(c) Let L be the language of all equations anX" + a1Xd-1+...+ ad-1X +ad = 0, with unknown X and integer coefficients ao, a1,...,ad-1, ad, which have a solution in the integers. (i) Show that L is semi-decidable by describing, in general mathematical terms, an algorithm that takes an cquation aoxd + a,xd-1 +...+ ad-1X + ag = 0 with ao, a2,..., ad-1,ad E Z as input and halts exactly when this equation has a solution in the integers. (ii) Using the fact that L is actually decidable, describe an algorithm which for any equation aoXd +a1Xd-1+...+ ad-1X +ad =0 with ao, a2,...,ad-1, ad E Z, decides whether or not it has integer solutions, and if it does, finds at least one such solution.
(c) Let L be the language of all equations anX" + a1Xd-1+...+ ad-1X +ad = 0, with unknown X and integer coefficients ao, a1,...,ad-1, ad, which have a solution in the integers. (i) Show that L is semi-decidable by describing, in general mathematical terms, an algorithm that takes an cquation aoxd + a,xd-1 +...+ ad-1X + ag = 0 with ao, a2,..., ad-1,ad E Z as input and halts exactly when this equation has a solution in the integers. (ii) Using the fact that L is actually decidable, describe an algorithm which for any equation aoXd +a1Xd-1+...+ ad-1X +ad =0 with ao, a2,...,ad-1, ad E Z, decides whether or not it has integer solutions, and if it does, finds at least one such solution.
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:(c) Let L be the language of all equations
aoXd + a1Xd- +...+ ad-1X +ad = 0,
with unknown X and integer coefficients ap, a1,...,ad-1, ad, which have a solution
in the integers.
(1) Show that L is semi-decidable by describing, in general mathematical terms,
an algorithm that takes an equation
+a,xd-1 +...+ ad-1X + ad = 0
with ao, a2,...,ad-1,ad E Z as input and halts exnctly when this equation has
a solution in the integers.
(ii) Using the fact that L is actually decidable, describe an algorithm which for any
equation
+ajXd-1 +...+ad-1X + ad = 0
with ao, a2,...,ad-1, ad € Z, decides whether or not it has integer solutions,
and if it does, finds at least one such solution.
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