CSCI 2670
Introduction to Theory of
Computing
October 5, 2004
Agenda
• This week
– Turing machines
– Read section 3.1
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Announcements
• Tests will be returned tomorrow
• Tutorial sessions are suspended until
further notice
– Extended office hours while tutorials
are suspended
• Monday 11:00 – 12:00
• Tuesday 3:00 – 4:00
• Wednesday 3:00 – 5:00
October 5, 2004
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Recap to date
• Finite automata (both deterministic
and nondeterministic) machines
accept regular languages
– Weakness: no memory
• Pushdown automata accept contextfree grammars
– Add memory in the form of a stack
– Weakness: stack is restrictive
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Turing machines
• Similar to a finite automaton
– Unrestricted memory in the form of a
tape
• Can do anything a real computer can
do!
– Still cannot solve some problems
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Touring machine schematic
Control
a b a ~
• Initially tape contains the input string
– Blanks everywhere else (denoted ~ in
class … different symbol in book)
• Machine may write information on the
tape
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Touring machine schematic
Control
a b a ~
• Can move tape head to read
information written to tape
• Continues computing until output
produced
– Output values accept or reject
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Touring machine schematic
Control
a b a ~
• Turing machine results
– Accept
– Reject
– Never halts
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Differences between TM and FA
1. TM has tape you can read from and
write to
2. Read-write head can be moved in
either direction
3. Tape is infinite
4. Accept and reject states take
immediate effect
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Example
• How can we design a Turing machine
to find the middle of a string?
– If string length is odd, return middle
symbol
– If string length is even, reject string
• Make multiple passes over string Xing
out symbols at end until only middle
remains
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Processing input
1.
Check if string is empty
 If so, return reject
2. Write X over first and last non-X
symbols
 After this, the head will be at the second X
3. Move left one symbol
 If symbol is an X, return reject (string is
even in length)
4. Move left one symbol
 If symbol is an X, return accept (string is
even in length)
5. Go to step 2
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Example
• 00110~
– First check if string is empty
– X first and last non-X symbols
• X011X~
– Move left one symbol
• X011X~
– Is symbol an X? No
– Move left one symbol
• X011X~
– Is symbol an X? No
– Write X over first and last non-X symbols
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Example
• XX1XX~
– Move left one symbol
• XX1XX~
– Is symbol an X? No
– Move left one symbol
• XX1XX~
– Is symbol an X? Yes
– Return accept
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Formal definition of a TM
Definition: A Turing machine is a 7tuple (Q,,,,q0,qaccept,qreject), where
Q, , and  are finite sets and
1. Q is the set of states,
2.  is the input alphabet not
containing the special blank symbol ~
3.  is the tape alphabet, where ~
and ,
4. : QQ{L,R} is the transition
function,
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Formal definition of a TM
Definition: A Turing machine is a 7tuple (Q,,,,q0,qaccept,qreject), where
Q, , and  are finite sets and
5. q0Q is the start state,
6. qacceptQ is the accept state, and
7. qrejectQ is the reject state, where
qrejectqaccept
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Computing with a TM
• M receives input w = w1w2…wn* on
leftmost n squares of tape
– Rest of tape is blank (all ~ symbols)
• Head position begins at leftmost
square of tape
• Computation follows rules of 
• Head never moves left of leftmost
square of the tape
– If  says to move L, head stays put!
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Completing computation
• Continue following  transition rules
until M reaches qaccept or qreject
– Halt at these states
• May never halt if the machine never
transitions to one of these states!
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TM configurations
• The configuration of a Turing machine
is the current setting
– Current state
– Current tape contents
– Current tape location
• Notation uqv
– Current state = q
– Current tape contents = uv
• Only ~ symbols after last symbol of v
– Current tape location
= first symbol of v
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