Computer Systems Lecture 19 NFAs and Regular Expressions CS 230 - - PowerPoint PPT Presentation

CS 230 – Introduction to Computers and Computer Systems Lecture 19 – NFAs and Regular Expressions CS 230 - Spring 2020 4-1

Non-deterministic Finite Automata (NFA)  The same as a DFA except that:  NFAs may have transitions from the same state on the same input to different states  DFAs can only have one transition per input per state  NFAs can include an ε (epsilon) transition  Move to a new state without consuming input  Easier to design than equivalent DFA  More complex to evaluate input  Can always create a DFA from an NFA  All DFAs are also legal NFAs CS 230 - Spring 2020 4-2

NFA Example b a c ε a  Start state has two a transitions  Bottom state has an epsilon transition to an accept state CS 230 - Spring 2020 4-3

NFA Example b a c ε a Lets consider the input string: acb CS 230 - Spring 2020 4-4

NFA Example b a c ε a Lets consider the input string: acb CS 230 - Spring 2020 4-5

NFA Example b a c ε a acb Lets consider the input string: Now we have two (?) choices so we create “clones” CS 230 - Spring 2020 4-6

NFA Example b a c ε a acb Lets consider the input string: Wait! There’s one more possibility! Add a clone. CS 230 - Spring 2020 4-7

NFA Example b a c ε a acb Lets consider the input string: Two clones got stuck and died! CS 230 - Spring 2020 4-8

NFA Example b a c ε a acb Lets consider the input string: At end and at least one clone is in an accept state CS 230 - Spring 2020 4-9

NFA Example b a c ε a acb Lets consider the input string: The NFA accepts acb CS 230 - Spring 2020 4-10

NFA Example 2 b b a ε  Middle state has two b transitions  Start state has an epsilon transition to the accept state CS 230 - Spring 2020 4-11

NFA Example 2 b b a ε abab Lets consider the input string: CS 230 - Spring 2020 4-12

NFA Example 2 b b a ε Lets consider the input string: abab CS 230 - Spring 2020 4-13

NFA Example 2 b b a ε Lets consider the input string: abab CS 230 - Spring 2020 4-14

NFA Example 2 b b a ε Lets consider the input string: abab CS 230 - Spring 2020 4-15

NFA Example 2 b b a ε abab Lets consider the input string: At end and at least one clone is in an accept state CS 230 - Spring 2020 4-16

NFA Example 2 b b a ε abab Lets consider the input string: The NFA accepts abab CS 230 - Spring 2020 4-17

Try it Yourself a b c c ε b bcbb Does this NFA accept the string: CS 230 - Spring 2020 4-18

Try it Yourself a b c c ε b bcbb Does this NFA accept the string: CS 230 - Spring 2020 4-19

Try it Yourself a b c c ε b bcbb Does this NFA accept the string: CS 230 - Spring 2020 4-20

Try it Yourself a b c c ε b bcbb Does this NFA accept the string: CS 230 - Spring 2020 4-21

Try it Yourself a b c c ε b bcbb Does this NFA accept the string: CS 230 - Spring 2020 4-22

Try it Yourself a b c c ε b bcbb Does this NFA accept the string: The NFA rejects bcbb CS 230 - Spring 2020 4-23

Try it Yourself Draw an NFA for the language of [an even number of lowercase c ] or [an odd number of lowercase d ]. CS 230 - Spring 2020 4-24

Try it Yourself Draw an NFA for the language of [an even number of lowercase c ] or [an odd number of lowercase d ]. ε ε CS 230 - Spring 2020 4-25

Try it Yourself Draw an NFA for the language of [an even number of lowercase c ] or [an odd number of lowercase d ]. d ε d ε CS 230 - Spring 2020 4-26

Try it Yourself Draw an NFA for the language of [an even number of lowercase c ] or [an odd number of lowercase d ]. d ε d ε c c CS 230 - Spring 2020 4-27

Regular Expression  Regular expressions (also called regexs ) are another way to define regular languages  Define set of strings over alphabet ∑  ∑ is the set of all legal characters in language  Constants:  empty set: Ø  empty string: ε  literal character: a ∈ ∑ CS 230 - Spring 2020 4-28

Basic Regex Operations  Alternation: R|S = R U S  U is the union operator  “R or S” (exclusive or)  Concatenation: RS = { αβ : α in R and β in S}  “R followed by S”  Kleene star: R* = smallest superset of R containing ε and closed under concatenation  “Zero or more copies of R” CS 230 - Spring 2020 4-29

