Mathematical Preliminaries Hw (p.13) 1, 4, 7, 8, 9, 13,  23, 26, 30, 32
<ul><li>Mathematical Preliminaries </li></ul><ul><li>Sets  </li></ul><ul><li>Functions </li></ul><ul><li>Relations </li></...
A set is a collection of elements SETS We write 1  is a member (or element) of set  A ship  is not a member (or element) o...
<ul><li>Set Representations </li></ul><ul><ul><li>C = { a, b, c, d, e, f, g, h, i, j, k } </li></ul></ul><ul><ul><li>C = {...
A = { 1, 2, 3, 4, 5 } Universal Set :  all possible elements  <ul><ul><ul><li>U = { 1 , … , 10 } </li></ul></ul></ul>1 2 3...
<ul><li>Set Operations </li></ul><ul><li>A = { 1, 2, 3 }  B = { 2, 3, 4, 5} </li></ul><ul><li>Union   </li></ul><ul><li>A ...
A <ul><li>Complement </li></ul><ul><ul><ul><ul><li>Universal set = {1, …, 7}  </li></ul></ul></ul></ul><ul><ul><ul><ul><li...
0 2 4 6 1 3 5 7 even { even integers }  =  { odd integers } odd Integers
DeMorgan’s Laws A U B = A  B U A  B = A U B U
Empty, Null Set: = { } S U  = S S  =  S -  = S - S = U = Universal Set
Subset A = { 1, 2, 3}  B = { 1, 2, 3, 4, 5 } Proper Subset: A B A  B U A  B U
Disjoint Sets A = { 1, 2, 3 }  B = { 5, 6}  A B A  B =  U
Set Cardinality <ul><li>For finite sets </li></ul>A = { 2, 5, 7 } |A| = 3 (set size)
Powersets A powerset is a set of subsets Powerset of S = the set of all the subsets of S S = { a, b, c } 2 S  = {  , {a}, ...
Cartesian Product A = { 2, 4 }  B = { 2, 3, 5 } A X B = { (2, 2), (2, 3), (2, 5), ( 4, 2), (4, 3), (4, 5) } A X B  is an  ...
Relation from sets A to B <ul><li>Ex., A = { 1, 2, …, 7 }, B={ 1, 2, …, 50} </li></ul><ul><ul><li>{(x, y): x  A, y  B, a...
FUNCTIONS domain 1 2 3 a b c range f : A -> B  A B If A = domain  then f is a total function otherwise f is a partial func...
GRAPHS A directed graph <ul><li>Nodes (Vertices)   </li></ul><ul><li>V = { a, b, c, d, e } </li></ul><ul><li>Edges (Ordere...
Labeled Graph a b c d e 1 3 5 6 2 6 2
Walk Walk is a sequence of  adjacent  edges (e, d), (d, c), (c, a)  is a walk from  e  to  a  of length 3  (or denoted as ...
Path a b c d e Path  is a walk where no edge is repeated Simple path : no node is repeated
Path a b c d e Path  is a walk where no edge is repeated Simple path : no node is repeated (e, b), (b, e), (e, d), (d, c),...
Cycle a b c d e 1 2 3 Cycle :   a walk from a node (base) to itself without  repeated edges Simple cycle :   only the base...
Find All  Simple Paths  starting from c  a b c d e origin The longest simple path has at most length 4. Since every vertex...
(c, a) (c, e) Step 1 a b c d e origin Starting from vertex c, list all outgoing edges as long as they do not lead to any v...
(c, a) (c, a), (a, b) (c, e) (c, e), (e, b) (c, e), (e, d) Step 2 a b c d e origin
Step 3 a b c d e origin (c, a) (c, a), (a, b) (c, a), (a, b), (b, e) (c, e) (c, e), (e, b) (c, e), (e, d)
Step 4 a b c d e origin (c, a) (c, a), (a, b) (c, a), (a, b), (b, e) (c, a), (a, b), (b, e), (e,d) (c, e) (c, e), (e, b) (...
Trees are connected directed graphs without cycles such that there is a special vertex called “root” having exactly one pa...
root leaf Level 0 Level 1 Level 2 Level 3 Height 3 The  level  associated with each vertex is the number of edges in the p...
