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Deterministic Automata The NFA shown in Fig.10 has more than one transition from state 0 on input a, that is, it may go to state 0 or 1. Similarly, the NFA of Fig.11 has two transitions on from state 0. These situations are the reason why it is hard to simulate an NFA with a computer program. The deterministic finite automata have at most one path from the start state labeled by any string. The finite automaton is deterministic if 1- It has no transitions on input 2- For each state s and input symbol a, there is at most one edge labeled a leaving s. Example: in Fig.12 below we see a Deterministic Finite Automata (DFA) accepting the language (ab)*abb, which is the same language as that accepted by the NFA of Fig.9 b b Start 0 b a 1 2 b 3 a a a Fig.12: DFA accepting (ab)*abb Since there is at most one transition out of any state on any symbol, a DFA is easier to simulate by a program than an NFA. How to Build a Lexical Analyzer Step1 Convert the Grammar into Transition Diagram. Step2 Convert the Regular Expression into Nondeterministic Finite State Automata. Step3 Convert the NFA into DFA. Step4 Minimize Finite State Automata. Step5 Write an efficient program for the minimized finite state automata, called (minimized finite state automata recognizer). 25 Convert a Grammar into Transition Diagram The regular grammars are the only type of grammars that can be converted into transition diagram. We take an example to explain this conversion. Formal definition for finite automaton M is a 5-tuple, (Q, Σ, δ, q0, F), consisting of a finite set of states (Q) a finite set of input symbols called the alphabet (Σ) a transition function (δ : Q × Σ → Q) a start state (q0 ∈ Q) a set of accept states (F ⊆ Q) Example: convert the following grammar into transition diagram: G=({S, R, U},{a, b, c}, P, {S}) where P is given by S R U abUcR abaUU bcS The productions R abaUU are not validate for regular grammar, so that we convert it into the following productions R aX bcS X bY Y aU So that grammar has the following productions: S abUcR R aX bcS X bY Y aU U bcS Therefore the transitions diagram for the grammar is shown in Fig.13 26 b c a Z S R c b b a c U a Y b X Fig.13: Transition Diagram The transition diagram may help the designer to build a lexical analyzer directly without using other steps. We use functions and procedures that read a character and check it according to some label. Example: If we have the following grammar for recognizing a real number <Real number>::= <integer> ● <integer> <integer>::= <digit><integer> <digit> <digit>::= 012…9 We use the following steps:1- Construct a separate transition diagram for every grammar. <Real number>: ● <integer> <integer> ------------- 1 < Integer>: <digit> ------------- 2 <digit> <digit>: 0 1 . . . 9 ------------- 3 27 2- Substitute every non terminal with its transition diagram (a) Substitute the non terminal <integer> in transition diagram 1 with the transition diagram 2 . <digit> <digit> <digit> <digit> (b) Substitute the non terminal <digit> with transition diagram 3 Start . 0…9 0 1 0…9 2 0…9 3 0…9 Convert a Regular Expression into an NFA We can generate an NFA from Regular expressions according to the following steps:1- Divide a regular expression R into its primitive components as follows:a- If there is in regular expression, we must build a start state (i) and a final state (f), and put as a label of the edge. i f b- If there exist a terminal Symbol 'a' in , we construct the NFA:- i1 a f1 c- If the regular expression R contains the expression PQ we build a start and a final state, and we decomposed P and Q into Sp and Fp for P and SQ and FQ for Q. 28 Sp Fp i f FQ SQ d- If the regular expression R contains the expression P.Q, then the transition diagram that represent this expression is given by:- Fp SQ Sp FQ e- If regular expression R contain P*, then the transition diagram that represent this expression is given by:- i P Sp fp f 2- Substitute the primitive components in regular expression to get an NFA Example: Convert the regular expression into an NFA. R= abcd* 1) The primitive component for this regular expression (a) a.b.c that can be converted into:1 a 2 b 3 c 4 29 d* that can be converted into:- 5 6 d 7 8 2) Substitute the primitive components to get:1 a 2 b 3 c 4 0 9 5 6 d 7 8 30