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Representation of Languages

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• Need finite descriptions of infinite sets of strings (=> specify languages).

• Discover and specify “regularity”.

• What is at all possible at all possible by the cardinality arguments?

• What is admissible admissible by the machines?

Fundamental Problems

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• The set of languages over a finite alphabet is uncountable, while the set of descriptions is countable (equi-numerous with N). So, all languages cannot be described finitely.

• Given that there are uncountable subsets of languages, which countably many subsets of these languages can be specified in any given formalism?

Fundamental Problems

Visualizing Languages : Cardinality vs Representation

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Uncountable Languages

Representable Countable Languages

Un-Representable Countable Languages

Spaces enclose Collections of

Languages

CFLs;RLs

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Language = set of strings

• Languages inherit operation on sets.• E.g., union, intersection, complement, etc.

• Operations on strings can be lifted to operations on languages.

• E.g., language concatenation, etc.

} | { YvXuuvXY

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Integer Power of a Language

n length of stringsBit

...}1,0}{11,10,01,00{

}11,10,01,00{

}1,0{

}{

}1,0{

23

2

1

0

nX

XXX

X

X

X

X

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Kleene Star and Kleene Plus

XX

XXXX

XXXXX

i

i

i

i

i

i

that Note

......

......

1

1

10

0

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Regular Sets

Family of languages • Seed elements:

• Empty language

• Language containing the empty string

• Singleton language for each letter in the alphabet

• Closure Operations:• Union: collects strings from languages

• Concatenation: generates longer strings

• Kleene Star: generates infinite languages

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Regular Sets over

• Basis: are regular sets over .

• Inductive Step: Let X and Y be regular sets over . Then so are:

• Closure:…

}{: and },{, aa

*X

XY

Y X

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Examples

• Bit strings containing at least a “1”

• Bit strings containing exactly one “1”

• Bit strings beginning or ending with a “1”

*})1{}0}({1{*})1{}0({

*}0}{1{*}0{

}1{*}1,0{ *}1,0}{1{

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Naming Languages

• Regular sets can be named using the derivation in terms of the seed elements and the closure operations. Regular expressions formalize this approach.

Regular sets :: Regular Expressions

:::: Numbers :: Numerals

:::: Semantics Semantics :: SyntaxSyntax

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• Basis: are regular expressions over .

• Inductive Step: Let x and y be regular expressions over . Then so are:

• Closure:…

Regular Expressions over aa : and ,,

. and , , *xxyyx

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Examples: Syntax Syntax vs SemanticsSemantics

• Regular Expressions • Regular sets

{},, }{

}0{) (0 0

0

0

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• Regular expressions for strings over {a,b} containing at least one “a”.

• Focus on the one “a”(a u b)*a(a u b)*

• Focus on the leftmost “a”b*a(a u b)*

• Focus on the “a”sb*ab*(ab*)*

• Further optimizationb*(ab*)+

(Motivates equivalence problem)

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• Equivalence of regular expressionsTwo regular expressions are equivalent if they represent the same regular set.

• “Irregularity” (to be taken on faith now ) There are non-regular languages such as

Informally, a finite amount of extra storage is sufficient for recognizing regular languages (or the patterns that characterize it).

Regular languages cannot capture “counting”.

}0 | ){( nnn

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• Complement of bit strings with at least one “1”

= bit strings containing no “1”s

= 0*• Complement of

bit strings with exactly one “1”

= bit strings containing no “1”s

U bit strings with at least two “1”s

= 0* U (0* 1 0* 1 0*)(0 U 1)*