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(a) Let n > 0 be an integer and Ln be the language of all linear equations 3. a1X1 + 02X2 + .+ an Xn + an+1 = 0 in n unknowns X1, X2,..., Xn and over integer coefficients a1,a2,..., an, an+1, which have a solution in the integers. (i) Show that Ln is semi-decidable by describing, in general mathematical terms, an algorithm that takes as input a linear equation a1X1 + a2X2 +…+ anXn+ an+1 = 0 %3D with a1, a2,..., an, an+1 in the integers, and halts exactly when this equation has a solution in the integers. (ii) We now use the fact (stated in the lectures) that Ln is actually decidable. Describe an algorithm, using a decider for Ln as a subroutine, which for any linear equation a,X1 + a2X2+ ...+ an Xn + an+1 = 0, with a1, a2,..., an, an+1 in the integers, decides whether or not it has an integer solution, and if it does, finds at least one such solution.

(a) Let n > 0 be an integer and Ln be the language of all linear equations 3. a1X1 + 02X2 + .+ an Xn + an+1 = 0 in n unknowns X1, X2,..., Xn and over integer coefficients a1,a2,..., an, an+1, which have a solution in the integers. (i) Show that Ln is semi-decidable by describing, in general mathematical terms, an algorithm that takes as input a linear equation a1X1 + a2X2 +…+ anXn+ an+1 = 0 %3D with a1, a2,..., an, an+1 in the integers, and halts exactly when this equation has a solution in the integers. (ii) We now use the fact (stated in the lectures) that Ln is actually decidable. Describe an algorithm, using a decider for Ln as a subroutine, which for any linear equation a,X1 + a2X2+ ...+ an Xn + an+1 = 0, with a1, a2,..., an, an+1 in the integers, decides whether or not it has an integer solution, and if it does, finds at least one such solution.

Computer Networking: A Top-Down Approach (7th Edition)
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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3. (a) Let n > 0 be an integer and L, be the language of all linear equations a1X1 + 02X2 + .. + an Xn + an+1 = 0 in n unknowns X1, X2,...,Xn and over integer coefficients a1,a2,..., an, an+1, which have a solution in the integers. (i) Show that Ln is semi-decidable by describing, in general mathematical terms, an algorithm that takes as input a linear equation a1X1 + a2X2 + .+ a,Xn + an+1 = 0 with a1, a2,..., an, an+1 in the integers, and halts exactly when this equation has a solution in the integers. (ii) We now use the fact (stated in the lectures) that L is actually decidable. Describe an algorithm, using a decider for Ln as a subroutine, which for any linear equation a1X1 + 42X2+ ..+ an Xn + an+1 = 0, with a1, a2,..., an, an+1 in the integers, decides whether or not it has an integer solution, and if it does, finds at least one such solution.
Transcribed Image Text:3. (a) Let n > 0 be an integer and L, be the language of all linear equations a1X1 + 02X2 + .. + an Xn + an+1 = 0 in n unknowns X1, X2,...,Xn and over integer coefficients a1,a2,..., an, an+1, which have a solution in the integers. (i) Show that Ln is semi-decidable by describing, in general mathematical terms, an algorithm that takes as input a linear equation a1X1 + a2X2 + .+ a,Xn + an+1 = 0 with a1, a2,..., an, an+1 in the integers, and halts exactly when this equation has a solution in the integers. (ii) We now use the fact (stated in the lectures) that L is actually decidable. Describe an algorithm, using a decider for Ln as a subroutine, which for any linear equation a1X1 + 42X2+ ..+ an Xn + an+1 = 0, with a1, a2,..., an, an+1 in the integers, decides whether or not it has an integer solution, and if it does, finds at least one such solution.
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