Binary Trees A  binary tree  is a tree in which no parent can have more than two children. A  binary tree  is a tree in wh...
PROOF TECHNIQUES <ul><li>Proof by induction </li></ul><ul><li>Proof by contradiction </li></ul>
Induction We have statements   P 1 , P 2 , P 3 , …  <ul><li>If we know </li></ul><ul><ul><ul><li>for some m that P 1 , P 2...
Proof by Induction <ul><li>Inductive basis </li></ul><ul><ul><ul><li>Find P 1 , P 2 , …, P m  which are true </li></ul></u...
Example Theorem:   A binary tree of height  n has at most  2 n   leaves.  (p.10) We want to show:   L(n)  ≦  2 n   for n =...
<ul><li>We want to show:   L(n)  ≦  2 n  for n = 0, 1, 2,….   </li></ul><ul><li>Inductive basis  </li></ul><ul><ul><ul><ul...
Induction Step From Inductive hypothesis: height k k+1 Let’s assume L(i)  ≦  2 i  for all i = 0, 1, …, k need to show that...
L(k)  ≦  2 k L(k+1)  ≦  2 · L(k)  ≦  2· 2 k   =  2 k+1 Induction Step height k k+1 (we add at most two nodes for every lea...
Remark <ul><li>Recursion is another thing </li></ul><ul><li>Example of recursive function: </li></ul><ul><ul><ul><li>f(n) ...
Proof by Contradiction <ul><li>We want to prove that a statement P is true </li></ul><ul><ul><ul><li>we assume that P is f...
Example <ul><li>Theorem:   is not rational </li></ul><ul><li>Proof: </li></ul><ul><ul><ul><li>Assume by contradiction that...
= n/m  2 m 2  = n 2   Therefore,  n 2   is even n is even n = 2 k 2 m 2  = 4k 2 m 2  = 2k 2 m is even m = 2 p Thus, m and ...
Languages
<ul><li>A language is a set of  strings </li></ul><ul><li>String:   A sequence of letters </li></ul><ul><ul><li>Examples: ...
Alphabets and Strings <ul><li>We will use small alphabets: </li></ul><ul><li>Strings </li></ul>
Empty String <ul><li>A string with no letters  </li></ul>
String Operations Concatenation w    = ?    w  = ?
Empty String <ul><li>Observations: </li></ul>
Reverse
String Length <ul><li>Length: </li></ul><ul><li>Examples: </li></ul>|    | = ?
Length of Concatenation <ul><li>Example:   </li></ul>
Example 1.8  (p.17)   | uv | = | u | + | v |   <ul><li>A recursive definition of the length of a string: </li></ul>|  a  |...
Example 1.8  (p.17)   | uv | = | u | + | v | <ul><li>For a string  u ,  consider the length of  uv , concatenation of  u  ...
Substring <ul><li>Substring of string:  a subsequence of  consecutive letters </li></ul><ul><li>String  Substring </li></ul>
Prefix and Suffix <ul><li>Prefixes  Suffixes </li></ul>prefix suffix
Another Operation <ul><li>Example: </li></ul>
Another Operation <ul><li>Definitions: </li></ul>
The * Operation <ul><li>: the set of all possible strings from </li></ul><ul><li>alphabet   </li></ul><ul><li>Example:  = ...
The  +  Operation : the set of all possible strings from alphabet  except
Languages Fall 2008 Automata
Note that: Sets Set size Set size String length
Another Example <ul><li>An infinite language </li></ul>
Operations on Languages <ul><li>The usual set operations </li></ul><ul><li>Complement: ?? </li></ul>
<ul><li>When we talk about a  language , we must know what ground does this language stands on ….. </li></ul>Languages We ...
Complement Example <ul><li>The complement of </li></ul>Universal Set?
