Answers to Sample Test

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Answers to Sample Test 2

• Now that you have had a chance to try the sample
  test, you can check out my answers on the
  bulletin board outside my office.
• Those of you who are not normally on campus can
  follow me back to my office to make a copy this
  afternoon.

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7a CFL Pumping Lemma Closure Decision Problems
8.1,8.2

• The Pumping Lemma for Context-Free Languages
• Languages that are NOT Context-Free
• SKIP Pumping Lemma for Linear Languages
• Closure of CFL under Union, Concatenation, and
• Non-Closure under Intersection and Complement
• Determining if L(G) is Empty, Infinite

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The Pumping Lemma for Context-Free
Languages--Theorem 8.1 page 206

• Let L be an infinite context-free language.
• Then there exists some positive integer m such
  that any w ? L with w ? m can be decomposed as
• w uvxyz
• with
• vxy ? m
• and
• vy ? 1
• such that
• uvixyiz ? L
• for all i 0, 1, 2, ...

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Pumping Example

• S ? aSb cAd
• A ? eBf g
• B ? hAi j
• Pick w acehgifdb
• S?aSb?acAdb?aceBfdb?acehAifdb?acehgifdb
• then uac veh xg yif zdb

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Where does "m" come from?

• Consider a grammar with 5 non-terminals and
  productions whose right-hand sides have at most 3
  symbols
• Any tree of height more than 5 must contain a
  "loop"
• Each level of the tree can increase the word
  length by a factor of at most 3
• Consider words of length 3(51) 729
• Forces tree height gt 5
• (recall that we can assume no ? or unit
  productions)
• Therefore, derivations of any word of length 729
  or greater forces a loop
• 729 could serve as our m (remember the joke)
• Note that m simply becomes a power of 2 for
  Chomsky Normal Form grammars

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Graphical Representation of CFL Pumping Lemma
S
A
u
z
A
v
y
x
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Languages that are NOT Context-Free

• L anbncn n ? 0
• Consider w ambmcm
• Proof by Cases What are the possible values for
  u,v,x,y and z recall that we must show that we
  get something not in L when we pump v and y, no
  matter what choices we make for u, v, x, y and z.
  
• L ww w ? a,b
• Consider w ambmambm

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Exercise 8.1.7(e) page 212L anbjck nltj, n
? k ? j

• Select w ambm1cm
• Case 1 v or y contains both a's and b's or both
  b's and c'sNo Good (violates a's first, b's
  second, c's last when pumped)
• Case 2 v within the a's, y within the c'sNo
  Good (vxy ? m)
• Case 3 v and y both entirely within one letter
  (a's, b's or c's)a's) No Good (pump up, get n gt
  j)b's) No Good (pump down, get j ? n)c's) No
  Good (pump up, get k gt j)
• Case 4 v in a's, y in b'sNo Good (pump up, get
  n gt k
• Case 5 v in b's, y in c'sNo Good (pump down,
  get k lt n)
• No other cases, therefore cannot pump!
  CONTRADICTION.

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Closure of CFL under Union

• Union L(S1)?L(S2)
• Given two context-free grammars
• G1 (V1, T1, S1, P1) and G2 (V2, T2, S2, P2)
• Create a new grammar G (V, T, S, P) whereS ?
  S1 S2
• V1 and V2 must be disjoint (Why?)

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Closure of CFL Concatenation, and

• Catenation L(S1)L(S2)
• S ? S1S2
• (This construction does NOT work for linear
  grammars.)
• Star L(S1)
• S ? ? S1 S1S1 S1S1S1 S1S1S1S1
• Using recursion to avoid infinitely many
  productions
• S ? ? S1S

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Non-Closure under Complement and Intersection

• Intersection
• Consider aibicj ? aibjcj i, j?0
• Can give a CFG for each part
• Complement follows from DeMorgan

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Complement of anbncn IS a CFL

• Hints on grammar
• More a's than b's, b's than a's, a's than c's,
  etc.
• c in front of a, b in front of a, c in front of
  b
• Makes an interesting test question

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Determining if L(G) is Empty, Infinite

• Assume ? not in L (or consider L - ?)
• Eliminate useless symbols and productions
• Eliminate ? and unit productions
• Empty if S is useless
• Infinite if grammar contains derivation of form A
  ? xAy, and x or y non-empty.

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David Hannays "Top Ten" List(with apologies to
David Letterman)

• 1) NFA º DFA
• 2) Unique minimal DFA equivalent to any DFA
• 3) Regular Expressions º Finite-State Machines
• 4) Pumping Lemmas (regular CFL)
• 5) Syntax Trees Ambiguous Context-Free
  Grammars
• 6) CFL º NPDA
• 7) Turing's Thesis
• 8) Universal Turing Machine
• 9) Halting Problem for Turing Machines
• 10) Recursive Functions º Turing Machines

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In-Class ExerciseExercise 8.1.8(b,c) page 212

• Work in groups of 2 or 3
• Hints on next slide (no peeking)
• First group done with both parts put (b) on board
• Second group done with both parts put (c) on board

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In-Class ExerciseAnswers

• 8.1.8(b) NO. consider ambmambm
• Give complete proof with all cases!
• 8.1.8(c) YES. give grammar
• Give actual grammar