Basic Regex Examples  a* = { ε , a, aa, aaa, ... }  b|a* = { b, ε , a, aa, aaa, ... }  (0|1)* = binary numbers, plus empty string  (h|c)at = { hat, cat }  (a|b)(c|d) = { ac, ad, bc, bd }  while = { while } CS 230 - Spring 2020 4-30

More Regex Operations  a+ = { a, aa, aaa, ... }  a? = { ε , a }  ab+ = { ab, abb, abbb, ... }  [a-dqh] = {a, b, c, d, q, h}  (h|c)?at = { hat, cat, at }  (a|b)+(c|d) = { ac, ad, bc, bd, aac, aad, bbc, bbd, aaac, aaad, bbbc, bbbd, abc, abd, abac ... }  wh?il[e-g] = { while, whilf, whilg, wile, wilf, wilg } CS 230 - Spring 2020 4-31

Regex Precedence [] > () > * > + > ? > concatenation > |  Regex operators have precedence  allows us to make regular expressions without overusing parenthesis  ab+ = {ab, abb, abbb, abbbb, ...}  c|de = {c, de} CS 230 - Spring 2020 4-32

Try it Yourself Write a regular expression for the language of all strings beginning with an optional lowercase a , followed by two or more copies of any other lowercase letter, followed by a lowercase a . CS 230 - Spring 2020 4-33

Try it Yourself Write a regular expression for the language of all strings beginning with an optional lowercase a , followed by two or more copies of any other lowercase letter, followed by a lowercase a . a? CS 230 - Spring 2020 4-34

Try it Yourself Write a regular expression for the language of all strings beginning with an optional lowercase a , followed by two or more copies of any other lowercase letter, followed by a lowercase a . a?[b-z][b-z] CS 230 - Spring 2020 4-35

Try it Yourself Write a regular expression for the language of all strings beginning with an optional lowercase a , followed by two or more copies of any other lowercase letter, followed by a lowercase a . a?[b-z][b-z]+ CS 230 - Spring 2020 4-36

Try it Yourself Write a regular expression for the language of all strings beginning with an optional lowercase a , followed by two or more copies of any other lowercase letter, followed by a lowercase a . a?[b-z][b-z]+a CS 230 - Spring 2020 4-37

Regular Expressions and DFA/NFA  Regular expressions define regular languages  NFAs and DFAs define regular languages  Simple conversion rules:  a* a? a a a a  a+ a|b a b CS 230 - Spring 2020 4-38

Regex to DFA Example  Regular expression: (ab)?(cd)+  We know we start with an optional ab so lets use that rule first: a b CS 230 - Spring 2020 4-39

Regex to DFA Example  Regular expression: (ab)?(cd)+  Now we add one or more copies of cd  But now how to we have the ab first then cd ? c c d a b CS 230 - Spring 2020 4-40

Regex to DFA Example  Regular expression: (ab)?(cd)+  We need to add a c arrow and remove the extra accept state. Now it works! c c d a c b CS 230 - Spring 2020 4-41

DFA to Regex Example b b b a a  Write out as many accepted strings as you can find until you see a pattern:  aa, bb, bbb, bbbb CS 230 - Spring 2020 4-42

DFA to Regex Example b b b a a  We can see a choice of either two a or at least two b  Regex: aa|bb+ or aa|bbb* CS 230 - Spring 2020 4-43

Try it Yourself Describe the language accepted by the following DFA then write a regular expression that accepts the same language. q q r q r q CS 230 - Spring 2020 4-44

Try it Yourself Describe the language accepted by the following DFA then write a regular expression that accepts the same language. q q r q r q Strings accepted: r, qr, qqr, qqqr, rq, rqq, rqqq, qrq, qrqq, qqrqq, qqqrqq, qqrqqq, qqqrqqq, ... CS 230 - Spring 2020 4-45

Try it Yourself Describe the language accepted by the following DFA then write a regular expression that accepts the same language. q q r q r q This DFA accepts the language of strings containing a single lowercase r and any number of lowercase q . CS 230 - Spring 2020 4-46

Try it Yourself Describe the language accepted by the following DFA then write a regular expression that accepts the same language. q q r q r q This DFA accepts the same language as the regular expression: q*rq* CS 230 - Spring 2020 4-47

Regex Syntax Extensions  Dot matches any single character  .at matches hat, cat, fat, mat, bat, 7at, Aat, etc.  Escape character  match actual brackets, dots, plus signs, etc.  [0-9]+\.[0-9]+ to match fractional numbers CS 230 - Spring 2020 4-48

Real-world Example  Search for all occurrences of a name in a file  Using unix command “ egrep ”  Different spelling for George Friedrich Händel:  Händel  Haendel  Handel  Hendel egrep "[Hh](ae|a|e|ä)ndel" ex.txt CS 230 - Spring 2020 4-49