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Grammars and Automata

Grammars and Automata. Review Questions (1/2). What is the size of the empty set? What is the size of the set containing just the empty string? Let L 2 = { λ , 00,0000} be defined over the alphabet ∑ = {0}. Describe the strings in the set L 2 *. Describe the complement of L 2.

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Grammars and Automata

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  1. Grammars and Automata

  2. Review Questions (1/2) • What is the size of the empty set? • What is the size of the set containing just the empty string? • Let L2 = {λ, 00,0000} be defined over the alphabet ∑ = {0}. Describe the strings in the set L2*. • Describe the complement of L2.

  3. Questions (2/2) • Let L3 = {awb | w  {a,b}*}. Define the language L3R • How would you prove that L3 = {a}.{a,b}*.{b}? • How would you prove that L3 ≠ L3R?

  4. Grammars • One formal way to present a solution to a problem. • Describes rules on how to form string for a language. • A grammar generates a language

  5. English Grammar • <sentence> -> <noun-phrase> <predicate> • <noun-phrase> -> <article> <noun> • <predicate> -> <verb> • <article> -> a | the

  6. Definition of a Grammar • A grammar G is a 4-tuple (V,T,S,P), where • V is a finite set of variables • T is a finite set of terminal symbols • S  V is the start variable • P is a finite set of productions

  7. Productions of a Grammar • A production has the form x -> y, where • x  (V  T)+ • y  (V  T)* • And can be applied to a string w = uxv by replacing x with y. • Productions are generally used to generate a string of terminals from the start symbol.

  8. Example • Let G=({S,A},{0,1},S,P), where P contains • S -> A1 • A -> 0A | 1A | λ

  9. Derivations • w => z if • w = uxv, and • z = uyv, and • x -> y is a production in the grammar • If w1 => w2 => w3 => … => wn then w1 =>* wn

  10. Language of a Grammar • The set of strings of terminals that can be generated from the start symbol of the grammar. • L(G) = { w  T* | S =>* w }

  11. Automata • Abstract model of a computer • May contain: • Input file • Temporary storage • Control unit – contains states and transitions • Output • Acceptor – output is yes or no • Transducer – output is a string

  12. Example - Acceptor b 1 a a 0 b 2 b a

  13. Example - Transducer 0/𝜆 0/𝜆 0/1 0/𝜆 0 1 2 3 4 1/0

  14. Example • Design automata that accepts passwords. • A password must contain at least one upper case character and at least one digit. • Assume that the only terminals are upper and lower case characters and digits.

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