Languages

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Languages
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Languages

• A language is a set of strings
• String A sequence of letters
• Examples cat, dog, house,
• Defined over an alphabet

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Alphabets and Strings

• We will use small alphabets
• Strings

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String Operations
Concatenation
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Reverse
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String Length

• 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

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Empty String

• A string with no letters
• Observations

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Substring

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

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Prefix and Suffix

• Prefixes Suffixes

prefix
suffix
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Another Operation

• Example
• Definition

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The Operation

• the set of all possible strings from
• alphabet

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The Operation
the set of all possible strings from
alphabet except
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Language

• A language is any subset of
• Example
• Languages

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Another Example

• An infinite language

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Operations on Languages

• The usual set operations
• Complement

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Reverse

• Definition
• Examples

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Concatenation

• Definition
• Example

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Another Operation

• Definition
• Special case

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More Examples

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Star-Closure (Kleene )

• Definition
• Example

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Positive Closure

• Definition

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Finite Automata
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Finite Automaton

Input
String
Output
Finite Automaton
String
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Finite Accepter

Input
String
Output
Accept or Reject
Finite Automaton
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Transition Graph

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

Input String
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Reading the Input

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Output accept
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Rejection

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Output reject
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Another Example
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Output accept
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Rejection
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Output reject
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Formalities

• Deterministic Finite Accepter (DFA)

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

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Set of States

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Initial State

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Set of Final States

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Transition Function

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Transition Function

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Extended Transition Function

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Observation There is a walk from to
with label
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Recursive Definition

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Languages Accepted by DFAs

• Take DFA
• Definition
• The language contains
• all input strings accepted by
• strings that drive to a
  final state

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Example

accept
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Another Example

accept
accept
accept
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Formally

• For a DFA
• Language accepted by

alphabet
transition function
initial state
final states
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Observation

• Language accepted by
• Language rejected by

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More Examples

trap state
accept
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all substrings with prefix
accept
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all strings without substring
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Regular Languages

• A language is regular if there is
• a DFA such that
• All regular languages form a language family

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Example

• The language
• is regular