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Lecture 5: REs, DFAs, NDFAs. 1

Regular Expressions

Let = {a, b}. The regular expressions over arecertain strings over the alphabet {a, b, (, ),,, }

Construction Rules:1., a and b are regular expressions2. If and are REs, then so are ( ), ( ), and

.

Examples:(ab)(((ab)c) (ba))

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Lecture 5: REs, DFAs, NDFAs. 2

What REs Do

Regular expressions (which are strings) represent languag(which are sets of strings), via the function L:

(1) L(a) = {a}(2) L(b) = {b}

(3) L() = (4) L(( )) = L()L( )(5) L(( )) = L() L( )(6) L( ) = L()

Example:

L((ab)) = L(a)L(b) (4)= L(a)L(b) (6)= {a}{b} (1, 2)

L is called the semantics .

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Lecture 5: REs, DFAs, NDFAs. 3

Syntactic Shorthand

Union and concatenation of languages are associativei.e., for any languages L1 , L2 , L3 :

(L1 L2 )L3 = L1(L2 L3 )(L1 L2 ) L3 = L1 (L2 L3 )

Thus, we can leave out many parentheses when writingREs, without changing the meaning [semantics]

For example, the following are all equivalent:((ab)c)(a(bc))abc

even though only the rst two are official REs

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Lecture 5: REs, DFAs, NDFAs. 4

Regular Languages

A language is regular iff there is a regular expressionthat represents it.

Examples:

Strings ending in a = (a b)a

Strings of even length = (aa abbabb)

Strings with even # of as = (baba)

= b(abab)

Strings with two as = b bab babab

= b(e a(b(e ab)))

Decimal numerals in standard form, no leading zeroes= 0 ((1 . . . 9)(0 . . . 9))

[Where + = and e = ]

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Lecture 5: REs, DFAs, NDFAs. 5

A more interesting regular language

L = All strings with an even # of as and an even #of bs

= (baba) (a bab)

but this isnt a regular expression

= ( aabb)

((abba)(aabb)

(abba)(aabb)

)

=

Claim: x L() iff x L( ) Clear that each member of L() has an even # of

as and even # of bs( ) If x has an even # of as and an even # of bs, then

x L()Pf: x = 1 1 . . . n n , i , i , n 0Each pair i i is either:

An equal pair aa or bb An unequal pair ab or baEach unequal pair must be matched with a subseque

unequal pair

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Lecture 5: REs, DFAs, NDFAs. 6

Deterministic Finite Automata (DFAs)

a b b a b a

1

23

4

Input tape

Start state marked with