Reverse <ul><li>Definition: </li></ul><ul><li>Examples: </li></ul>
Reverse Hw # 10 (a) Prove or Disprove:  i.e.,  w R    L     w     L R
Concatenation <ul><li>Definition: </li></ul><ul><li>Example:  </li></ul>
Concatenation Hw 8. Prove
Another Operation <ul><li>Definition: </li></ul><ul><li>Special case:  </li></ul>
More Examples
Star-Closure (Kleene *) <ul><li>Definition: </li></ul><ul><li>Example: </li></ul>
Positive Closure <ul><li>Definition: </li></ul>If   L then L +    L* - {  } It is  not  necessary that
True or False
True or False How to prove your answer?
Try Hw#  9 & 10(b) on p.28 What does  w  L 2  mean? What does  w  L * mean?
More Examples <ul><li>Consider a language on   = { a, b } </li></ul>What is  L 2   ?   L * ? e.g.
Grammars
Another Example <ul><li>Grammar: </li></ul><ul><li>Derivation of sentence  : </li></ul>
<ul><li>Grammar: </li></ul><ul><li>Derivation of sentence  : </li></ul>
<ul><li>Other derivations for  </li></ul>Grammar:
<ul><li>Language of the grammar </li></ul>
More Notation <ul><li>Grammar   </li></ul>Set of variables Set of terminal symbols Start variable Set of Production rules ...
Example <ul><li>Grammar   : </li></ul>
More Notation <ul><li>Sentential Form: </li></ul><ul><li>A sentence that contains  </li></ul><ul><li>variables </li></ul><...
<ul><li>We write: </li></ul><ul><li>Instead of: </li></ul>
<ul><li>In general we write: </li></ul><ul><li>If: </li></ul>
<ul><li>By default: </li></ul>
Example Grammar Derivations
Another Grammar Example <ul><li>Grammar  : </li></ul>Derivations: From  A   aAb   and  A    , we know A   ,   ab ,  a...
Language defined by a Grammar <ul><li>For a grammar  </li></ul><ul><li>with start variable  :  </li></ul>String of terminals
Example <ul><li>For grammar  : </li></ul>Since Pf: show L(G)   {a n b n+1 }  &  L(G)    {a n b n+1 }  from A   aAb we g...
A Convenient Notation In general, we need to give a proof that a given language indeed generated by a certain grammar. Bac...
Example <ul><li>For grammar  : </li></ul>To  show Pf: show L(G)   {a n b n+1 }  &  L(G)    {a n b n+1 }  from A   aAb w...
More Examples on Grammars <ul><li>Find grammars for  L  on { a, b } and give brief arguments to explain why they work. </l...
1. Problems on p.27  <ul><li>You should be able to do </li></ul><ul><li>2 ~ 17, 21 </li></ul><ul><li>Hand in:  9, 10, 11c,...
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  • Fall 2008 Automata
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  • Class1

    1. 1. Mathematical Preliminaries Hw (p.13) 1, 4, 7, 8, 9, 13, 23, 26, 30, 32
    2. 2. <ul><li>Mathematical Preliminaries </li></ul><ul><li>Sets </li></ul><ul><li>Functions </li></ul><ul><li>Relations </li></ul><ul><li>Graphs </li></ul><ul><li>Proof Techniques </li></ul>
    3. 3. A set is a collection of elements SETS We write 1 is a member (or element) of set A ship is not a member (or element) of set B Membership of a given set
    4. 4. <ul><li>Set Representations </li></ul><ul><ul><li>C = { a, b, c, d, e, f, g, h, i, j, k } </li></ul></ul><ul><ul><li>C = { a, b, …, k } </li></ul></ul><ul><ul><li>S = { 2, 4, 6, … } </li></ul></ul><ul><ul><li>S = { j : j > 0, and j = 2k for some integer k > 0 } </li></ul></ul><ul><ul><li>S = { j : j is nonnegative and even } </li></ul></ul>finite set infinite set
