It is a well-established fact that each regular expression can be transformed into a nondeterministic finite automaton.

1. Mr.Keshav Tambre
Information Technology Department
International Institute of Information Technology, I²IT
www.isquareit.edu.in
Regular Expressions To Finite Automata

2.  Lexical analyzer recognizes tokens present in the source
program
 Compiler defines tokens using RE
 These indicate what characters may go into a lexeme
belonging to a particular token and how they go together
 Lexical analyzer is implemented as FSM
2
Regular Expressions

3.  Design of lexical analyzer involves:-
 Represent tokens by RE
 Synthesis of finite state machines that recognize desired
strings generated by RE
 Simulating the behavior of FSM which generates the lexical
analyzer
3
Regular Expressions

4.  RE are notations for specifying patterns
 Alphabet/Character class
 Denotes a finite set of symbols
 String
 Strings are defined over some alphabets, is a finite sequence
of symbols drawn from the alphabet
 Empty string denoted by €
 Language
 Set of strings over some alphabet
4
Specification of Tokens

5.  ∑={a,b}
 a|b denotes the set {a,b}
 RE (a|b) (a|b) denotes the set {aa,ab,ba,bb}
 (a|b)*
 a|a*b
 Identifiers in C
 letter|(letter | digit)*
5
Regular Expression

6.  Names given to RE
 If ∑ is an alphabet of basic symbols then RE is a sequence
of definitions of the form:
d1 → r1……….. dn → rn
where di is the distinct name and ri is the RE
 Short hand notations:
 *
 +
 ?
6
Regular Definitions

7.  Intermediate steps in the construction of the analyzer
 Depicts the action of the lexical analyzer when called
upon by the parser to get the next token
 Positions are represented as circles called as states
 States connected by arrows called edges
 Edges have labels indicating the input characters that
can appear next after the transition diagram has
reached states
 Assumption –TD are deterministic in nature
7
Transition diagrams

8.  When RE is compiled as a recognizer for a language by
constructing a generalized transition diagram called finite
automata
 Deterministic and Non deterministic
 NFA
 Mathematical model that consists of
 Set of states , s
 Set of input symbol ∑
 Transition function move that maps state symbol pairs to set of states
 A start state
 A final state
8
Finite automata

9.  Diagrammatically represented as a labeled directed graph
called transition graph
 Implementation-State Transition Table
 DFA
 Special case of NFA in which
 No state has € transition
 For each state s and input symbol a is at most one edge labeled a
leaving s
9
Finite automata

10.  RE are notations for specifying patterns
 Alphabet/Character class
 Denotes a finite set of symbols
 String
 Strings are defined over some alphabets, is a finite sequence
of symbols drawn from the alphabet
 Empty string denoted by €
 Language
 Set of strings over some alphabet
10
Specification of Tokens

11.  RE to DFA
 We augment the grammar the RE with #
 Construct nullable,firstpos ,lastpos and followpos which is done by
making traversal over the tree
 Construct DFA from followpos
 Firstpos: at what positions the string can start
 Lastpos:at what positions the string can end
 Follopos:what positions can follow position I in the syntax tree
 Nullable:whether an empty string can be generated at the node (sub
expression)
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RE to DFA

12. Node n Nullable(n) Firstpos(n) Lastpos(n)
N is a leaf labeled έ true ø ø
if N is | node with left child
c1 and right child c2
Nullable(c1) or
Nullable (c2)
Firstpos (C1) U Firstpos(C2) Lastpos (C1) U Lasttpos(C2)
N is a leaf labeled with
position i
false { i } { i }
if N is . Node node with left
child c1 and right child c2
Nullable(c1) and
Nullable (c2)
If nullable (C1) then
Firstpos (C1) U Firstpos(C2)
else Firstpos(C1)
If nullable (C2) then
Lastpos (C1) U Lastpos(C2)
else Lastpos(C2)
if N is the node with only c1 as
child
true Firstpos (C1) Lastpos (C2)
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RE to DFA

13.  Computation of Followpos : Followpos(i) tells us what
positions can follow position i in the syntax tree. This can be
computed as follows.
 if n is a ‘.’ Node, with a left child C1 and right child C2 and i
is a position in the Lastpos(C1), then all positions in
Firstpos(C2) are in Followpos(i)
 if n is a * Nodeand i is a position in the Lastpos(n), then all
positions in Firstpos(n) are Followpos(i)
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RE to DFA

14. Algorithm for construction of DFA transition table
Initially , the only unmarked state in Dstates is firstpos(root), where root is the root of a
syntax tree.
While there is an unmarked state T in Dstates do
Begin
Mark T
For each input symbol a do
Begin
Let U be the set of positions that are in Followpos(P) for some P in T,
such that the symbol at position P is a.
If U is not empty and is not in Dstates then add U as an unmarked
state to Dstates
Dtran [T,a] = U
End
End
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RE to DFA

15. Thank You
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