MC 306 Theory of Computation Thursday, 111303

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MC 306 Theory of ComputationThursday, 11/13/03

• Last time
• Using closure properties to prove languages are
  and are not context-free
• Pumping Lemma for context-free languages
• Turing Machine Basics
• QUESTIONS??
• Todays Class
• More on TMs formal definitions, examples
• Church-Turing Thesis

• Correction Pumping Lemma assignments should have
  referred to p. 116 3.38 (not 3)
• Exercises 3.38 a b
• Hand-in 3.38 c d
• Reading Still 4.1
• New exercises from text
• pp. 151-152 4.1, 4.3b
• From handout sheet
• Exercises 7ab, 10 (just outline of machine)
• Hand-in No new ones yet
• New hand-in HW will be given on Tuesday

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Definition of a Turing Machine

• Note My definition is slightly different from
  text definition
• A Turing Machine is a 6-tuple
• Q is the set of states
• ?0 is the input alphabet ?0 must not contain the
  symbols ? or ?
• ? is the tape alphabet ?0 ? ?, ? must contain
  the symbols ? (left end marker) and ? (blank cell
  marker)
• Neither ?0 nor ? may contain the symbols ? and ?
• s ? S is the initial state
• H?? S is the set of halting states
• ? is the transition function
• For every non-halting state, and every tape
  symbol, there is a transition that (1) selects a
  new state, and (2) either writes a new symbol in
  the current tape square or moves left or right.
• In addition, ?(q, ?) (q,?). The machine always
  moves right when it sees the left-end marker.
• In addition, if ?(q, x) (q,y), then y ? ?. The
  machine never writes the left-end marker.

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Variations on the standard TM definition

• One alphabet for both input and tape (our text)
• Two-way infinite tape (JFLAP)
• Machine writes and moves in single transition
  (JFLAP)
• ? is a partial function, meaning some arrows
  are missing
• Say the machine stalls if there is no transition
  to apply
• Can get a total function by adding arrows to a
  dead state
• May use states to accept or writing to accept.
• These are all equivalent in power to our
  definition.

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Church-Turing Thesis

• Church-Turing Thesis Every computer algorithm
  can be implemented as a Turing Machine.
• In other words, any program that can be
  implemented in any computer language, on any
  actual computer, can also be implemented as a TM.
• So TMs can simulate loops, if-thens, function
  calls, data structures, etc.

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Example

• Consider a TM with the following transitions
• (q0, a), (q1, b)
• (q0, b), (q1, a)
• (q0, ?), (h, ?)
• (q1, x), (q0, ?),for any x
• What does this TM do?
• See JFLAP file MC306031113.jfl

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Configurations and Computations

• A configuration tells us the current state of a
  TM while its computing
• Current state
• Current contents of tape
• Current position of read/write pointer
• We usually write this C (q, ? x ?), where q is
  the current state, x is the current symbol
  pointed at, and the tape contents comprise the
  string ? x ?
• We usually begin a computation on string ? x ?
  in the starting configuration C0 (s, ? x ?)
• Example Do computation of previous machine on ?
  aab.

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Tasks for TMs

• Language accepters (recognizers)
• L set of strings on which machine computation
  goes to a halting state view halting states as
  favorable states
• Language Deciders
• Always end in a halting state
• Give answer by writing 1 (yes) or 0 (no) on tape
• C0 (s, ??) ? (s, ?1) means ? is in L
• C0 (s, ??) ? (s, ?0) means ? is not in L
• Transducers (compute functions)
• Given function f ?0 ??0, the TM computes f(?)
  ?
• C0 (s, ??) ? (s, ??)

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Examples

• Design a TM to accept (ab)
• Design a TM to decide L set of strings
  containing at least one a.
• Design a TM to accept anbn