Finishing up contextfree languages

1
Finishing up context-free languages

• Review
• properties of cfls
• material covered in Chapters 5-8

2
Context-free Languages

• Review
• A language is context-free if it can be generated
  by a context-free grammar (cfg) or recognized by
  a pushdown automaton (pda)
• A context-free grammar is one in which every
  production is of the form X ? wthat is, of
  the form in which the left-hand side is a single
  variable and the right-hand side is any string of
  variables and terminals.

3
Context-free Languages

• Review
• a cfg production X ? xXy generates from the
  center and produces an equal number of xs and
  ys
• the cfg productions X ? xY, Y ? xYy will generate
  more xs than ys.
• the cfg productions X ? aXa bXb generates
  palindromic strings.
• x ? y is the same as x gt y or x lt y

4
Context-free Languages

• Review
• Derivation trees
• Derivations
• Left-most and right-most derivations
• Ambiguity (grammars)
• Inherent Ambiguity (languages)
• anbmcm ? anbncm is INHERENTLY AMBIGUOUS!
• There has to be at least two ways to generate abc
  since it could be in the language because the b's
  equal the c's, or because the a's equal the b's

5
Context-free Languages

• Review
• ? rules (X ? ?)
• unit rules
• useless productions

• we can get rid of a ? rule by crossing it out
  and adding new rules with the erased variable
  (X) removed for each case where it appears on a
  right-hand side

• in general, hard to remove because of cycles
  (X?Y, Y?X). But in simple cases can be removed
  by merely crossing them out and adding
  productions of the form X ? w where there is a
  production of the form Y ? w

• a variable is said to be useless if the
  variable can not be generated from the start
  symbol or if no string containing only terminal
  symbols can be generated from it. Any
  production mentioning a useless variable anywhere
  in it is said to be useless and can be removed.

6
Context-free Languages

• Review
• Chomsky Normal Form
• all rules are of the form X ? YZ where X, Y and Z
  are all variables. (They dont have to be
  different variables) or X ? a. We also allow the
  rule S ? ?
• I expect you to understand this and to be able to
  transform simple cfgs into Chomsky Normal Form.
• The technique is to create new variables
  wherever a terminal appears on the right-hand
  side and then to break up X ? WYZ into X ? VZ, V
  ? WY

7
Context-free Languages

• Review
• Greibach Normal Form
• all rules are of the form X ? aY...Z where a is a
  terminal and Y...Z is any string of zero or more
  variables. We also allow the rule S ? ?
• I expect you to understand this and to be able to
  transform simple cfgs into Greibach Normal Form.
• This is generally hard to do (when the right-hand
  side begins with a variable), but in simple
  cases, technique is to remove unit rules and add
  variables corresponding to any terminal on the
  right-hand side that is not the first symbol.

8
Context-free Languages

• Review
• Pushdown automata
• A pushdown automaton is merely a finite automaton
  with a stack added to it.
• The stack allows for unbounded memorization.
• The transition function maps (q, a, s) to a
  finite number of possible pairs (q, s). We also
  allow ? transitions. The transition (q, a, s) ?
  (r, uts) means that if we are in state q and the
  next input symbol is a and the symbol s is on top
  of the stack, then go into state r and replace
  the symbol s on the stack with the symbols u, t
  and s (one at a time) that is, push ut. The
  stack string read from left to right is the same
  from top downward.

9
Context-free Languages

• Review
• Pushdown automata
• A string is accepted by a pda if after reading
  that string, the pda could wind up in an
  accepting state. The contents of the stack is
  irrelevant. We always start the stack with a
  special stack symbol z0 to avoid the situation of
  having an empty stack and therefore cant write a
  transition rule for the situation.

10
Context-free Languages

• Review
• Pushdown automata
• The language anbmcnm
• why does this work?

11
Context-free Languages

• Review
• Pushdown automata
• The language anbm n? m
• why does this work?

?, x, x b, z0, z0
too many as
?, z0 , z0 ? ,x, x
a, z0 , xz0 a, x, xx
b, x, ?
too many bs
12
Context-free Languages

• Review
• Pushdown automata
• The language of balanced parentheses
• why does this work?

?, z0 , z0
(, z0 , (z0 (, (, (( ), (, ?
13
Context-free Languages

• Review
• Pushdown automata
• A deterministic pushdown automaton (dpda) only
  allows one transition for each triple (q, a, s)
  and if there are ? transitions (q, ?, s) then
  there are no corresponding non-? transitions (q,
  c, s) for any c
• Not every context-free language can be recognized
  by a deterministic push-down automaton.
• How would a dpda generate the language wwR, the
  language of unmarked palindromes? It cant. Why?

