Lexical Analysis Part 2 Professor Yihjia Tsai Tamkang University

2 Regular Expressions and FA are Equivalent Regular expressions NFA DFA For every regular expression, there is a deterministic finite-state machine that defines the same language, and vice versa

3 Closure Operator TheClosure operator is defined as Closure(s) = { s } U states reachable from s using  transitions. Example: Closure(1) = {1,3} b  a a/b a start

4 Subset Construction for NFA-> DFA 1.Compute A = Closure(start) 2.Compute the set of states reachable from A on transition a, call this new set S ’ 3.Compute Closure(S ’ ) – this is the new state and label it with the next available label 4.Continue for all possible transitions from the current state for all applicable elements of  5.Repeat steps 2-4 for each new state

5 Comparison from Last Time a b c     NFA: D A C B a b b c c DFA:

6 Example: aa*b | aab* NFA: 3 1 a b 2 a 5 4 a a b

7 Example: aa*b | aab* DFA State NFA Set of States ab A{1}B B{2,4}CD C{2,5}EF D{3} E{2}ED F{3,5}G G{5}G

8 Resultant DFA D A a b B a C E a b b b F G b a How is this result different than before?

9 Generalized NFA Generalized Nondeterministic Finite Automaton M=(Q, , , q start, q accept ) with Q finite set of states  the input alphabet q start the start state q accept the accept state  :(Q\{q accept })  (Q\{q start })  R the transition function ( R is the set of regular expressions over  )

10 Example GNFA qSqS qAqA 01* 0 0* |  

11 Observation: This GNFA: recognizes the language L(R) Characteristics of GNFA ’ s   :(Q\{q accept })  (Q\{q start })  R The interior Q\{q accept,q start } is fully connected by  From q start only ‘outgoing transitions’ To q accept only ‘ingoing transitions’ Impossible q i  q j transitions are “  (q i,q j ) =  ” qSqS qAqA RRRR

12 DFA M  Equivalent GNFA M ’ Let M have k states Q={q 1,…,q k } - Add two states q accept and q start qSqS q1q1  - Connect q start to earlier q 1 : qiqi  qjqj - Complete missing transitions by qAqA qjqj  - Connect old accepting states to q accept - Join multiple transitions : qiqi 0 qjqj 1 becomes qiqi 0|1 qjqj

13 Remove Internal state of GNFA If the GNFA M has more than 2 states, rip internal q rip to get equivalent GNFA M’ by: - Removing state q rip : Q’=Q\{q rip } - Changing the transition function  by  ’(q i,q j ) =  (q i,q j ) | (  (q i,q rip )(  (q rip,q rip ))*  (q rip,q j )) for every q i  Q’\{q accept } and q j  Q’\{q start } qiqi R 4 |(R 1 R 2 *R 3 ) qjqj qiqi R2R2 qjqj R4R4 q rip R1R1 R3R3 =

14 Try at Home What is the regular expression for –Binary numbers, multiples of 2 –Strings of a’s and b’s with a at start and finish –Strings of a’s and b’s, no consecutive a’s

15 Try at Home What is the NFA for the following RE? ((a|b)*c) | (a b c*)

16 References Aho, A.V., R. Sethi, and J.D. Ullman, Compilers Principles, Techniques and Tools, Addison-Wesley, ISBN Appel, A., Modern Compiler Implementation In Java (2 nd Ed), Cambridge University Press, ISBN X.