ICS312 Set 29
Deterministic Finite Automata
Nondeterministic Finite Automata
Deterministic Finite Automata
• A regular expression can be represented (and recognized) by a
machine called a deterministic finite automaton (dfa).
• A dfa can then be used to generate the matrix (or table) used by
the scanner (or lexical analyzer).
• Deterministic finite automata are frequently also called simply
finite automata (fa).
Example of a DFA for Recognizing
Identifiers
Examples
A dfa for regular expressions on the alphabet
S = { a, b, c }
a. Which have exactly one b:
Examples (Cont. 1)
b. Which have 0 or 1 b's:
Examples (Cont. 2)
A dfa for a number with an optional fractional
part (assume S = { 0,1,2,3,4,5,6,7,8,9,+,-,. }:
Constructing DFA
• Regular expressions give us rules for recognizing
the symbols or tokens of a programming language.
• The way a lexical analyzer can recognize the
symbols is to use a DFA (machine) to construct a
matrix, or table, that reports when a particular kind
of symbol has been recognized.
• In order to recognize symbols, we need to know
how to (efficiently) construct a DFA from a regular
expression.
How to Construct a DFA from a
Regular Expression
• Construct a nondeterministic finite automata
(nfa)
• Using the nfa, construct a dfa
• Minimize the number of states in the dfa to
get a smaller dfa
Nondeterministic Finite Automata
• A nondeterministic finite automata (NFA) allows
transitions on a symbol from one state to possibly
more than one other state.
• Allows e-transitions from one state to another
whereby we can move from the first state to the
second without inputting the next character.
• In a NFA, a string is matched if there is any path
from the start state to an accepting state using that
string.
NFA Example
This NFA accepts strings such as:
abc
abd
ad
ac
Examples
a f.a. for ab*:
a f.a. for ad
To obtain a f.a. for: ab* | ad
We could try:
but this doesn't work, as it matches strings such as abd
Examples (Cont. 1)
So, then we could try:
It's not always easy to construct a f.a. from a regular expressio
It is easier to construct a NFA from a regular expression.
Examples (Cont. 2)
Example of a NFA with epsilon-transitions:
This NFA accepts strings such as ac, abc, ...
Algorithm to employ in getting a
computer program to construct a
NFA for any regular expression
Basic building blocks:
(1) Any letter a of the alphabet is recognized by:
(2) The empty set  is recognized by:
Note: it is possible to avoid including some of the ε-productions
employed by the algorithm, but the increase in speed, if any, is
negligible.
(3) The empty string e is recognized by:
(4) Given a regular expression for R and S, assume these boxe
represent the finite automata for R and S:
(5) To construct a nfa for RS (concatenation):
(6) To construct a nfa for R | S (alternation):
(7) To construct a nfa for R* (closure):
NOTE: In 1-3 above we supply finite automata for
some basic regular expressions, and in 4-6 we
supply 3 methods of composition to form finite
automata for more complicated regular expressions.
These, in particular, provide methods for
constructing finite automata for regular expressions
such as, e.g.:
R+
= RR*
R?
= R|ε
[1-3ab]
= 1|2|3|a|b
Example
Construct a NFA for an identifier using the above mechanical me
for the regular expression: letter ( letter | digit )*
First: construct the nfa for an identifier: ( letter | digit )
Example (Cont.1)
Next, construct the closure: ( letter | digit )*
e
1
e
3
e
2
letter
5
e
7
e
4
digit
e
6
e
e
8
Example (Cont.2)
Now, finish the construction for: letter ( letter | digit )*