14
Context-free grammars and Pushdown Automata

• Context-free grammars recognize exactly the same
  class of languages as non-deterministic pushdown
  automata.
• Proof
• Consider any context-free grammar. There is an
  equivalent Greibach Normal Form grammar.
  Construct an npda that takes the form below

?, z0, Sz0
15
Context-free grammars and Pushdown Automata

• Proof (continued)
• Consider any context-free grammar. There is an
  equivalent Greibach Normal Form grammar. For
  each production Y ? aX1X2...Xn, where n 1 , add
  the label a, Y, X1X2...Xn to the loop on the
  middle state. For each production Y ? a , add a,
  Y, ? to the transition going to the accepting
  state. This machine accepts the same language as
  the original context-free grammar!!!

?, z0, Sz0
16
Context-free grammars and Pushdown Automata

• ExampleConsider the grammar
• S ? aXY
• X ? aXB aB
• Y ? bYA bA
• A ? a
• B ? b

L(G) aanbnbmam m, n gt 0
Consider aaabbba
B
X
a, S, XY
b, Y, YA
a, X, XB
X
B
b, Y, A
a, X, B
S
Y
A
a, A, ?
z0
?, z0, Sz0
b, B, ?
17
Context-free grammars and Pushdown Automata

• Context-free grammars recognize exactly the same
  class of languages as non-deterministic pushdown
  automata.
• Proof
• We leave the proof that any push-down automaton
  can be converted into a context-free grammar to
  the "reader."

18
The Pumping Lemma for Context-Free Languages

• Let L be context-free. There exists some
  integer m such that for all w in L with w m,
• w uvxyz with vxy m and vy gt 0 such that
• uvixyiz ? L for all i 0, 1, 2, 3 ....

19
The Pumping Lemma for Context-Free Languages

• This is similar to our previous Pumping Lemma,
  except we are allowed to pump equally in two
  different spots in the middle of the string.
  These two spots must be within a distance of m
  from one another.
• We use this Pumping Lemma for a very similar
  reason to show that a language is NOT
  context-free

20
The Pumping Lemma for Context-Free Languages

• Show that L anbncn n 0 is not
  context-free.
• Proof Assume that L is context-free.
• Then the Pumping Lemma applies.
• Let m be the magic number.
• Consider the string ambmcm which is clearly in L
  and is longer than m.
• By the Pumping Lemma,

21
The Pumping Lemma for Context-Free Languages
u
x
z
v
y
aaaaa....aaaaabbb...bbbbbccc...ccccc
m
m
m
22
The Pumping Lemma for Context-Free Languages

• ambmcm can be written as uvxyz with vxy m and
  vy gt 0 such that uvixyiz is in L for all i 0,
  1, 2, 3 ....
• There are only a limited number of possibilities
  for v and y.
• v crosses the border between the as and bs (or
  y crosses the border between bs and cs)
• v is all as and y is all bs
• v is all bs and y is all cs
• etc. etc.

23
The Pumping Lemma for Context-Free Languages

• Take each case separately. Clearly if v or y
  crosses a boundary uv2xy2z will not be of the
  form as followed by bs followed by cs
• But if v is all a's then y can not contain c's.
  So pumping up will leave too few c's.
• If v is all b's then pumping up will leave too
  few a's.
• Therefore the language is not context-free.

24
Context-free Languages

• What are the properties of Context-Free
  Languages?
• There is an algorithm to determine if a given
  Context-Free Language is empty.
• Proof Construct any grammar for the language.
  Remove all useless productions. If S is found to
  be useless, then the language is empty otherwise
  it is not.

25
Context-free Languages

• What are the properties of Context-Free
  Languages?
• There is an algorithm to determine if a given
  Context-Free Language is infinite
• Proof Let G be a grammar for L that has no unit
  rules, ? productions or useless productions. Let
  G have n variables. For each variable, consider
  all the derivations of length n starting with
  that variable. If any sentential form in any of
  these derivations contains its starting symbol,
  the language is infinite.

26
Context-free Languages

• What are the properties of Context-Free
  Languages?
• There exists an algorithm to determine if a
  string belongs to a given context-free language.
• Proof Construct a Greibach Normal Form grammar.
  Construct the corresponding npda. If the npda
  accepts the string the string is in the language.
  If it doesn't, the string it isn't in the
  language.

