This document discusses variants of Turing machines that increase their computational power and proves that any language accepted by a variant Turing machine is also accepted by a standard Turing machine. It covers variants such as multiple tapes, two-way infinite tapes, and nondeterminism. For each variant, it provides a proof that the standard Turing machine is equally powerful by simulating the variant with the standard model.

1. Variants of Turing
Machines
Lecture 26
Section 3.2
Mon, Oct 22, 2007

2. Increasing the Power of
a Turing Machine
• It is hard to believe that
something as simple as a Turing
machine could be powerful
enough for complicated
problems.

3. Increasing the Power of
a Turing Machine
• We can imagine a number of
improvements.
• Multiple tapes
• Two-way infinite tape
• Two-dimensional tape
• Addressable memory
• Nondeterminism
• etc.

4. Multiple Tapes
• Would a Turing machine with k
tapes, k > 1, be more powerful
than a standard Turing
machine?
• Each tape could be processed
independently of the others.

5. Multiple Tapes
• In other words, each transition
would read each tape, write to
each tape, and move left or
right independently on each
tape.

6. Multiple Tapes
• Theorem: Any language that is
accepted by a multitape Turing
machine is also accepted by a
standard Turing machine.

7. Multiple Tapes
• Sketch of the proof:
• On a single tape, we could write
the contents of all k tapes.
• If tape i contains xi1xi2xi3…, for each
i, then write
#x11x21…xk1#x12x22…xk2#...
on the single tape.

8. Multiple Tapes
• To show the current location on
each tape, put a special mark on
one of that tapes symbols:
#x11x21…xk1#x12x22…xk2#...
• Now the Turing machine scans the
tape, locating the current symbol
on each “tape.”

9. Multiple Tapes
• It then makes the appropriate
transition.
• Write a symbol over each of the
current symbols.
• Move the location of the current
symbol one space left or right for each
“tape.”
• Of course, the devil is in the
details.

10. Two-way Infinite Tape
• We can use a two-tape machine
to simiulate the two-way infinite
tape.
• The right half of the two-way tape
is stored on tape 1.
• The left half is stored on tape 2.
• Transitions are modified to handle
the transition from tape 1 to tape
2.

11. Two-way Infinite Tape
• Theorem: Any language
accepted by a two-way infinite
tape is also accepted by a
standard Turing machine.

12. Other Variants
• Metatheorem: Any language
accepted by a Turing machine
with any variant that anyone
has ever thought of is also
accepted by a standard Turing
machine.

13. Nondeterminism
• A nondeterministic Turing
machine is defined like a
standard Turing machine except
for the transition function.
δ : Q′ × Γ → ℘(Q × Γ × {L, R})
where Q′ = Q – {qacc, qrej}.

14. Nondeterminism
• That is, δ(q, a) may result in any
of a number of actions.
• If any sequence of transitions
leads to the accept state, then
the input is accepted.
• If all sequences of transitions
lead to the reject state or to
looping, then the input is not
accepted.

15. Nondeterminism
• Theorem: Any language
accepted by a nondeterministic
Turing machine is also
accepted by a standard Turing
machine.

16. Nondeterminism
• Proof:
• We may use a three-tape machine
to simulate a nondeterministic
Turing machine.
• Tape 1 preserves a copy of the original
input.
• Tape 2 contains a “working” copy of
the input.
• Tape 3 keeps track of the current state
in the nondeterministic machine.

17. Nondeterminism
• Start with the input w on Tape 1
and with Tapes 2 and 3 empty.
• Copy w from Tape 1 to Tape 2.
• For Tape 3, imagine the transitions
starting from the start state as
forming a tree.
• Each state has child states.

18. Nondeterminism
• Let b be the largest number of
children of any node.
• Number the children of each state
using the numbers 1, 2, …, b (as
many as needed).

19. Nondeterminism
• Now each finite string of numbers
from {1, 2, …, b} represents a
particular path through the
nondeterministic Turing machine,
or else represents no path at all.
• Beginning by writing the empty
string on Tape 3, representing no
moves at all.

20. Nondeterminism
• If that leads to acceptance, then
quit.
• If not, then replace ε with its
lexicographical successor.
• For that string, follow the
sequence of transitions that it
describes.

21. Nondeterminism
• If that sequence leads to
acceptance, then quit and accept.
• If not, then continue in the same
manner.

22. Nondeterminism
• If w is accepted by the
nondeterministic Turing machine,
then some sequence of transitions
leads to the accept state.
• Eventually that sequence will be
written on Tape 3 and tried.

23. Nondeterminism
• On the other hand, if no sequence
of transitions leads to the accept
state, then the deterministic
Turing machine will loop.