Regular Expressions

1
Regular Expressions
Programming Language Translators

• Prepared by
• Manuel E. Bermúdez, Ph.D.
• Associate Professor
• University of Florida

2
Regular Expressions

• A compact, easy-to-read language description.
• Use operators to denote the language constructors
  described earlier, to build complex languages
  from simple atomic ones.

3
Regular Expressions

• Definition A regular expression over an alphabet
  S is recursively defined as follows
• ø denotes language ø
• e denotes language e
• a denotes language a, for all a ? S.
• (P Q) denotes L(P) U L(Q), where P, Q are
  r.e.s.
• (PQ) denotes L(P)L(Q), where P, Q are r.e.s.
• P denotes L(P), where P is r.e.
• To prevent excessive parentheses, we assume left
  associativity, with the following operator
  precedence hierarchy, from most to least binding
  , ,

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Regular Expressions

• Examples
• (O 1) any string of Os and 1s.
• (O 1)1 any string of Os and 1s, ending
  with a 1.
• 1O1 any string of 1s with a single O
  inserted.
• Letter (Letter Digit) an identifier.
• Digit Digit an integer.
• Quote Char Quote a string.
• Char Eoln a comment.
• Char another comment.
• Assuming that Char does not contain quotes,
• eolns, or .

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Regular Expressions

• Conversion from Right-linear grammars to regular
  expressions
• Example
• S ? aS R ? aS
• ? bR
• ? e
• What does S ? aS mean?
• L(S) ? aL(S)
• S ? bR means L(S) ? bL(R)
• S ? e means L(S) ?e

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Regular Expressions

• Together, they mean that
• L(S) aL(S) bL(R) e
• or S aS bR e
• Similarly, R ? aS means R aS.
• Thus, S aS bR e
• R aS
• System of simultaneous equations, in which the
  variables are nonterminals.

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Regular Expressions

• Solving systems of simultaneously equations.
• S aS bR e
• R aS
• Back substitute R aS
• S aS baS e
• (a ba) S e
• Question What to do with equations of the form
• X ?X ß ?

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Regular Expressions

• Answer ß ? L(x), so aß ? L(x),
• aaß ? L(x),
• aaaß ? L(x),
• Thus aß L(x).
• In our case,
• S (a ba) S e
• (a ba) e
• (a ba)

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Regular Expressions

• Right-linear regular grammar
• ?
• regular expression
• 1. A a1 a2 an if A ? a1
• ? a2
• .
• .
• .
• ? an

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Regular Expressions

• If equation is of the form X a, where X does
  not appear in a, then replace every occurrence of
  X with a in all other equations, and delete
  equation X a.
• If equation is of the form X aX ß, where
  X does not occur in either a or ß, then replace
  the equation with X aß.
• Note Some algebraic manipulations may be needed
  to obtain the form X aX ß.
• Important Catenation is not commutative!!

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Regular Expressions

• Example
• S ? a R ? abaU U ? aS
• ? bU ? U ? b
• ? bR
• S a bU bR
• R abaU U (aba e) U
• U aS b
• Back substitute R
• S a bU b(aba e) U
• U aS b

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Regular Expressions

• Back substitute U
• S a b(aS b) b(aba e)(aS b)
• a baS bb babaaS babab baS bb
• (ba babaa)S (a bb babab)
• therefore
• S (ba babaa)(a bb babab)

repeats
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Regular Expressions

• Summarizing
• RGR RGL Minimum
• DFA
• RE NSA DFA

Done
Soon
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Regular Expressions

• Regular Expression
• ?
• NFA
• Recursively build the FSA, mimicking the
  structure of the regular expression. Each FSA
  built has one start state, and one final state.
• Conversions
• if ø

2
1
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Regular Expressions

• if e
• if a
• if P Q
• if P Q
• or

1
a
1
2
P
e
e
1
2
e
e
Q
e
P
Q
e
e
e
1
P
Q
2
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Regular Expressions
e

• if P
• Example (b (aba e) a)
• (b (aba e) a)
• (b (aba e) a)
• (b (aba e) a)

e
e
1
P
2
e
b
1
2
a
3
4
b
5
6
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Regular Expressions
a

• (b (aba e) a)
• (b (aba e) a)
• (b (aba e) a)
• (b (aba e) a)

7
8
9
a
10
11
a
b
e
3
4
5
6
e
7
8
a
18
Regular Expressions

• (b (aba e) a)
• (b (aba e) a)

a
b
e
3
4
5
6
e
e
12
7
9
8
13
e
a
e
e
b
2
1
e
a
b
e
3
4
5
6
e
e
12
7
9
8
13
e
a
e
e
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Regular Expressions

• (b (aba e) a)

b
2
1
e
a
b
e
3
e
4
5
6
e
12
7
9
8
13
e
a
e
e
e
10
a
11
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Regular Expressions