    5. 5. A = { 1, 2, 3, 4, 5 } Universal Set : all possible elements <ul><ul><ul><li>U = { 1 , … , 10 } </li></ul></ul></ul>1 2 3 4 5 A U 6 7 8 9 10
    6. 6. <ul><li>Set Operations </li></ul><ul><li>A = { 1, 2, 3 } B = { 2, 3, 4, 5} </li></ul><ul><li>Union </li></ul><ul><li>A U B = { 1, 2, 3, 4, 5 } </li></ul><ul><li>Intersection </li></ul><ul><ul><ul><li>A B = { 2, 3 } </li></ul></ul></ul><ul><li>Difference </li></ul><ul><ul><ul><li>A - B = { 1 } </li></ul></ul></ul><ul><ul><ul><li>B - A = { 4, 5 } </li></ul></ul></ul>U 2 3 1 4 5 2 3 1 Venn diagrams A B
    7. 7. A <ul><li>Complement </li></ul><ul><ul><ul><ul><li>Universal set = {1, …, 7} </li></ul></ul></ul></ul><ul><ul><ul><ul><li>A = { 1, 2, 3 } A = { 4, 5, 6, 7} </li></ul></ul></ul></ul>1 2 3 4 5 6 7 A A = A
    8. 8. 0 2 4 6 1 3 5 7 even { even integers } = { odd integers } odd Integers
    9. 9. DeMorgan’s Laws A U B = A B U A B = A U B U
    10. 10. Empty, Null Set: = { } S U = S S = S - = S - S = U = Universal Set
    11. 11. Subset A = { 1, 2, 3} B = { 1, 2, 3, 4, 5 } Proper Subset: A B A B U A B U
    12. 12. Disjoint Sets A = { 1, 2, 3 } B = { 5, 6} A B A B = U
    13. 13. Set Cardinality <ul><li>For finite sets </li></ul>A = { 2, 5, 7 } |A| = 3 (set size)
    14. 14. Powersets A powerset is a set of subsets Powerset of S = the set of all the subsets of S S = { a, b, c } 2 S = { , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} } Observation: | 2 S | = 2 |S| ( 8 = 2 3 )
    15. 15. Cartesian Product A = { 2, 4 } B = { 2, 3, 5 } A X B = { (2, 2), (2, 3), (2, 5), ( 4, 2), (4, 3), (4, 5) } A X B is an ordered set, i.e. A X B ≠ B X A |A X B| = |A|·|B| Generalizes to more than two sets A X B X … X Z
    16. 16. Relation from sets A to B <ul><li>Ex., A = { 1, 2, …, 7 }, B={ 1, 2, …, 50} </li></ul><ul><ul><li>{(x, y): x  A, y  B, and y= x 2 } </li></ul></ul><ul><ul><li>{(x, y): x  A, y  B, and x < y} </li></ul></ul><ul><ul><li>Are relations from A to B. </li></ul></ul><ul><li>A relation on set A </li></ul><ul><ul><li>Equivalence relation </li></ul></ul><ul><ul><li>a partition on set A </li></ul></ul>
    17. 17. FUNCTIONS domain 1 2 3 a b c range f : A -> B A B If A = domain then f is a total function otherwise f is a partial function f(1) = a 4 5 In general, we mean this.
    18. 18. GRAPHS A directed graph <ul><li>Nodes (Vertices) </li></ul><ul><li>V = { a, b, c, d, e } </li></ul><ul><li>Edges (Ordered pairs) </li></ul><ul><li>E = { (a,b), (b,c), (b,e),(c,a), (c,e), (d,c), (e,b), (e,d) } </li></ul>node edge a b c d e
    19. 19. Labeled Graph a b c d e 1 3 5 6 2 6 2
    20. 20. Walk Walk is a sequence of adjacent edges (e, d), (d, c), (c, a) is a walk from e to a of length 3 (or denoted as e-d-c-a ) Length = # of edges a b c d e
    21. 21. Path a b c d e Path is a walk where no edge is repeated Simple path : no node is repeated
    22. 22. Path a b c d e Path is a walk where no edge is repeated Simple path : no node is repeated (e, b), (b, e), (e, d), (d, c), (c, a) is a path from e to a but it is not a simple path.
    23. 23. Cycle a b c d e 1 2 3 Cycle : a walk from a node (base) to itself without repeated edges Simple cycle : only the base node is repeated Loop: an edge from a node to itself base
    24. 24. Find All Simple Paths starting from c a b c d e origin The longest simple path has at most length 4. Since every vertex can only be visited at most once, and there are 4 other vertices.
