Advanced Discrete Math

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Advanced Discrete Math

• Intro to Formal Languages

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Languages

• A language is a set of strings of symbols drawn
  from some finite set S called the alphabet
• Example S 0,1, L x Î S x has an even
  number of 0s
• Example S a, b, , z, Lx x is a word in
  the Merriam-Webster unabridged dictionary

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Languages

• Languages have many uses in computer science
• Programming languages (e.g., Java)
• Descriptions of sets (e.g., set of valid US
  telephone numbers)
• Descriptions of expression syntax (e.g., the set
  of valid arithmetic expressions)

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Languages

• For a given alphabet S, how many different
  languages can be defined?
• Let S represent the set of all strings (of any
  length) over S
• The set of languages is the power set of S,
  written 2 S
• This set is uncountably infinite!

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Languages

• If a finite set of rules is used to write down a
  description of a language, then the set of
  languages described is countably infinite
• So we cant hope to come up with a set of rules
  to describe all languages!
• However, certain sets of language-writing rules
  have interesting properties and are widely used

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Language Classes

• Regular languages widely used to represent
  syntax of strings (telephone numbers, arithmetic
  expressions, etc.)
• Context-free languages most programming
  languages belong to this class
• Recursive languages those sets that can be
  recognized by some computer program that always
  terminates

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Regular Languages

• A language over S is regular if it can be
  described using the following recursive rules
• The empty set is a regular language
• Any set a where a Î S is a regular language
• The union or concatenation of two regular
  languages is a regular language
• The Kleene star of a regular language is a
  regular language

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Regular Languages

• Concatenation of sets of strings is Cartesian
  product
• a,bc,d ac, ad, bc, bd
• Kleene star of a set is the set of all
  concatenations of the set with itself (including
  the set of no concatenations, producing empty
  string e)
• a,b e, a, b, aa, ab,

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Regular Languages

• We can describe a regular language using a
  shorthand known as a regular expression
• Dont use s for sets
• Ex a represents a, e represents e
• Ex a(bUc) represents the set a, ab, ac, abb,
  abc, acb, acc, abbb,

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Regular Languages

• Regular expressions are used widely in computer
  science
• Support built in to Java 1.5, Perl, JavaScript,
• System tools such as grep (generalized regular
  expression processor)
• Emacs regexp searches

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Regular Languages

• A graphical representation of regular languages
  finite state automaton (FSA)

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Regular Languages

• Theorem (Kleene) a language is regular iff it is
  accepted by an FSA
• Proof
• There is a procedure that converts any regular
  expression w into a nondeterministic FSA that
  accepts L(w)
• There is a procedure that converts any
  nondeterministic FSA to an FSA
• There is a procedure that converts any FSA M to a
  regular expression representing L(M)

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Regular Languages

• Writing regular expressions
• Often easiest to start with non-deterministic FSA
• Follow procedure for converting to equivalent
  regular expression

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Regular Languages

• Many interesting languages are not regular
• Set of bit strings having an equal number of 0s
  and 1s
• Intuition cannot remember how many 0s have been
  seen with a finite number of states
• Proving that a language is not regular formalizes
  this intuition Pumping lemma

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Context-free Languages

• A CFL is described using a grammar
• A context-free grammar (CFG) consists of a set of
  replacement rules.
• Left side of each rule is a single non-terminal
  symbol (symbol not in S)
• Right side is a string of terminal symbols (in S)
  and nonterminals
• One nonterminal is the start symbol

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Context-free Languages