
Language
  • A language is any subset of
  • Examples
  • A string is called sentence

17
Another Example
  • An infinite language

18
Operations on Languages
  • The usual set operations
  • Complement

19
Reverse
  • Definition
  • Examples

20
Concatenation
  • Definition
  • Example

21
Another Operation
  • Definition
  • Example
  • Special case

22
Example
23
Star-Closure (Kleene )
  • Definition
  • Example

24
Positive Closure
  • Definition

25
Finite Automata

Input
String
Output
Finite Automaton
String
26
Finite Accepter

Input
String
Output
Accept or Reject
Finite Automaton
27
Transition Graph

Abba -Finite Accepter
initial state
final state accept
transition
state
28
Initial Configuration

Input String
29
Reading the Input

30

31

32

Output accept
33
Rejection

34

35

36

Output reject
37
Formalities
  • Deterministic Finite Accepter (DFA)

set of states
input alphabet
transition function
initial state
set of final states
38
Input Aplhabet

39
Set of States

40
Initial State

41
Set of Final States

42
Transition Function

43
Transition Function

44
Extended Transition Function

45
Recursive Definition

46
Languages Accepted by DFAs
  • Take DFA
  • Definition
  • The language accepted by
  • contains all input strings accepted by
  • In other words
  • strings that drive to a
    final state

47
Example

accept
48
Another Example

accept
accept
accept
49
Formally
  • For a DFA
  • Language accepted by

alphabet
transition function
initial state
final states
50
Observation
  • Language accepted by
  • Language rejected by

51
More Examples

trap state
accept
52

all substrings with prefix
accept
53

all strings without substring
54
Regular Languages
  • Definition
  • A language is regular
  • if there is a DFA such that
  • All the regular languages form a family

55
Example
  • The language
  • is regular
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