27
Context-free Languages

• What are the properties of Context-Free
  Languages?
• Context-free Languages are closed under union.
• Proof Let G1 be a grammar for L1 and G2 a
  grammar for L2. We can modify one of these to
  ensure that they have no variables in common
  without affecting the language. Change the start
  symbol of G1 to S1, and the start symbol of G2 to
  S2. Add S ? S1 S2. Voila!

28
Context-free Languages

• What are the properties of Context-Free
  Languages?
• Context-free languages are not closed under
  intersection.
• Proof Consider the following two grammars
• L1 anbncm, n gt 0, m gt 0
• L2 anbmcm, n gt 0, m gt 0
• L1 ? L2 anbncn, n gt 0. We will shortly show
  that this language is not context-free.

29
Context-free Languages

• Review
• Context-free languages
• What kinds of languages are beyond the scope of
  context-free languages?
• anbncn
• ww
• anbmcnm

30
Homework

• 8.1/3, 7c and d, 88.2/12, 18
• You are NOT responsible for any material
  regarding linear languages.

31
Where do we go from here?

• From the earlier example, we see that not all
  languages are context-free.
• Is there a mathematical model for even more
  complex computations?
• There are many.
• We will study one in particular an automaton
  called a Turing machine.

32
Turing machines

• A Turing machine (tm) is a finite automaton with
  a twist
• The tape head can move in either direction (left
  OR right)
• A square on the input tape can be overwritten
  with another symbol
• By the way, there is no stack!

33
Turing machines
"guts" of the machine
on/off switch
accept or reject
tape read head
input tape
Remember our finite automaton?
34
Turing machines
"guts" of the machine
on/off switch
accept or reject
tape head can move left or right.
tape read head
input tape
tape head can read or write!
A Turing machine
35
Turing machines

• A Turing machine (tm) is a finite automaton with
  a twist
• The tape head can move in either direction (left
  OR right)
• A square on the input tape can be overwritten
  with another symbol
• By the way, there is no stack!

36
Turing machines

• What can a Turing machine do? Clearly it can
  accept any regular language. Just limit the
  instructions to read from left to right and
  never change the contents of the input tape.
• Can it accept the language anbn? How?
• How about wcwR?
• How about wwR?
• How about ww?

37
Turing machines

• Can it accept the language anbn? How?
• Lay out a strategy
• Let's mark off each a and matching b with an X
• Let's work with the a's and b's from left to
  right
• Whenever we mark off an a, move right until we
  find a b. Mark off the b and head back left
  looking for the left-most remaining a. Repeat as
  long as you can.
• If there are no more a's, go to the right and
  check that there are no more b's.
• Let the machine die if there is no matching b for
  an a.

38
Turing machines
L anbn, n 0
a, a, R
X, X, R
a, X, R
X, X, R
X, X, L
?, ?, R
b, b, L
X, X, R
a, a, L
?, ?, R
?, ?, R
a, a, L
X, X, R
39
Turing machines

• Can it accept the language wcwR? How?
• Lay out a strategy
• Let's find the center c.
• Find the next unmarked symbol on the left. Head
  right, remembering whether it was an a or a b.
• If the next unmarked symbol on the right matches,
  head back to the middle and start again.
• If from the center c, there are no more unmarked
  symbols on the left, make sure there are no more
  on the right as well.

40
Turing machines
X,X,R
X,X,L
X,X,R

• How about wcwR?

X,X,L
a,X,L
c,c,R
a,X,R
c,c,L
c,c,L
c,c,L
b,X,R
?,?,R
c,c,R
a,a,R b,b,R
b,X,L
c,c,R
X,X,R
X,X,L
?,?,R
X,X,R
X,X,R
X,X,R
41
Turing machines

• How about wwR?
• Describe in words how a Turing machine can
  recognize this language.
• Instead of going to the middle and working
  outwards like the last example, start with the
  outermost symbols and keep matching until you hit
  the middle.

42
Turing machines

• How about anbncn?
• YES!!!!!!!!!!!
• consider our first machine and how it can be
  extended to handle matching cs as well.
• Turing machines are more powerful than pushdown
  automata!

43
Turing machines

• The Church-Turing Thesis
• If something can be computed it can be computed
  by a Turing machine.
• Note that this is called a Thesis, not a theorem.
• It cant be proved, because the term can be
  computed is too vague.
• But it is universally accepted as a true
  statement.

44
Turing machines

• Given the Church-Turing Thesis
• What does this say about "computability"?
• Are there things even a Turing machine can't do?
• If there are, then there are things that simply
  can't be "computed."
• Not with a Turing machine
• Not with your laptop
• Not with a supercomputer
• There ARE things that a Turing machine can't do!!!