    25. 25. (c, a) (c, e) Step 1 a b c d e origin Starting from vertex c, list all outgoing edges as long as they do not lead to any vertex already used in the path. At this point, we have all paths of length one starting at c . For all vertices a , e reached by c , we list all outgoing edges originating at a or e according the same way we did before.
    26. 26. (c, a) (c, a), (a, b) (c, e) (c, e), (e, b) (c, e), (e, d) Step 2 a b c d e origin
    27. 27. Step 3 a b c d e origin (c, a) (c, a), (a, b) (c, a), (a, b), (b, e) (c, e) (c, e), (e, b) (c, e), (e, d)
    28. 28. Step 4 a b c d e origin (c, a) (c, a), (a, b) (c, a), (a, b), (b, e) (c, a), (a, b), (b, e), (e,d) (c, e) (c, e), (e, b) (c, e), (e, d)
    29. 29. Trees are connected directed graphs without cycles such that there is a special vertex called “root” having exactly one path to every other vertices. root leaf parent child
    30. 30. root leaf Level 0 Level 1 Level 2 Level 3 Height 3 The level associated with each vertex is the number of edges in the path form the root to the vertex. The height of the tree is the largest level number of any vertex.
    31. 31. Binary Trees A binary tree is a tree in which no parent can have more than two children. A binary tree is a tree in which no parent can have more than two children. (p.10) Example 1.5. Prove that a binary tree of height n has at most 2 n leaves.
    32. 32. PROOF TECHNIQUES <ul><li>Proof by induction </li></ul><ul><li>Proof by contradiction </li></ul>
    33. 33. Induction We have statements P 1 , P 2 , P 3 , … <ul><li>If we know </li></ul><ul><ul><ul><li>for some m that P 1 , P 2 , …, P m are true </li></ul></ul></ul><ul><ul><ul><li>for any k >= m that </li></ul></ul></ul><ul><ul><ul><ul><ul><li>P 1 , P 2 , …, P k imply P k+1 </li></ul></ul></ul></ul></ul><ul><li>Then </li></ul><ul><li>Every P i is true </li></ul>
    34. 34. Proof by Induction <ul><li>Inductive basis </li></ul><ul><ul><ul><li>Find P 1 , P 2 , …, P m which are true </li></ul></ul></ul><ul><li>Inductive hypothesis </li></ul><ul><ul><ul><li>Let’s assume P 1 , P 2 , …, P k are true, </li></ul></ul></ul><ul><ul><ul><li>for any k >= m </li></ul></ul></ul><ul><li>Inductive step </li></ul><ul><ul><ul><li>Show that P k+1 is true </li></ul></ul></ul>
    35. 35. Example Theorem: A binary tree of height n has at most 2 n leaves. (p.10) We want to show: L(n) ≦ 2 n for n = 0, 1, 2,…. Proof by induction: let L(i) be the maximum number of leaves of any subtree at height i
    36. 36. <ul><li>We want to show: L(n) ≦ 2 n for n = 0, 1, 2,…. </li></ul><ul><li>Inductive basis </li></ul><ul><ul><ul><ul><li>L(0) =1 ≦ 2 0 (the root node : height=0) </li></ul></ul></ul></ul><ul><li>Inductive hypothesis </li></ul><ul><ul><ul><ul><li>Let’s assume L(i) ≦ 2 i for all i = 0, 1, …, k </li></ul></ul></ul></ul><ul><li>Induction step </li></ul><ul><ul><ul><ul><li>we need to show that L(k + 1) ≦ 2 k+1 , k ≧0 </li></ul></ul></ul></ul>
    37. 37. Induction Step From Inductive hypothesis: height k k+1 Let’s assume L(i) ≦ 2 i for all i = 0, 1, …, k need to show that L(k + 1) ≦ 2 k+1 0 … L(k) ≦ 2 k
    38. 38. L(k) ≦ 2 k L(k+1) ≦ 2 · L(k) ≦ 2· 2 k = 2 k+1 Induction Step height k k+1 (we add at most two nodes for every leaf of level k) … need to show that L(k + 1) ≦ 2 k+1 To get a binary tree of height k+1 from one of height k , we can create at most 2 leaves in place of each previous one
