Finite Automata - basic computational model: limited amount of - - PowerPoint PPT Presentation

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finite automata
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Finite Automata - basic computational model: limited amount of - - PowerPoint PPT Presentation

Oct 18, 2023 299 likes •425 views

[Section 1.1] Finite Automata - basic computational model: limited amount of memory - example: controller for an automatic door [Section 1.1] Finite Automata Formal definition: A finite automaton (FA) is a 5-tuple (Q, , ,q 0 ,F), where -

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SLIDE 1

Finite Automata

[Section 1.1]

  • basic computational model: limited amount of memory
  • example: controller for an automatic door
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SLIDE 2

Finite Automata

[Section 1.1]

Formal definition: A finite automaton (FA) is a 5-tuple (Q,Σ,δ,q0,F), where

  • Q is a finite set of states
  • Σ is a (finite) alphabet
  • δ:QxΣ  Q is the transition function
  • q0 ∈ Q is the start state
  • F ⊆ Q is the set of accept states

Pictorial representation: state diagram

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SLIDE 3

Finite Automata

[Section 1.1]

Formal definition: A finite automaton (FA) is a 5-tuple (Q,Σ,δ,q0,F), where

  • Q is a finite set of states
  • Σ is a (finite) alphabet
  • δ:QxΣ  Q is the transition function
  • q0 ∈ Q is the start state
  • F ⊆ Q is the set of accept states

Pictorial representation: state diagram

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SLIDE 4

Finite Automata

[Section 1.1]

Another (more abstract) example:

  • accept all strings over {0,1} that start with 1 and end with 0
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SLIDE 5

FA: computation & language

[Section 1.1]

Let M=(Q,Σ,δ,q0,F) be a FA. The language of M (accepted / recognized by M) is L(M). Formally: need the definition of computation: M accepts w=w1w2…wn if there exist states r0,r1,…,rn in Q such that

  • r0 = ?
  • δ
  • rn

A language is regular if there exists a FA that recognizes it.

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SLIDE 6

Designing FAs

[Section 1.1]

Examples – languages over {0,1} consisting of strings:

  • with odd number of 1’s
  • that contain 001 as a substring
  • that are even length and do not contain 00 as a substring

A language that cannot be accepted by a FA?

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SLIDE 7

Designing FAs

[Section 1.1]

A language that cannot be accepted by a FA?

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Regular operations

[Section 1.1]

Let A and B be languages. The following three language

  • perations are called the regular operations:
  • union: A ∪ B
  • concatenation: A.B
  • star: A*

The natural numbers are closed under multiplication but not division. What about the class of regular languages ?

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SLIDE 9

Regular languages are closed under …

[Section 1.1]

Thm 1.25: The class of regular languages is closed under the union operation.

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SLIDE 10

Regular languages are closed under …

[Section 1.1]

Thm 1.25: The class of regular languages is closed under the union operation.

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SLIDE 11

Regular languages are closed under …

[Section 1.1]

Thm 1.26: The class of regular languages is closed under the concatenation operation.