Computability
Catch up last lecture.
Turing machines. Variations
Homework: Check out videos.
Describe certain TMs
What next?
• It wasn't done this way, but, how can we define
a machine more powerful than PDA (which was
more powerful than FSM)
• Still talking about recognizers: something that
accepts a string as a member of a language
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Put input on tape
Add scanning back and forth
Add writing on the tape
Keep finite number of states
Keep finite rules
Turing Machines
Informal definition:
• Finite number of states
• Tape without end. Finite input starts at one end. A
special symbol stands for blank. Initial tape holds input
and then blanks.
• TM can write on the tape and
• TM can go back and forth.
• TM accepts a string (which may no longer be on the
tape) if it stops in a state labeled as an accepting state.
• TM rejects a string if it stops in a state labeled as a
rejecting state.
• TM may not stop!
Turing Machines
Formal definition: A TM is a 7-tuple:
Q set of states
A input alphabet
G tape alphabet, including a blank character and A is
contained in G. The TM can write and read extra
symbols.
Transition function: f: QxG  QxGx {L,R}, meaning for each
pair of states and symbols in tape alphabet, there is a
[possibly new] state, a possibly, new symbol written on
the tape, and the TM head moves left or right.
q0 is the initial state
qaccept is a state in Q that is the accepting state.
qreject is a state in Q that is the rejecting state. Not equal to
accepting state.
Alternative formalism
• TM is a set of quintuples: <qj,ai,qk,bn,M> where
M is L for left or R for right or H for halt and
interpretation for <qj,ai,qk,bn,R> would be: in
state qj reading ai, change to state qk, write bn
and move right on the tape.
• [Note: some books use quadruples: you write or
move, not both. Can simulate the quintuple way
by specifying additional states.]
Comment
• Using quintuples may be easier way to
think about the transition function,
especially when adding non-determinism.
• Think: is there a quintuple starting with
qj,ai, ? or Take all the quintuples …
Advance notice
• Later, define TM for computing functions
– input is encoded on tape
• (n+1) 1s standing for the the number n
– output is what remains on the tape after TM
halts.
FSM can be simulated by TM
• How?
FSM
• Define a TM with the same states as the FSM plus 1
designated accepting state and 1 designated reject
state.
• Starting state is the TM starting state..
• Nearly same transition function: state q scanning symbol
a goes to state p and move right if state q goes to state p
in FSM. Never write on the tape.
• When machine sees blank, if in one of accepting states
for the FSM, go to new designated accepting state for
TM. If NOT in one of the FSM accepting states, go to the
new designated reject state.
Pushdown automata
What to do?
• Add new tape symbols for start S and end E of
stack and create additional states, including start
state, that scans to end of input string and
places S and E
– 1011 input. add 1011SE
• Using instructions of PDA, for every instruction
that has state q, scanning a, add b to stack,
change to state p
– ???
FSM examples directly
• {1n | n is divisible by 3 }
• { larry, curly, moe}
• { all strings w that contain larry, curly, moe
somewhere }
• ?
PDA example directly
• {wwR | w a string in {0,1}* }
{ww
| w a string in {0,1}* }
• So we don't have a stack, but we can write
on tape, specifically we can erase….
Classwork/Homework
Describe TMs to recognize
• {0n1n | n>=0}
• {anbncn | n >= 0}
• {0p | p = 2n for n>=0} See Sipser text.
For each compare to what is allowed and
what is not allowed (available) in FSM or
PDA.
Homework
• Video clip of sections from one of Turing's
papers:
http://www.youtube.com/watch?v=mPec64
RUCsk&feature=related
• Post response on moodle