    39. 39. Remark <ul><li>Recursion is another thing </li></ul><ul><li>Example of recursive function: </li></ul><ul><ul><ul><li>f(n) = f(n-1) + f(n-2) </li></ul></ul></ul><ul><ul><ul><li>f(0) = 1, f(1) = 1 </li></ul></ul></ul>
    40. 40. Proof by Contradiction <ul><li>We want to prove that a statement P is true </li></ul><ul><ul><ul><li>we assume that P is false </li></ul></ul></ul><ul><ul><ul><li>then we arrive at an incorrect conclusion </li></ul></ul></ul><ul><ul><ul><li>therefore, statement P must be true </li></ul></ul></ul>
    41. 41. Example <ul><li>Theorem: is not rational </li></ul><ul><li>Proof: </li></ul><ul><ul><ul><li>Assume by contradiction that it is rational </li></ul></ul></ul><ul><ul><ul><li>= n/m </li></ul></ul></ul><ul><ul><ul><li>n, m are nonzero integers and without </li></ul></ul></ul><ul><ul><ul><li>common factors </li></ul></ul></ul><ul><ul><ul><li>We will show that this is impossible </li></ul></ul></ul>
    42. 42. = n/m 2 m 2 = n 2 Therefore, n 2 is even n is even n = 2 k 2 m 2 = 4k 2 m 2 = 2k 2 m is even m = 2 p Thus, m and n have a common factor 2 Contradiction!
    43. 43. Languages
    44. 44. <ul><li>A language is a set of strings </li></ul><ul><li>String: A sequence of letters </li></ul><ul><ul><li>Examples: “cat”, “dog”, “house”, … </li></ul></ul><ul><ul><li>Defined over an alphabet: </li></ul></ul>Non-empty and finite
    45. 45. Alphabets and Strings <ul><li>We will use small alphabets: </li></ul><ul><li>Strings </li></ul>
    46. 46. Empty String <ul><li>A string with no letters </li></ul>
    47. 47. String Operations Concatenation w  = ?  w = ?
    48. 48. Empty String <ul><li>Observations: </li></ul>
    49. 49. Reverse
    50. 50. String Length <ul><li>Length: </li></ul><ul><li>Examples: </li></ul>|  | = ?
    51. 51. Length of Concatenation <ul><li>Example: </li></ul>
    52. 52. Example 1.8 (p.17) | uv | = | u | + | v | <ul><li>A recursive definition of the length of a string: </li></ul>| a | =1, | wa | = | w | + 1 For all a  , w is any string from  <ul><li>Fix the string u and consider all possible strings v ( the length of v can be 1, 2, …. ( 0 is trivial) ) </li></ul><ul><li>The proof is done by induction on the length of v ( for any given u ) </li></ul>
    53. 53. Example 1.8 (p.17) | uv | = | u | + | v | <ul><li>For a string u , consider the length of uv , concatenation of u with a string v </li></ul><ul><li>Basis </li></ul><ul><li>Inductive Step </li></ul><ul><li>Induction Assumption </li></ul>
    54. 54. Substring <ul><li>Substring of string: a subsequence of consecutive letters </li></ul><ul><li>String Substring </li></ul>
    55. 55. Prefix and Suffix <ul><li>Prefixes Suffixes </li></ul>prefix suffix
    56. 56. Another Operation <ul><li>Example: </li></ul>
    57. 57. Another Operation <ul><li>Definitions: </li></ul>
    58. 58. The * Operation <ul><li>: the set of all possible strings from </li></ul><ul><li>alphabet </li></ul><ul><li>Example: = { a, b } then = ? </li></ul>
    59. 59. The + Operation : the set of all possible strings from alphabet except
    60. 60. Languages Fall 2008 Automata
    61. 61. Note that: Sets Set size Set size String length
    62. 62. Another Example <ul><li>An infinite language </li></ul>
    63. 63. Operations on Languages <ul><li>The usual set operations </li></ul><ul><li>Complement: ?? </li></ul>
    64. 64. <ul><li>When we talk about a language , we must know what ground does this language stands on ….. </li></ul>Languages We should know the ALPHABETS that constitute the language
    65. 65. Complement Example <ul><li>The complement of </li></ul>Universal Set?
