proving statements involving primitive recursive functions and relations and (not) computable functions
$begingroup$
I am learning about primitive recursive functions and primitive recursive relations (whereby a relation $R subseteq mathbb{N}^n$ is called primitive recursive if its characteristic function is primitive recursive).
I am not managing to prove the following, so I'd very much appreciate your help and ideas:
a) If $E subseteq mathbb{N}^2$ is a primitive recursive relation and $f, g: mathbb{N} rightarrow mathbb{N}$ are primitive recursive, then also ${(n,k): (f(n), g(k)) in E}$ is primitive recursive.
b) There is a not computable total function $g: mathbb{N} times mathbb{N} rightarrow mathbb{N}$, such that the functions $f_0, f_1, cdots, h_0, h_1, cdots: mathbb{N} rightarrow mathbb{N}$ defined by $f_n(k): = g(n,k)$ and $h_k(n): = g(n,k)$ are all computable (even primitive recursive).
c) There is a computable partial function $f: mathbb{N}timesmathbb{N} rightarrow mathbb{N}$ such that the partial function $g(x): = min{y: f(x,y) = 0}$ is not computable.
Hint: a function f such that $f(x,1) = 0 ; forall x$ can be found.
I don't know how to aproach a) and neither do I have any idea for b) and c) nor do I know how to approach these kinds of problems, so I'm really looking forward to your replies. Thanks in advance!
functions recursion computability
asked Dec 15 '18 at 20:59
StudentuStudentu
1279
$endgroup$
add a comment |
$begingroup$
I am learning about primitive recursive functions and primitive recursive relations (whereby a relation $R subseteq mathbb{N}^n$ is called primitive recursive if its characteristic function is primitive recursive).
I am not managing to prove the following, so I'd very much appreciate your help and ideas:
a) If $E subseteq mathbb{N}^2$ is a primitive recursive relation and $f, g: mathbb{N} rightarrow mathbb{N}$ are primitive recursive, then also ${(n,k): (f(n), g(k)) in E}$ is primitive recursive.
b) There is a not computable total function $g: mathbb{N} times mathbb{N} rightarrow mathbb{N}$, such that the functions $f_0, f_1, cdots, h_0, h_1, cdots: mathbb{N} rightarrow mathbb{N}$ defined by $f_n(k): = g(n,k)$ and $h_k(n): = g(n,k)$ are all computable (even primitive recursive).
c) There is a computable partial function $f: mathbb{N}timesmathbb{N} rightarrow mathbb{N}$ such that the partial function $g(x): = min{y: f(x,y) = 0}$ is not computable.
Hint: a function f such that $f(x,1) = 0 ; forall x$ can be found.
I don't know how to aproach a) and neither do I have any idea for b) and c) nor do I know how to approach these kinds of problems, so I'm really looking forward to your replies. Thanks in advance!
functions recursion computability
asked Dec 15 '18 at 20:59
StudentuStudentu
1279
$endgroup$
$begingroup$
Is anything unclear? If so, please let me know. The definition of primitive recursive relations is the one you find on Wikipedia.
$endgroup$
– Studentu
Dec 16 '18 at 18:34
add a comment |
$begingroup$
I am learning about primitive recursive functions and primitive recursive relations (whereby a relation $R subseteq mathbb{N}^n$ is called primitive recursive if its characteristic function is primitive recursive).
I am not managing to prove the following, so I'd very much appreciate your help and ideas:
a) If $E subseteq mathbb{N}^2$ is a primitive recursive relation and $f, g: mathbb{N} rightarrow mathbb{N}$ are primitive recursive, then also ${(n,k): (f(n), g(k)) in E}$ is primitive recursive.
b) There is a not computable total function $g: mathbb{N} times mathbb{N} rightarrow mathbb{N}$, such that the functions $f_0, f_1, cdots, h_0, h_1, cdots: mathbb{N} rightarrow mathbb{N}$ defined by $f_n(k): = g(n,k)$ and $h_k(n): = g(n,k)$ are all computable (even primitive recursive).
c) There is a computable partial function $f: mathbb{N}timesmathbb{N} rightarrow mathbb{N}$ such that the partial function $g(x): = min{y: f(x,y) = 0}$ is not computable.
Hint: a function f such that $f(x,1) = 0 ; forall x$ can be found.
I don't know how to aproach a) and neither do I have any idea for b) and c) nor do I know how to approach these kinds of problems, so I'm really looking forward to your replies. Thanks in advance!
functions recursion computability
asked Dec 15 '18 at 20:59
StudentuStudentu
1279
$endgroup$
I am learning about primitive recursive functions and primitive recursive relations (whereby a relation $R subseteq mathbb{N}^n$ is called primitive recursive if its characteristic function is primitive recursive).
I am not managing to prove the following, so I'd very much appreciate your help and ideas:
a) If $E subseteq mathbb{N}^2$ is a primitive recursive relation and $f, g: mathbb{N} rightarrow mathbb{N}$ are primitive recursive, then also ${(n,k): (f(n), g(k)) in E}$ is primitive recursive.
b) There is a not computable total function $g: mathbb{N} times mathbb{N} rightarrow mathbb{N}$, such that the functions $f_0, f_1, cdots, h_0, h_1, cdots: mathbb{N} rightarrow mathbb{N}$ defined by $f_n(k): = g(n,k)$ and $h_k(n): = g(n,k)$ are all computable (even primitive recursive).
c) There is a computable partial function $f: mathbb{N}timesmathbb{N} rightarrow mathbb{N}$ such that the partial function $g(x): = min{y: f(x,y) = 0}$ is not computable.
Hint: a function f such that $f(x,1) = 0 ; forall x$ can be found.
I don't know how to aproach a) and neither do I have any idea for b) and c) nor do I know how to approach these kinds of problems, so I'm really looking forward to your replies. Thanks in advance!
functions recursion computability
functions recursion computability
asked Dec 15 '18 at 20:59
StudentuStudentu
1279
asked Dec 15 '18 at 20:59
StudentuStudentu
1279
asked Dec 15 '18 at 20:59
StudentuStudentu
1279
asked Dec 15 '18 at 20:59
StudentuStudentu
1279
asked Dec 15 '18 at 20:59
StudentuStudentu
1279
1279
$begingroup$
Is anything unclear? If so, please let me know. The definition of primitive recursive relations is the one you find on Wikipedia.
$endgroup$
– Studentu
Dec 16 '18 at 18:34
add a comment |
$begingroup$
Is anything unclear? If so, please let me know. The definition of primitive recursive relations is the one you find on Wikipedia.
$endgroup$
– Studentu
Dec 16 '18 at 18:34
$begingroup$
Is anything unclear? If so, please let me know. The definition of primitive recursive relations is the one you find on Wikipedia.
$endgroup$
– Studentu
Dec 16 '18 at 18:34
$begingroup$
Is anything unclear? If so, please let me know. The definition of primitive recursive relations is the one you find on Wikipedia.
$endgroup$
– Studentu
Dec 16 '18 at 18:34
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3041950%2fproving-statements-involving-primitive-recursive-functions-and-relations-and-no%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3041950%2fproving-statements-involving-primitive-recursive-functions-and-relations-and-no%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
Is anything unclear? If so, please let me know. The definition of primitive recursive relations is the one you find on Wikipedia.
$endgroup$
– Studentu
Dec 16 '18 at 18:34