Lecture 2 Alphabets, Strings and Languages

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Lecture 2Alphabets, Strings and Languages

CSCE 355 Foundations of Computation

• Topics
• What I forgot from last lecture
• Proof techniques continued
• Alphabets, strings, languages
• Automata

Aug 27, 2007
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What I forgot from last lecture

• Slides skipped or not well covered
• 3 Course outcomes
• 8 Examples of Relations
• 9 POSETS
• 10 Equivalence relations
• 15 Equality of Sets
• 17 If and only if
• 25 Structural Induction Proof

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http//infolab.stanford.edu/ullman/ialc/slides/sl
ides1.pdf
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Recursive Def of Arithmetic Expressions

• Basis a number or a variable is an expression.
• If E and F are expressions then a new expression
  G can be formed by applying one of the three
  rules
• G E F
• G E F
• G ( E )

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Graphs Visual representation of relations
(binary)

• a R b if and only if a?b in the graph

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The Pigeon Hole Principle

• Proof techniques Continued
• If you have n boxes and more than n balls to put
  in the boxes then you must be two balls (at least
  two) in the same box.
• Formally If A and B are sets with A gt B then
  there is no 1-1 function from A to B.

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Mutual Inductions

• Example 1.23 On-Off pushbutton automaton

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Languages

• Alphabet a finite set of symbols
• String finite sequence of characters from an
  alphabet
• Empty string, length of string
• Language (over an alphabet)

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Operations on Strings
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Examples of Languages
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Operations on Languages

• Suppose S and T are languages (sets of strings)
• Union, intersection, complement
• concatenation

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Powers, Kleene Closure

• S1 S
• Sn S Sn-1
• How would you prove Sn Sm Snm ?
• What is S0?
• S

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Some Special Languages
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Finite Automata - Informally
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Finite Automata - formally

• A Deterministic Finite Automata (DFA) is a
  5-tuple

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Transition Diagrams Transition Tables

• Conventions
• Notes
• Number of out-arcs
• Dead state

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Path determined by a string
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Language accepted by a DFA
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Example L(M) for DFA M
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Given L find DFA for it
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Important application of Pigeon Hole Principle to
DFAs
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Regular Expressions
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Homework

• Whats wrong with Ullmans proof of if a
  complete binary tree has n leaves then it has
  2n-1 nodes. (extra credit)
• Operations on StringsGiven strings s abc and
  t12
• What are st , s2, and s3?
• Is e (empty string) in every language?
• Is ? a language? It is a sublanguage of every
  language?
• Operations on Languages. Given S a, b, ab and
  T 1, 22
• What is ST?
• What is TS?
• What is T2? T3?

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• 3. DFA recognizing the following languages
• w in a,b each a is immediately preceded
  by a b
• w in a,b neither aa nor bb is a substring
  of w
• 4. What language does the DFA below recognize

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References Mathematical Foundations

• http//infolab.stanford.edu/ullman/ialc.html -
  The website for the textbook
• Extended Proof techniques
• http//www.maths.uwa.edu.au/berwin/humour/invalid
  .proofs.html
• Fair Use Books Online
• http//fair-use.org/bertrand-russell/the-principle
  s-of-mathematics/
• Books
• Dr. Euler's Fabulous Formula Cures Many
  Mathematical Ills