A 2-dimensional Turing machine has an infinite 2-dimensional grid as its storage device (one cell for every address (i, j) € Z x Z). Instcad of L and R, there are 5 options for the head to move from (i, j): Left (to (i – 1, j), Right (to (i + 1,j)), Up (to (i, j+1)), Down (to (i,j – 1)), and Stay (remain at (i, j)) . Let us assume, that our 2-dimensional TM M has an additional 1– dimensional tape where the input is located as in a l-tape TM. The 2-dimensional storage device contains a blank symbol in every cell at the start of the computation. Show that a 2-dimensional TM is not more powerful, i.c., every lan- guage accepted by a 2-dimensional TM M is accepted by one of our standard 1-dimensional TMs. You don't have to simulate directly with a l-tape TM, as you are allowed to use what we know about k-tapc (1-dimensional) TMs without proof. Describe the moves of a simulating TM M' in plain English. For every possible move of M, you have to design a sequence of moves of M' that simulates that move of M. Hints: Your simulation does not have to be efficient. You may store in- formation about any nonempty cell of M anywhere and in any format on the tape of M'. But you have to make sure that your stored infor- mation is consistent and can be found by M'. The simulating TM M' might use coordinates to decfine locations on the 2-dimensional tape.
A 2-dimensional Turing machine has an infinite 2-dimensional grid as its storage device (one cell for every address (i, j) € Z x Z). Instcad of L and R, there are 5 options for the head to move from (i, j): Left (to (i – 1, j), Right (to (i + 1,j)), Up (to (i, j+1)), Down (to (i,j – 1)), and Stay (remain at (i, j)) . Let us assume, that our 2-dimensional TM M has an additional 1– dimensional tape where the input is located as in a l-tape TM. The 2-dimensional storage device contains a blank symbol in every cell at the start of the computation. Show that a 2-dimensional TM is not more powerful, i.c., every lan- guage accepted by a 2-dimensional TM M is accepted by one of our standard 1-dimensional TMs. You don't have to simulate directly with a l-tape TM, as you are allowed to use what we know about k-tapc (1-dimensional) TMs without proof. Describe the moves of a simulating TM M' in plain English. For every possible move of M, you have to design a sequence of moves of M' that simulates that move of M. Hints: Your simulation does not have to be efficient. You may store in- formation about any nonempty cell of M anywhere and in any format on the tape of M'. But you have to make sure that your stored infor- mation is consistent and can be found by M'. The simulating TM M' might use coordinates to decfine locations on the 2-dimensional tape.
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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Transcribed Image Text:A 2-dimensional Turing machine has an infinite 2-dimensional grid as
its storage device (one cell for every address (i, j) E Z ×Z). Instead of
L and R, there are 5 options for the head to move from (i, j): Left (to
(i – 1, j), Right (to (i+1,j)), Up (to (i, j+ 1)), Down (to (i,j – 1)),
and Stay (remain at (i, j)).
Let us assume, that our 2-dimensional TM M has an additional 1-
dimensional tape where the input is located as in a 1-tape TM. The
2-dimensional storage device contains a blank symbol in every cell at
the start of the computation.
Show that a 2-dimensional TM is not more powerful, i.e., cvery lan-
guage accepted by a 2-dimensional TM M is accepted by one of our
standard 1-dimensional TMs. You don't have to simulate directly with
a l-tape TM, as you are allowed to use what we know about k-tape
(1-dimensional) TMs without proof.
Describe the moves of a simulating TM M' in plain English. For every
possible move of M, you have to design a sequence of moves of M'
that simulates that move of M.
Hints: Your simulation does not have to be efficient. You may store in-
formation about any nonempty cell of M anywhere and in any format
on the tape of M'. But you have to make sure that your stored infor-
mation is consistent and can be found by M'. The simulating TM M'
might use coordinates to define locations on the 2-dimensional tape.
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