1. Consider the problem of conputing the XOR of two binary and determining if the result is equal to a third binary number. Suppose the input tape contains three binary numbers, separated by $ symbols. Each binary number consists of one or more bits with one bit per cell. E.g.,
# 0 1 1 0 $ 0 1 0 1 $ 0 0 1 1 # #
(a) Describe a Turing Machine that accepts if the XOR of the first two binary numbers is equal to the third binary number and rejects otherwise. Note: You just need
to describe how the TM would function as we have done in lectures. You do not need
to specify the TM in full. Feel free to describe a multi-tape Turing Machine if that is
easier.
(b) [5 marks] Describe a Random Access Machine that accepts if the XOR of the first two
binary numbers is equal to the third binary number and rejects otherwise. Note: You
just need to describe how the RAM would function as we have done in lectures.
2. Consider the problem of checking if an XOR of n binary numbers is equal to 0. The input
tape contains n binary numbers, separated by $ symbols.
(a) Describe a Turing Machine that accepts if the XOR of the binary numbers
is 0. Note: You just need to describe how the TM would function as we have done in
lectures. You do not need to specify the TM in full. Feel free to describe a multi-tape
Turing Machine if that is easier.
(b) Describe a Random Access Machine that accepts if the XOR of the binary
numbers is 0 and rejects otherwise. Note: You just need to describe how the RAM would
function as we have done in lectures.
Marking Scheme
Marking scheme for Question 1a, 1b, 2a, and 2b.
3 points 2 points 1 points 0 points
Correctness Machine would work as
described
Machine may not work
on some inputs
Reasonable attempt
made No answer provided
Description Description is suffi-
ciently detailed
Some key steps are miss-
ing
Very poor description or
no answer provided
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