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 -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 expression.

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-productionsemployed by the algorithm, but the increase in speed, if any, is negligible.

(3) The empty string is recognized by:

(4) Given a regular expression for R and S, assume these boxes 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 method 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 )*

1 2

3

4

5

6

7 8

letter

digit

Example (Cont.2)

Now, finish the construction for: letter ( letter | digit )*