(c) Let L be the language of all equations + a1Xd-1 +...+ ad-1X + ad = 0, with unknown X and integer coefficients ao, a1,..., ad-1, ɑd, 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 aoxd + a,xd-1+...+ ad-1X + ad = 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 € 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 + a1Xd-1 +...+ ad-1X + ad = 0, with unknown X and integer coefficients ao, a1,..., ad-1, ɑd, 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 aoxd + a,xd-1+...+ ad-1X + ad = 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 € Z, decides whether or not it has integer solutions, and if it does, finds at least one such solution.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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
aoxd + a,Xd-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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