Outline for Test

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Outline for Test 2

• CSc-140 Tests
• Web Page

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6b Pushdown Automata 7.1,7.2,7.3

• Outline for Test 2
• Deterministic Pushdown Simulator
• Define Non-Deterministic Pushdown Automata (npda)
• State Graphs, State Tables for npda (not in text)
• Instantaneous Description of a npda
• Acceptance of a Language by a npda
• Constructing a npda for a Given Language
• npda Corresponding to an Arbitrary Context-Free
  Grammar
• Context-Free Grammar for a npda
• Deterministic Context-Free Languages

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Block Diagram of Pushdown Acceptor
Input Tape
Pushdown Store
Finite-State Control
z
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Pushdown Machine Simulation

• http//cs.union.edu/csc140/simulators/
• Simulates only deterministic machines
• Accepts by Final State AND Empty Stack
• (different than textbook and written HW)
• Handout Sample Pushdown Machines
• Unusual Simulation will get extra credit
  Simulate SOMETHING for regular credit

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Define Non-Deterministic Pushdown Automaton (npda)

• M (Q, ?, ?, ?, q0 , z, F) ordered septuple
• Q finite set of states
• ? input alphabet
• ? stack alphabet
• q0 ? Q initial state
• z ? ? stack start symbol
• F ? Q set of final states

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Transition Function

• ? Q ??? ? ? 2Q ?
• ?(q0,a,z) (q1,0z), (q0,z)
• ?(q0,b,z) (q1,z)
• ?(q1,b,0) (q1,?)
• ?(q1,?,z) (qf,z) qf final state

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Trace the Action of an npda

• ?(q0,a,z) (q1,0z), (q0,z)
• ?(q0,b,z) (q1,z)
• ?(q1,b,0) (q1,?)
• ?(q1,?,z) (qf,z) qf final state
• input aab
• watch stack...

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State Graphs for npda (not in text)

b,0/?
?,z/z
q0
qf
a,z/0z a,0/00
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Instantaneous Description of a npda

• represents a move
• (q0,aabb,z) (q0,abb,0z) (q0,bb,00z)
• (q1,b, 0z) (q1,?,z) (qf,?,z)

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Acceptance of a Language by a npda

• L(M) w in ? (q0,w,z) M (p,?,u) where p
  ? F, u ? ?
• Called acceptance by final state. Equivalently we
  could have defined it by empty stack or both
  final state and empty stack

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Constructing a npda for a Given Language

• Key issue in design is the use of the stack
• na(w) nb(w) page 179-80
• wwR page 180-81
• aibi i ? 0

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npda Corresponding to an Arbitrary Context-Free
Grammar

• First Convert language to Greibach form
• Corresponding npda will have three states
• q0, q1, qf, q0 is initial state, qf is only
  final state
• Add transitions
• ?(q0, ?, z) (q1, Sz) and
• ?(q1, ?, z) (qf, z)
• Non-terminals become stack symbols
• not states as with right-linear grammars
• Add transitions for each production in grammar
• see page 194

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Sample Conversioncfg to npda

• S ? aAS bAB aB
• A ? bBB aS a
• B ? bA a

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Context-Free Grammar for a npda

• To be converted, npda has two restrictions
• single final state, qf, that is entered if and
  only if the stack is empty
• each move either increases (or decreases) the
  stack content by one symbol
• Other npdas can easily be converted to
  equivalent machines with these restrictions(i.e.
  I won't ask you to convert on test)

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npda Conversion to cf Grammar(grammar
"simulates" machine)

• "Plug and Chug" from pages 189-90
• All transitions are of one of the following two
  types
• ?(qi,a,A) c1, c2, , cn where either
• (7.5) ci (qj, ?)
• (7.6) ci (qj, BC)
• For transitions of type (7.5) add productions
• (qiAqj) ? a
• For transitions of type (7.6) add productions
• (qiAqk) ? a(qjBql)(qlCqk) for all qk and ql in Q
• Finally, take (q0zqf) as the S in the grammar

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Sample Conversion
b,0/?
?,z/?
From (7.6) (q0zq0) ?a(q00q0)(q0zq0)(q0zq0)
?a(q00qf)(qfzq0) (q0zqf) ?a(q00q0)(q0zqf)(q0zqf)
?a(q00qf)(qfzqf) (q00q0) ?a(q00q0)(q00q0)(q00q0
) ?a(q00qf)(qf0q0) (q00qf) ?a(q00q0)(q00qf)(q00q
f) ?a(q00qf)(qf0qf)
q0
qf
a,z/0z a,0/00
From (7.5)(q00q0) ? b(q0zqf) ? ?
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(qfzq0), (qfzqf), (qf0q0) and (qf0qf) are
undefined(q0zq0) and (q00qf) are nonterminating

• (q00q0) ? b(q0zqf) ? ?
• (q0zq0) ?a(q00q0)(q0zq0)(q0zq0)
  ?a(q00qf)(qfzq0)(q0zqf) ?a(q00q0)(q0zqf)(q0zqf)
  ?a(q00qf)(qfzqf)
• (q00q0) ?a(q00q0)(q00q0)(q00q0)
  ?a(q00qf)(qf0q0)(q00qf) ?a(q00q0)(q00qf)(q00qf)
  ?a(q00qf)(qf0qf)

• Replacing (q0zqf) with S
• and (q00q0) with B we get
• S ? ? aBS
• B ? b aBB

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Deterministic Context-Free Languages

• Given w ? a,b, considerwwR vs. wcwR
• What about wcw? (trick questionlooks simpler
  than wcwR)
• It turns out that wcw is not context free
• Stay tuned for proof (next class)

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In-Class ExerciseJavaScript Pushdown Simulator

• Work in groups of 2 or 3
• (state graph of first 2 groups finished on board)
• Link to
• cs.union.edu/csc140/simulators
• Given
• L anbn n?1
• L wcwR w in a,b
• Create a machine to accept
• L set of matched parentheses, ? (,),,
• See language of exercise 5.1.21 page 135
• Recall To accept, simulator needs final state
  and empty stack.