    66. 66. Reverse <ul><li>Definition: </li></ul><ul><li>Examples: </li></ul>
    67. 67. Reverse Hw # 10 (a) Prove or Disprove: i.e., w R  L  w  L R
    68. 68. Concatenation <ul><li>Definition: </li></ul><ul><li>Example: </li></ul>
    69. 69. Concatenation Hw 8. Prove
    70. 70. Another Operation <ul><li>Definition: </li></ul><ul><li>Special case: </li></ul>
    71. 71. More Examples
    72. 72. Star-Closure (Kleene *) <ul><li>Definition: </li></ul><ul><li>Example: </li></ul>
    73. 73. Positive Closure <ul><li>Definition: </li></ul>If  L then L +  L* - {  } It is not necessary that
    74. 74. True or False
    75. 75. True or False How to prove your answer?
    76. 76. Try Hw# 9 & 10(b) on p.28 What does w  L 2 mean? What does w  L * mean?
    77. 77. More Examples <ul><li>Consider a language on  = { a, b } </li></ul>What is L 2 ? L * ? e.g.
    78. 78. Grammars
    79. 79. Another Example <ul><li>Grammar: </li></ul><ul><li>Derivation of sentence : </li></ul>
    80. 80. <ul><li>Grammar: </li></ul><ul><li>Derivation of sentence : </li></ul>
    81. 81. <ul><li>Other derivations for </li></ul>Grammar:
    82. 82. <ul><li>Language of the grammar </li></ul>
    83. 83. More Notation <ul><li>Grammar </li></ul>Set of variables Set of terminal symbols Start variable Set of Production rules p.21
    84. 84. Example <ul><li>Grammar : </li></ul>
    85. 85. More Notation <ul><li>Sentential Form: </li></ul><ul><li>A sentence that contains </li></ul><ul><li>variables </li></ul><ul><li>Example: </li></ul>sentence Sentential Forms
    86. 86. <ul><li>We write: </li></ul><ul><li>Instead of: </li></ul>
    87. 87. <ul><li>In general we write: </li></ul><ul><li>If: </li></ul>
    88. 88. <ul><li>By default: </li></ul>
    89. 89. Example Grammar Derivations
    90. 90. Another Grammar Example <ul><li>Grammar : </li></ul>Derivations: From A  aAb and A   , we know A   , ab , aabb , aaabbb , … * * * *
    91. 91. Language defined by a Grammar <ul><li>For a grammar </li></ul><ul><li>with start variable : </li></ul>String of terminals
    92. 92. Example <ul><li>For grammar : </li></ul>Since Pf: show L(G)  {a n b n+1 } & L(G)  {a n b n+1 } from A  aAb we get A  a n Ab n when it is applied n times. Together with A   , we get A  a n b n for n = 0, 1, 2, ….  : w  L(G), i.e. S  * w
    93. 93. A Convenient Notation In general, we need to give a proof that a given language indeed generated by a certain grammar. Back to last Example
    94. 94. Example <ul><li>For grammar : </li></ul>To show Pf: show L(G)  {a n b n+1 } & L(G)  {a n b n+1 } from A  aAb we get A  a n Ab n when it is applied n times. Together with A   , we get A  a n b n for n = 0, 1, 2, ….  : w  L(G), i.e. S  * w L ( G )  L & L ( G )  L Need to show
    95. 95. More Examples on Grammars <ul><li>Find grammars for L on { a, b } and give brief arguments to explain why they work. </li></ul>L contains all strings with exactly one a L contains all strings with at least one a L 3 At Least: S  BaB; B  aB | bB | 
    96. 96. 1. Problems on p.27 <ul><li>You should be able to do </li></ul><ul><li>2 ~ 17, 21 </li></ul><ul><li>Hand in: 9, 10, 11c, 14ef, 15c, 17 </li></ul>2. Read P. 37~ 41 , and try to describe L ( M ) in Fig. 2.6. Homework for next week.

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