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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 )*