ass1_2_2006

ass1_2_2006 - m,n computes the remainder of m after...

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MATH 335 001 Computational Algebra Winter 2006 Assignment 2. Part 1. 1) Find a polynomial bijective enumeration π : N × N N . [ Hint : Look at the Cantor’s enumeration of pairs] 2) a) Let N 1 = h N , + , 0 i be the systems of natural numbers equipped with ad- dition. Write down a program for a RAM over N 1 that for given numbers m,n N computes their greatest common divisor gcd ( m,n ). [ Hint : Use the Euclidean algorithm] b) Estimate the time complexity of the algorithm in a), i.e., for given m,n estimate how many steps are required for the RAM to find gcd ( m,n ). 3) a) Let N 2 = h N , + , - , · ,rest, 0 i be the systems of natural numbers equipped with addition, subtraction, multiplication, and the function rest ( m,n ) that for given
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Unformatted text preview: m,n computes the remainder of m after division by n . Write down a program for a RAM over N 2 that for given numbers m,n ∈ N computes their greatest common divisor gcd ( m,n ). [ Hint : Use the Euclidean algorithm] b) Estimate the time complexity of the algorithm in a). 4) Let γ : M → γ ( M ) ∈ N be the enumeration of Turing machine programs M discussed in the class. Find the program M such that γ ( M ) = 24. 5) a) Enumerate effectively all programs for Random Access Machines (RAM). b) Explain how one can try to write down a program for universal RAM for one variable functions....
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