CMSC 203 / 0201 Fall 2002

1
CMSC 203 / 0201Fall 2002

• Week 15 2/4/6 December 2002
• Prof. Marie desJardins

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MON 12/2FSMs WITH NO OUTPUT (10.3)
3
Concepts/Vocabulary

• Language concatenation
• Kleene closure A concatenation of 0 or more
  strings from A
• Finite-state automaton (FSA) M(S,I,f,s0,F)
  states S, input alphabet I, transition function
  f S?I?S, initial state s0, final states F
• Recognize a string (series of inputs) that
  results in a series of transitions starting at s0
  and ending in any s?F
• Nondeterministic FSA M(S,I,f,s0,F) transition
  function f S?I?P(S) power set of S
• Recognizes a string that can result in some
  series of transitions starting at s0 and ending
  in any s?F
• For any language recognized by a nondeterministic
  FSA, there is a deterministic FSA that recognizes
  the same language

4
Examples

• Exercise 10.3.5 Describe the elements of the set
  A for the followingvalues of A
• (a) 10
• (c) 0, 01
• (d) 1, 101
• Exercise 10.3.15 Find the language recognized by
  the given deterministic FSA

1
0
s1
Start
s2
s0
0
1
0,1
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Examples II

• Exercise 10.3.21 Find the language recognized by
  the given nondeterministic FSA

1
0,1
1
0
s1
Start
s2
s0
s3
0
0
0
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Examples III

• Exercise 10.3.27/28 Find a deterministic FSA
  that recognizes each of the following sets, and a
  nondeterministic FSA that recognizes the set, and
  has fewer states than the dFSA (if possible)
• (a) 0
• (b) 1, 00
• (c) 1n n2, 3, 4,

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WED 12/4LANGUAGE RECOGNITION (10.4)
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Concepts/Vocabulary

• Regular expressions ?, ? ?, x?I x, (AB)
  concatenation, (A?B) union, and A Kleene
  closure
• Regular set Any set that can be represented by a
  regular expression
• Can be recognized using (deterministic)
  finite-state automata (Kleenes Theorem)
• if part proved by constructive induction
• only if part left as exercise 20
• Regular set regular (type 3) grammar!
• (More powerful automata Pushdown automaton,
  linear bounded automata)

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Examples

• Exercise 10.4.3 Express each of the following
  sets using a regular expression
• (a) the set of strings of one or more 0s followed
  by a 1
• (c) the set of strings with either no 1 preceding
  a 0 or no 0 preceding a 1
• (d) the set of strings containing a string of 1s
  so that the number of 1s equals 2 modulo 3,
  followed by an even number of 0s
• Construct a FSA for (d) above

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Examples II

• Exercise 10.4.8 Construct a nondeterministic FSA
  that recognizes the language generated by the
  regular grammar G(V,T,S,P) where V0,1,S,A,B,
  T0,1, S is the start symbol, and the set of
  productions is
• (b) S?1A, S?0, S??, A?0B, B?1B, B?1

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FRI 12/6TURING MACHINES (10.5)
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Concepts/Vocabulary

• Turing machine general model of computation
• Inventor Alan Turing
• T(S,I,f,s0) states S, alphabet I that includes
  blank symbol B, partial function f S?I ?
  S?I?R,L, and start state s0?S
• Control unit has states S read/write tape is
  infinite in both directions single read/write
  head takes input from the tape, writes to the
  tape, and moves left or right
• Specify as 5-tuples (s, x, s, x, d) in state
  s, if you read x, transition to state s, output
  x, and then move one step in direction d

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Concepts/Vocabulary II

• Halting and language recognition
• T halts if f is undefined (i.e., no 5-tuple) for
  (s, x)
• A final state is a state that no 5-tuple begins
  with (i.e., no transitions are defined from the
  state)
• A string is recognized if T halts in a final
  state
• A string is not recognized if T doesnt halt, or
  halts in a state that isnt final
• Any problem that can be solved, or algorithm that
  can be written, with a digital computer, can also
  be solved with a Turing machine, despite its
  simplicity!
• Church-Turing thesis Any problem that can be
  solved with an effective algorithm can be solved
  with a Turing machine

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Examples

• Example 10.5.2 Find a Turing machine that
  recognizes the set of bit strings that have a 1
  as their second bit (that is, the regular set
  (0?1)1(0?1)).
• Example 10.5.3 FInd a Turing machine that
  recognizes the set 0n1n n ? 1