Go Over 1'1'3 from HW

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Go Over 1.1.3 from HW 1

• Let S1m, S2n
• Induction on S2
• Basis S21 S1?aS1S1?a, a?U.
• Assume theorem true for S2n, show for n1
• Consider any set S3 with n1 elements.
• S3S2?a for some S2n, a?U, a ? S2.
• S1?S3S1?S2S1?aS1?S2S1
• mnm m(n1) S1?S3!

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Roman Numeral Acceptor

• Long line of states, transitions on
• MMMMDCCCCLXXXXVIIII
• Jump ahead on, say, X if in M's, etc.
• If you get C after X, for example, you need to
  reject the word (no matter what else comes in)
• MUCH easier problem if we allow non-determinism
• Straight line of states, l or letter transitions
• (stay tuned for more details next class)

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2a Deterministic Finite Acceptors 2.1

• JavaScript Finite-State Machine Simulator
• Deterministic Finite Acceptor (dfa)
• Languages and dfas
• Regular Languages
• Not All Languages are Regular

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JavaScript dfa Simulator

• http//cs.union.edu/csc140/
• Loading a canned machine
• Running the simulation
• Entering a new machine
• Printout of your own simulation is required
• If you want extra credit (and to make the "list")
  you need to do something creative

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Problems suitable for dfa

• See Finite-State HW.doc in course folder

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Formal Definition of dfa

• M (Q, ?, ?, q0, F) ordered quintuple
• Q finite set of states
• ? input alphabet
• ? Q x ? ? Q transition function
• total function in this text
• q0 ? Q initial state
• F ? Q set of final states (accepting)

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Specifying the Transitions

• Functions
• ?(even,0) even ?(even,1) odd?(odd,0)
  odd ?(odd,1) even
• Tables
• Graphs
• Arrow from nowhere into initial state
• Double circle for final state

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Extended Transition Function

• ? Q x ? ? Q
• What state after 3 inputs? 7 inputs? etc.
• even/odd machine after 011?
• after 0011100?

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Language of a dfa

• L(M) w ? ? ?(q0,w) ? F
• Language of even/odd machine set of all words
  over 0,1 with an odd number of 1s
• dfa Accepting Complement of L(M)
• must start with dfa and total transition function
• Why? (we will come back to this in exercise
  2.3.6)
• make all non-final states final, and all final
  states non-final

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Trap State

• Any words with prefixes that take you to a trap
  state will not be accepted
• Example Machine accepting words over 0,1 that
  start with 0

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Trap States in Roman Numeral dfa

• More than 4 of the same symbol in a row
• X followed by C, I followed by M, etc.

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? 0,1, accept if and only if odd parity

• For all languages, it is important to accept ALL
  words in the desired language, but also NOT to
  accept words that are not in the language
• one-state dfa with loops on 0 and 1 accepts this
  language plus a whole lot more
• likewise S?0S1S? generates this language, but
  also lots of other words
• likewise (01) includes this language

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Accept if number of 1's 1 mod 5

• Like Odd/Even machine only more so!
• 5 states (q0-q4), stay put on 0, move clockwise
  around the circle on 1
• q1 will be the only accepting state

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Words with odd a's, even b's

• 4 states EaEb, EaOb, OaEb, OaOb
• Could draw it 2-dimensionally
• as move horizontally
• bs move vertically

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Exercises 2.1.2(c,d) page 45

• (c) 5 states
• 0 as, 1 a, 2 as, 3 as, more-than-3 as
• all accepting state except last one (trap)
• (d) 2 rows of 4 states
• top row no as, bottom row got an a
• first column no bs, next column 1 b, third
  column 2 bs, fourth column more
• only accepting state bottom row 3rd column

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Exercise 2.1.7(a) page 45

• 3 states
• go around the triangle in same direction on both
  a and b
• initial state is only final state

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? a,b, "aaa" is not a substring

• (note the need for a trap state)
• 4 states 0, 1, 2, 3 as in a row
• b from any state except 3 takes you back to the
  start
• 3 is the only non-accepting state

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Exercise 2.1.9(a) page 46

• again note the need for a trap
• 4 states no 0s, one 0 in a row, two in a row,
  three or more in a row
• do not accept if you just saw two 0s in a row
  (need the trailing 1), nor if more than two 0s
  in a row (trap)

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Exercise 2.1.9(c) page 46

• How can we possibly remember the first symbol
  all the way until we get to the last symbol?
• Only way we have to remember information in a
  finite-state machine is by the state we are in
  (no auxiliary storage)

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Regular Languages

• Family of Languages
• Accepted by some dfa
• To prove regular, give dfa accepting it

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Regular Language Examples

• Exercise 2.1.20 page 47 page 43
• Given L awa w ? a,b, show that L is
  regular.
• What is L?
• What is L? L ? what?
• Exercise 2.1.22
• If L is regular not containing ?, show that
  truncate(L) is also regular. Where truncate drops
  last symbol from each word. construct new dfa
  from original
• Illustrate with 1(01),101, abc machines

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Not all Languages are Regular

• L 0i1i i ? 0 is a non-regular language
• L ? Li, i ? 0, where
• L0 ?
• L1 01
• L2 0011
• etc.
• Machine for each Li exists
• Machine for union exists for any finite i

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Related Languages Regular?

• L 0i1j i, j?0
• Proof?
• L (01)i i?0
• Proof?

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In-Class ExerciseJavaScript Simulator

• Work in groups of 2 or 3
• Use Internet Explorer to link to
• cs.union.edu/csc140/simulators
• Run the built-in Finite-State simulations
• Addition Modulo 5
• Pattern Recognition
• Create a machine to search for one of 2 words
• Be sure you dont accept extra words

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In-Class ExerciseAnswer

• Make sure your machine accepts the desired
  pattern anywhere in the input.
• Make sure your machine does NOT accept strings
  without the pattern.