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

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

Apr 11, 2023 187 likes •686 views

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

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CS 230 – Introduction to Computers and Computer Systems Lecture 19 – NFAs and Regular Expressions

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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

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NFA Example

a b ε a c

 Start state has two a transitions  Bottom state has an epsilon transition to an

accept state

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NFA Example

a b ε a

Lets consider the input string: acb

c

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NFA Example

a b ε a

Lets consider the input string: acb

c

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NFA Example

a b ε a

Lets consider the input string: Now we have two (?) choices so we create “clones”

acb

c

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NFA Example

a b ε a

Lets consider the input string: Wait! There’s one more possibility! Add a clone.

acb

c

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NFA Example

a b ε a

Lets consider the input string: Two clones got stuck and died!

acb

c

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NFA Example

a b ε a

Lets consider the input string: At end and at least one clone is in an accept state

acb

c

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NFA Example

a b ε a

Lets consider the input string: The NFA accepts acb

acb

c

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NFA Example 2

a b

 Middle state has two b transitions  Start state has an epsilon transition to the

accept state

ε b

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NFA Example 2

a b ε b

Lets consider the input string:

abab

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NFA Example 2

a b ε b

Lets consider the input string: abab

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NFA Example 2

a b ε b

Lets consider the input string: abab

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NFA Example 2

a b ε b

Lets consider the input string: abab

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NFA Example 2

a b ε b

Lets consider the input string: At end and at least one clone is in an accept state

abab

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NFA Example 2

a b ε b

Lets consider the input string: The NFA accepts abab

abab

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Try it Yourself

a ε b c c b

Does this NFA accept the string:

bcbb

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Try it Yourself

a ε b c c b

Does this NFA accept the string:

bcbb

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Try it Yourself

a ε b c c b

Does this NFA accept the string:

bcbb

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Try it Yourself

a ε b c c b

Does this NFA accept the string:

bcbb

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Try it Yourself

a ε b c c b

Does this NFA accept the string:

bcbb

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Try it Yourself

a ε b c c b

Does this NFA accept the string: The NFA rejects bcbb

bcbb

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Try it Yourself

Draw an NFA for the language of [an even number of lowercase c] or [an odd number of lowercase d].

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Try it Yourself

ε

Draw an NFA for the language of [an even number of lowercase c] or [an odd number of lowercase d].

ε

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Try it Yourself

ε

Draw an NFA for the language of [an even number of lowercase c] or [an odd number of lowercase d].

ε d d

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Try it Yourself

c ε

Draw an NFA for the language of [an even number of lowercase c] or [an odd number of lowercase d].

c ε d d

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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 ∈ ∑

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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”

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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 }

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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 }

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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}

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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.

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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?

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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]

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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]+

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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

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Regular Expressions and DFA/NFA

 Regular expressions define regular languages  NFAs and DFAs define regular languages  Simple conversion rules:

 a* a?  a+ a|b

a a a a a b

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Regex to DFA Example

 Regular expression: (ab)?(cd)+  We know we start with an optional ab so lets

use that rule first:

a b

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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 ?

a b d c c

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Regex to DFA Example

 Regular expression: (ab)?(cd)+  We need to add a c arrow and remove the extra

accept state. Now it works!

a b d c c c

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DFA to Regex Example

a b

 Write out as many accepted strings as you can find

until you see a pattern:

 aa, bb, bbb, bbbb

a b b

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DFA to Regex Example

a b

 We can see a choice of either two a or at least two b  Regex: aa|bb+ or aa|bbb*

a b b

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Try it Yourself

r q

Describe the language accepted by the following DFA then write a regular expression that accepts the same language.

q r q q

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Try it Yourself

r q

Describe the language accepted by the following DFA then write a regular expression that accepts the same language. Strings accepted: r, qr, qqr, qqqr, rq, rqq, rqqq, qrq, qrqq, qqrqq, qqqrqq, qqrqqq, qqqrqqq, ...

q r q q

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Try it Yourself

r q

Describe the language accepted by the following DFA then write a regular expression that accepts the same language. This DFA accepts the language of strings containing a single lowercase r and any number of lowercase q.

q r q q

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Try it Yourself

r q

Describe the language accepted by the following DFA then write a regular expression that accepts the same language. This DFA accepts the same language as the regular expression: q*rq*

q r q q

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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

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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