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3) Construct a standard, deterministic, one-tape Turing machine M to compute each of the following functions: a) The function sub3, which is defined as follows: sub3(n) = n-3 if n > 2 0 if n ? 2. Specifically, compute sub3 of a natural number represented in binary. For example, on input 10111, M should output 10100. On input 11101, M should output 11010. (Hint: you may want to define a subroutine.) b) Addition of two binary natural numbers (as described in Example 17.13). Specifically, given the input string <x>; <y>, where <x> is the binary encoding of a natural number x and <y> is the binary encoding of a natural number y, M should output <z>, where z is the binary encoding of x + y. For example, on input 101; 11, M should output 1000. c) Multiplication of two unary numbers. Specifically, given the input string <x>; <y>, where <x> is the unary encoding of a natural number x and <y> is the unary encoding of a natural number y, M should output <z>, where z is the unary encoding of xy. For example, on input 111; 1111, Mshould output 111111111111. d) The proper subtraction function monus, which is defined as follows: monus(n, m) = \begin{cases} n-m & \text{if } n > m \\ 0 & \text{if } n ? m \end{cases} Specifically, compute monus of two natural numbers represented in binary. For example, on input 101; 11, M should output 1. On input 11; 101, M should output 0.

          3) Construct a standard, deterministic, one-tape Turing machine M to compute each of the following functions:
a) The function sub3, which is defined as follows:
sub3(n) = n-3 if n > 2
0 if n ? 2.
Specifically, compute sub3 of a natural number represented in binary. For example, on input 10111, M
should output 10100. On input 11101, M should output 11010. (Hint: you may want to define a
subroutine.)
b) Addition of two binary natural numbers (as described in Example 17.13). Specifically, given the input string
<x>; <y>, where <x> is the binary encoding of a natural number x and <y> is the binary encoding of a natural
number y, M should output <z>, where z is the binary encoding of x + y. For example, on input 101; 11, M
should output 1000.
c) Multiplication of two unary numbers. Specifically, given the input string <x>; <y>, where <x> is the unary
encoding of a natural number x and <y> is the unary encoding of a natural number y, M should output <z>,
where z is the unary encoding of xy. For example, on input 111; 1111, Mshould output 111111111111.
d) The proper subtraction function monus, which is defined as follows:
monus(n, m) = \begin{cases} n-m & \text{if } n > m \\ 0 & \text{if } n ? m \end{cases}
Specifically, compute monus of two natural numbers represented in binary. For example, on input 101; 11,
M should output 1. On input 11; 101, M should output 0.

3) Construct a standard, deterministic, one-tape Turing machine M to compute each of the following functions:
a) The function sub3, which is defined as follows:
sub3(n) = n-3 if n > 2
0 if n ? 2.
Specifically, compute sub3 of a natural number represented in binary. For example, on input 10111, M
should output 10100. On input 11101, M should output 11010. (Hint: you may want to define a
subroutine.)
b) Addition of two binary natural numbers (as described in Example 17.13). Specifically, given the input string
<x>; <y>, where <x> is the binary encoding of a natural number x and <y> is the binary encoding of a natural
number y, M should output <z>, where z is the binary encoding of x + y. For example, on input 101; 11, M
should output 1000.
c) Multiplication of two unary numbers. Specifically, given the input string <x>; <y>, where <x> is the unary
encoding of a natural number x and <y> is the unary encoding of a natural number y, M should output <z>,
where z is the unary encoding of xy. For example, on input 111; 1111, Mshould output 111111111111.
d) The proper subtraction function monus, which is defined as follows:
monus(n, m) = n-m if n > m
0 if n ? m
Specifically, compute monus of two natural numbers represented in binary. For example, on input 101; 11,
M should output 1. On input 11; 101, M should output 0.

Added by Heather H.

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