Lecture 10: Decidability
David Dill and C
´
esar S
´
anchez
Department of Computer Science
1
Outline
•
Multiple tapes
•
Nondeterministic TMs
•
Weakening the TM model
–
semiinfinite tape
–
multistack PDAs
–
threecounter machines
–
twocounter machines
•
P vs. NP
•
Recursively enumerable and Recursive languages
2
Extensions to Turing Machines
Since Turing machines are already as powerful as any real model of computation
can be, adding stuff to them doesn’t increase their power.
It rarely even speeds them up all that much.
Multitape Turing Machine
It has
k
tapes, each with a separate head.
The input is on the first tape with the head on the first nonblank. All other tapes
are blank, and the heads are in arbitrary positions (doesn’t matter).
The transition function depends on the state and currently scanned symbol on
each tape, and updates the state and writes a new symbol on each tape and
moves each head.
Each head can also remain stationary instead of moving left or right.
Everything else is the same.
3
Equivalence of Single and MultiTape Turing Machines
(I’ll present the basic ideas. Details are in the book.)
Obviously, an MTTM can do anything a singletape TM can do.
Construction in the other direction: Create a TM
N
that simulates a MTTM
M
as
follows:
N
is a multitrack (but single tape) machine.
Use 2 tracks for each simulated tape. One track has the tape contents, the other
has a marker for the head position.
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 Winter '08
 Motwani,R
 Halting problem, Turing Machines, tape, polynomial time

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