Regular Expressions

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Regular Expressions

• CIS 361

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Fundamental Problems

• Need finite descriptions of infinite sets of
  strings.
• Discover and specify regularity.
• The set of languages over a finite alphabet is
  uncountable, while the set of descriptions is
  countable

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Regular Expressions

• Language L is regular if there exists a finite
  acceptor for it
• Any language that is described by a regular
  expression can be accepted by some finite
  automaton

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Regular Expressions

• Regular expressions
• Combination of strings of symbols from some
  alphabet, parentheses and operators U, .,
• U is union (some literature uses )
• . (or nothing) is concatenation
• is star closure or Kleene star
• superscripted
• repetition, 0 or more times
• is closure
• superscripted
• repetition, 1 or more times

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Specifying Lexical Structure Using Regular
Expressions

• Have some alphabet ? set of symbols
• Regular expressions are built from
• ? - empty string
• Any letter from ?
• r1r2 String r1 followed by r2 (concatenation)
• r1 U r2 (r1 r2) either regular expression r1
  or r2 (union)
• r - iterated sequence and choice ? r r r
• Parentheses to indicate grouping/precedence

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Regular Expressions

• Operations
• Union
• Complement
• Intersection
• Difference
• Concatenation
• Repetition
• Kleene star
• Plus operator

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Regular Expressions

• Union
• L ? M
• The union of two regular expressions Q and R is Q
  U R
• In terms of automata A and B, respectively
• create a new initial state q
• connect it to the initial states of A and B by ?
  transitions

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Regular Expressions

• Complement
• ? - L
• To construct the complement of a regular
  expression L, inspect the automaton that accepts
  its strings
• convert the automaton for L to a deterministic
  automaton
• flips favorable and nonfavorable states
• construct a regular expression for strings
  accepted by the updated automaton

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Regular Expressions

• Complement of
• bit strings with at least one 1
• bit strings containing no 1s
• 0
• Complement of
• bit strings with exactly one 1
• bit strings containing no 1s
• U bit strings with at least
  two 1s
• 0 U (0 1 0 1 0)(0 U 1)

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Regular Expressions

• Intersection
• L ? M
• Apply DeMorgans law
• Union of the complements of L and M

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Regular Expressions

• Difference
• L M
• Can be expressed as the intersection of languages
  L and ? - M

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Regular Expressions

• Concatenation
• Strings u and v over alphabet ? is string uv
• Languages L1 and L2 concatenated
• L1L2 uvu ? L1, v ? L2
• Can be extended to any finite number of languages

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Regular Expressions

• Concatenation
• LM
• Algorithm connects every favorable state of L to
  the initial state of M by an arrow labeled ?
• Favorable states of L become non-favorable
• Favorable states of M become favorable states of
  the new automaton

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Regular Expressions

• Kleene star
• L
• In terms of automaton
• connect every favorable state of L to the initial
  state of L by a transition labeled ?
• create a new initial state s, make it the only
  favorable state and connect it to the old initial
  state by ? transition

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Regular Expressions

• Plus ()
• L
• In terms of automaton
• connect every favorable state of L to the initial
  state of L by a transition labeled ?
• Thats it. This gets one or more times to a
  favorable state

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Naming Languages

• Regular sets can be named using the derivation in
  terms of the seed elements and the closure
  operations. Regular expressions formalize this
  approach.
• Regular sets ? Regular Expressions
• Numbers ? Numerals
• Semantics ? Syntax

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Example

• Regular expressions for strings over a,b
  containing at least one a.
• Focus on the one a
• (a u b)a(a u b)
• Focus on the leftmost a
• ba(a u b)
• Focus on the as
• bab(ab)
• Further optimization
• b(ab)

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Equivalence of regular expressions

• Two regular expressions are equivalent if they
  represent the same regular set.

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Concept of Language Generated by Regular
Expressions

• Set of all strings generated by a regular
  expression is the language of the regular
  expression
• In general, a language may be (countably)
  infinite
• A string in a language is often called a token

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Examples of Languages and Regular Expressions

• ? 0, 1, .
• (0 U 1).(0 U 1) - Binary floating point numbers
• (00) - even-length all-zero strings
• 1(0101) - strings with even number of zeros
• ? A,,Z, a,,z, 0,,9,_
• (A U U z)(A U U z U 0 U U 9 U _)
  identifiers
• (1 U U 9)(0 U U 9) natural numbers (no
  negatives)
• (012) - trinary (base 3) numbers

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Finite-State Automata

• Alphabet ?
• Set of states with initial and accepting states
• Transitions between states, labeled with symbol(s)

(0 1).(01)
1
1
0
0