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

ExamplesA dfa for regular expressions on the alphabet S = { a, b, c }

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 DFARegular 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 ExpressionConstruct 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 ExampleThis NFA accepts strings such as: abc abd ad ac

Examplesa 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 expressionBasic 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 -productionsemployed 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 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 | [1-3ab] = 1|2|3|a|b

ExampleConstruct 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 )* 12345678

letterdigit

Example (Cont.2)Now, finish the construction for: letter ( letter | digit )*