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Lexical Analysis Part 2

Lexical Analysis Part 2. Professor Yihjia Tsai Tamkang University. Regular Expressions and FA are Equivalent. For every regular expression, there is a deterministic finite-state machine that defines the same language, and vice versa. NFA. Regular expressions. DFA. e Closure Operator.

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Lexical Analysis Part 2

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  1. Lexical Analysis Part 2 Professor Yihjia Tsai Tamkang University

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

  3. eClosure Operator • The eClosure operator is defined as eClosure(s) = { s } U states reachable from s using e transitions. • Example: eClosure(1) = {1,3} a  start 1 5 3 a a/b b 2 4

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

  5. e a b e e c 1 6 3 2 5 4 e b b a A C B D c c Comparison from Last Time NFA: DFA:

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

  7. Example: aa*b | aab*

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

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

  10. Example GNFA 0110 0  qS qA  01* 0* | 11

  11. Characteristics of GNFA’s  • :(Q\{qaccept})(Q\{qstart})  R The interior Q\{qaccept,qstart} is fully connected by  From qstart only ‘outgoing transitions’ To qaccept only ‘ingoing transitions’ Impossible qiqj transitions are “(qi,qj) = ” Observation: This GNFA:recognizes the language L(R) RR qS qA

  12. qj DFA M  Equivalent GNFA M’ Let M have k states Q={q1,…,qk} - Add two states qaccept and qstart  q1 qS - Connect qstart to earlier q1:  qj qA - Connect old accepting states to qaccept  - Complete missing transitions by qi - Join multiple transitions: 1 becomes 0|1 qj qi qj qi 0

  13. Remove Internal state of GNFA If the GNFA M has more than 2 states, ripinternal qrip to get equivalent GNFA M’ by: - Removing state qrip: Q’=Q\{qrip} - Changing the transition function  by’(qi,qj) = (qi,qj) | ((qi,qrip)((qrip,qrip))*(qrip,qj)) for every qiQ’\{qaccept} and qjQ’\{qstart} R1 qrip R2 R4|(R1R2*R3) qj qi qi = R3 qj R4

  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, 1988. ISBN 0-201-10088-6. • Appel, A., Modern Compiler Implementation In Java (2nd Ed), Cambridge University Press, 2002. ISBN 052182060X.

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