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Lect 8 - October 21, 2008 - ECS 20 - Fall 2008 - Phil Rogaway
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Today: o Regular languages, continued
       o Relations

- SCRIBE  (none today, use your own notes for the class notes) 

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1. Reviews strings, languages, and their operators 
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 Strings  emptystring
          concatenation
          length 
          reversal
          substring extraction
          x^k 

 Language concatenation operator
          L^k
          Kleene closure
          union, intersect, complement

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2. Regular languages - the languages of regular expressions
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 Formally, given an alphabet Sigma, the following are the regular
 expressions over Sigma:

 -a, for all a\in\Sigma
 -emptystring symbol
 -emptyset symbol
 -(x o y), (x u y), (x*)  for any regular expressions x,y 


 Omit parenthesis with the understanding that o binds most strongly,
 then *, then u, and things group left-to-right, 

 L(\alpha) - the language of a regular expression - defined
 recursively in the natural way.

 Exercise 1: write a regular expression for all strings over {a,b}
 whose length is divisible by 3. 
       (aub)(aub)(aub))*

 Exercise 2: write a regular expression for all strings over {a,b}
 whose length is NOT divisible by 3. 
       (aub)(aub)(aub))*(aub) u 
       (aub)(aub)(aub))*(aub)(aub)

 Exercise 3: write a regular expression for all strings over {0,1}
 that contain _some_ '111'.
       (0u1)* 111 (0u1)*

 Exercise 4: write a  regular expression for all strings over {0,1}
       that contain an even # of 0's and an even # of 1's.
              Hmm......   sounds HARD!
              Can it be done?!  
 
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3. DFAs
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 Describe pictorially (no actual definition). 
 What they capture: a computer with a finite amount of memory.

 Make a DFA for each of the languages above.


 Theorem (ecs 120): L is the language of a regular expression
                    iff 
                    L is the language of a DFA 

 So now we _know_ that example 4 does have a regular expression,
 just our own stupidity not to be able to write it.

 Nice new ways to describe languages: give a DFA. 

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4. Relations
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(Change of topics.  But do define some relations on 
strings, regular languages, and DFAs to tie the 
two topics together.)

DEF: A and B sets.  Then a *relation* R is subset of A x B.

R \subseteq A x B

Variant notation:  x R y  for (x,y) \in R

May use a symbol like ~ or < for a relations
                    x ~ y  if (x,y) \in ~

Relations in arithmetic, where A and B are both natural numbers:
           = < <= > >=
           |   divides

     what about succ, +, *   NO: function symbols, not relations

In set theory:
           in   = subseteq  
           what about emptyset  NO: constant symbol

Relations are useful for things other than numbers and sets and the like:

S = all UCD students for F08
C = all UCD classes  for F08
P = all UCD professors for F08

E: enrolled relation \subseteq S x C
s E c  (ie, (s,c)\in E) - x is taking class y 

T: teaches relation \subseteq C x P 
c T p  (ie, (c,p)\in T) - professor p is teaching class c this term

You can *compose* relations

what _should_ 
   E o T 
mean,do you think
    E o T \subseteq S x P        S x C   C x P -> S x P
    s EoT p  if there exists c in C such that s E c and c T p --
             student s is taking some course that p is teaching -- 
             p is s's teacher this term

What I've just given is the general definition
R \subseteq X x Y
S \subseteq Y x Z  then R o S \subseteq X x Z is {(x,z): \exists y in Y xRy and yRz}

DRAW A GRAPH -- and give the directed-graph interpretation of 

            Students         Classes         Instructors
            --------         -------         ------------

            John             ECS20           Prof. Rogaway

            Nina             Math118         Prof. Hamann

            Till             ECS20           Prof. Morrison

            Elizabeth        Lin 10          Prof. Samuelson
            
            Ali



What _should_  E^{-1} should be?
   formalize
   
   if R\subseteq X x Y is a relation that R^{-1} is the relation on Y x X
   where (y,x) R^{-1} iff x R y.


More examples:
    Often X = Y is the *same* set
      Relations on natural numbers, real numbers, strings, etc.

    X = set of strings 

    x <= y    "is a substring of y"


    alpha and bet are regular expressions.
    alpha ~ beta if L(alpha) = L(beta)


T/F:   (0u1)^*(0u1)^*  ~ (00 u 01 u 10 u 11)^*    TRUE
       emptystring    ~  emptystring*             TRUE
       0(0u1)0       ~ 1(0u1)1                    FALSE


    M and M' are DFAs.  
    M ~ M' if L(M) = L(M')
   
    M subseteq M' if L(M) subseteq L(M')



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5. Properties of Relations
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Reflexive:  x R x                    for all x
  Symmetric:  x R y -> y R x           for all x,y
  Transitive: x R y and y R x -> x R z for all x, y, z
  
if R has all three properties, R is said to be an equivalence relation

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