Module 14

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Module 14

• Regular languages
• Inductive definitions
• Regular expressions
• syntax
• semantics

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Regular Languages(Regular Expressions)
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Regular Languages

• New language class
• Elements are languages
• We will show that this language class is
  identical to LFSA
• Language class to be defined by Finite State
  Automata (FSA)
• Once we have shown this, we will use the term
  regular languages to refer to this language
  class

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Inductive Definition of Integers

• Base case definition
• 0 is an integer
• Inductive case definition
• If x is an integer, then
• x1 is an integer
• x-1 is an integer
• Completeness
• Only numbers generated using the above rules are
  integers

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Inductive Definition of Regular Languages

• Base case definition
• Let S denote the alphabet
• is a regular language
• l is a regular language
• a is a regular language for any character a in
  S
• Inductive case definition
• If L1 and L2 are regular languages, then
• L1 union L2 is a regular language
• L1 concatenate L2 is a regular language
• L1 is a regular language
• Completeness
• Only languages generated using above rules are
  regular languages

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Proving a language is regular

• Prove that aa, bb is a regular language
• a and b are regular languages
• base case of definition
• aa aa is a regular language
• concatenation rule
• bb bb is a regular language
• concatenation rule
• aa, bb aa union bb is a regular language
• union rule
• Typically, we will not go through this process to
  prove a language is regular

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Regular Expressions

• How do we describe a regular language?
• Use set notation
• aa, bb, ab, ba
• aa,bb
• Use regular expressions R
• Inductive def of regular languages and regular
  expressions on page 72
• (aabbabba)
• a(ab)b

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R and L(R)

• How we interpret a regular expression
• What does a regular expression R mean to us?
• aaba represents the regular language aaba
• f represents the regular language
• aabb represents the regular language aa, bb
• We use L(R) to denote the regular language
  represented by regular expression R.

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Precedence rules

• What is L(abc)?
• Possible answers
• a(b union c
• (ab,c)
• (ab union c)
• ab union c
• Must know precedence rules
• first, then concatenation, then

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Precedence rules continued

• Precedence rules similar to those for arithmetic
  expressions
• abc2
• (a times b) (c times c)
• exponentiation first, then multiplication, then
  addition
• Think of Kleene closure as exponentiation,
  concatenation as multiplication, and union as
  addition and the precedence rules are identical

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Regular expressions are strings

• Let L be a regular language over the alphabet S
• A regular expression R for L is just a string
  over the alphabet S union (, ), , , f, l
  which follows certain syntactic rules
• That is, the set of legal regular expressions is
  itself a language over the alphabet S union (,
  ), ,
• f, aaba are strings in the language of legal
  reg. exp.
• )(, a are strings NOT in the language of legal
  reg. exp.

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Semantics

• We give a regular expression R meaning when we
  interpret it to represent L(R).
• aaba is just a string
• we interpret it to represent the language
  aaba.
• We do similar things with arithmetic expressions
• 1072 is just a string
• We interpret this string to represent the number
  59

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Key fact

• A language L is a regular language iff there
  exists a reg. exp. R such that L(R) L
• When I ask for a proof that a language L is
  regular, rather than going through the inductive
  proof we saw earlier, I expect you to give me a
  regular expression R s.t. L(R) L

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Summary

• Regular expressions are strings
• syntax for legal regular expressions
• semantics for interpreting regular expressions
• Regular languages are a new language class
• A language L is regular iff there exists a
  regular expression R s.t. L(R) L
• We will show that the regular languages are
  identical to LFSA