• (b (aba e) a)

e
a
e
e
b
2
14
12
3
4
1
e
e
e
e
5
9
11
e
e
a
6
13
15
7
8
10
a
e
e
e
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Regular Expressions

• Regular Expression
• ?
• NFA
• Start With

ALGORITHM 2
E
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Regular Expressions

• Apply Rules

a
a
e
e
ab
a
b
a
a b
b
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Regular Expressions

• Algorithm 1
• Builds FSA bottom up
• Good for machines
• Bad for humans
• Algorithm 2
• Builds FSA top down
• Bad for machines
• Good for humans

Arguable
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Regular Expressions

• Example (Algorithm 2)

(a b) (aa bb)
(a b)
aa bb
aa
e
e
bb
a b
a
a
e
e
b
b
a
b
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Regular Expressions

• Example (Algorithm 2)

ba(a b) ab
a
b
a
e
e
a
b
b
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Regular Expressions

• Deterministic Finite-State Automata (DFAs)
• Definition A deterministic FSA is defined just
  like an NFA, except that
• d Q x S ? Q, rather than
• d Q x S union e? 2Q
• Thus, both
• and
• are impossible.

e
a
a
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Regular Expressions

• Every transition of a DFA consumes a symbol.
  Fortunately, DFAs are just as powerful as NFAs.
• Theorem For every NFA there exists an equivalent
  (accepting the same language) DFA.

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Regular Expressions

• Conversion from NFAs to DFAs
• Simulate all moves of the NFA with the DFA.
• The start state of the DFA is the start state of
  the NFA (say, S), together with states that are
  e-reachable from S.
• Each state in the DFA is a subset of the set of
  states of the NFA the notion of being in any
  one of a number of states.
• New states in the DFA are constructed by
  calculating the sets of states that are reachable
  through symbols, after the start state.
• The final states in the DFA are those that
  contain any final state of the NFA.

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Regular Expressions

• Example ab ba

a
e
3
2
b
e
6
a
NFA
1
b
e
4
5
e
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Regular Expressions

• DFA
• Input
• State a b
• 123 23 456
• 23 23 6
• 456 56 ---
• 6 --- ---
• 56 56 ---

a
6
23
b
a
123
a
b
456
56
a
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Regular Expressions

• In general, if NFA has N states, the DFA can have
  as many as 2N states.
• Example ba (a b) ab

e
a
5
4
e
e
b
a
e
8
3
0
1
2
e
e
6
7
b
NFA
e
11
10
9
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Regular Expressions

• DFA
• Input
• State a b
• 0 --- 1
• 1 234689 ---
• 234689 34568910 346789
• 34568910 34568910 34678911
• 346789 34568910 346789
• 34678911 34568910 346789

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Regular Expressions
a
34568910
b
a
a
b
a
234689
34678911
a
0
1
b
346789
b
b
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Regular Expressions

• State Minimization
• Theorem Given a DFA M, there exists an
  equivalent DFA M that is minimal, i.e. no other
  equivalent DFA exists with fewer states than M.
• Definition A partition of a set S is a set of
  subsets of S such that every element of S appears
  in exactly one of the subsets.

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Regular Expressions

• Example S 1, 2, 3, 4, 5
• ?1 1, 2, 3, 4, 5
• ?2 1, 2, 3,, 4, 5
• ?3 1, 3, 2, 4, 5
• Note ?2 is a refinement of ?1 , and ?3 is a
  refinement of ?2.

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Regular Expressions

• Minimization Algorithm
• Remove all undefined transitions by introducting
  a TRAP state, i.e. a state from which no final
  state is reachable.
• Partition all states into two groups (final
  states and non-final states).
• Complete the Next State table for each group,
  by specifying transitions from group to group.
• Form the next partition split groups in which
  Next State table entries differ.
• Repeat 3 until no further splitting is possible.
• Determine start and final states.

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Regular Expressions
a
b

• Example
• ?0 1, 2, 3, 4, 5
• State a b
• 1 1234 1234
• 2 1234 1234
• 3 1234 1234
• 4 1234 5
• 5 1234 1234

b
Split 4 from partition 1,2,3,4
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Regular Expressions

• ?1 1, 2, 3, 4, 5
• State a b
• 1 123 123
• 2 123 4
• 3 123 123
• 4 123 5
• 5 123 123
• Split 2 from partition 1,2,3

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Regular Expressions

• ?2 1, 3, 2, 4, 5
• State a b
• 1 2 13
• 3 2 13
• 2 2 4
• 4 2 5
• 5 2 13
• No more splitting
  Minimal DFA

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Regular Expressions

• Summary of Regular Languages
• Smallest class in the Chomsky hierarchy.
• Appropriate for lexical analysis.
• Four representations RGR , RGL , RE and FSA.
• All four are equivalent there are algorithms to
  perform transformations among them.
• Various advantages and disadvantages among these
  four, for language designer, implementor, and
  user.
• FSAs can be made deterministic, and minimal.