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
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