How can I prove given language is not regular?
My first post here, so glad I found this great place. Hoping I could improve and learn a lot from you and contribute in the future if I can.
I have a problem with the following scenario:
Given $Sigma= {a,b,c,d}$, prove that $L$ is not regular, where $$L = {a^ib^jcd^k mid i geq 0; k > j > 0}$$
using: 1) pumping lemma 2) closure properties.
Regarding 1)
Assuming that $L$ is regular, using the pumping lemma there should exist some integer $n$ (pumping length). Choosing a word $w in L$ (not sure if I chose the right word). If $w=a^ib^jcd^k$ ($|w|>n$), but I cannot find a way to obtain $xy^i z notin L$, would very appreciate an explanation or example of how you found the term so it can be concluded that $L$ is not regular using the pumping lemma.
Regarding 2)
Assuming that $L$ is regular, it means that $a^ib^jcd^k$ is also regular. So it should be closed under intersection, however, the form of ${a^ib^jcd^k mid igeq 0;k>j>0}$ is not regular, and it can be achieved using completion, so contradiction and because of that $L$ is not regular.
Hoping to learn from my mistakes and improve.
Thank you very much for your aid.
computer-science automata regular-language
edited Dec 4 at 6:20
J.-E. Pin
18.3k21754
asked Dec 2 at 11:27
mathnoobie
93
add a comment |
My first post here, so glad I found this great place. Hoping I could improve and learn a lot from you and contribute in the future if I can.
I have a problem with the following scenario:
Given $Sigma= {a,b,c,d}$, prove that $L$ is not regular, where $$L = {a^ib^jcd^k mid i geq 0; k > j > 0}$$
using: 1) pumping lemma 2) closure properties.
Regarding 1)
Assuming that $L$ is regular, using the pumping lemma there should exist some integer $n$ (pumping length). Choosing a word $w in L$ (not sure if I chose the right word). If $w=a^ib^jcd^k$ ($|w|>n$), but I cannot find a way to obtain $xy^i z notin L$, would very appreciate an explanation or example of how you found the term so it can be concluded that $L$ is not regular using the pumping lemma.
Regarding 2)
Assuming that $L$ is regular, it means that $a^ib^jcd^k$ is also regular. So it should be closed under intersection, however, the form of ${a^ib^jcd^k mid igeq 0;k>j>0}$ is not regular, and it can be achieved using completion, so contradiction and because of that $L$ is not regular.
Hoping to learn from my mistakes and improve.
Thank you very much for your aid.
computer-science automata regular-language
edited Dec 4 at 6:20
J.-E. Pin
18.3k21754
asked Dec 2 at 11:27
mathnoobie
93
add a comment |
My first post here, so glad I found this great place. Hoping I could improve and learn a lot from you and contribute in the future if I can.
I have a problem with the following scenario:
Given $Sigma= {a,b,c,d}$, prove that $L$ is not regular, where $$L = {a^ib^jcd^k mid i geq 0; k > j > 0}$$
using: 1) pumping lemma 2) closure properties.
Regarding 1)
Assuming that $L$ is regular, using the pumping lemma there should exist some integer $n$ (pumping length). Choosing a word $w in L$ (not sure if I chose the right word). If $w=a^ib^jcd^k$ ($|w|>n$), but I cannot find a way to obtain $xy^i z notin L$, would very appreciate an explanation or example of how you found the term so it can be concluded that $L$ is not regular using the pumping lemma.
Regarding 2)
Assuming that $L$ is regular, it means that $a^ib^jcd^k$ is also regular. So it should be closed under intersection, however, the form of ${a^ib^jcd^k mid igeq 0;k>j>0}$ is not regular, and it can be achieved using completion, so contradiction and because of that $L$ is not regular.
Hoping to learn from my mistakes and improve.
Thank you very much for your aid.
computer-science automata regular-language
edited Dec 4 at 6:20
J.-E. Pin
18.3k21754
asked Dec 2 at 11:27
mathnoobie
93
My first post here, so glad I found this great place. Hoping I could improve and learn a lot from you and contribute in the future if I can.
I have a problem with the following scenario:
Given $Sigma= {a,b,c,d}$, prove that $L$ is not regular, where $$L = {a^ib^jcd^k mid i geq 0; k > j > 0}$$
using: 1) pumping lemma 2) closure properties.
Regarding 1)
Assuming that $L$ is regular, using the pumping lemma there should exist some integer $n$ (pumping length). Choosing a word $w in L$ (not sure if I chose the right word). If $w=a^ib^jcd^k$ ($|w|>n$), but I cannot find a way to obtain $xy^i z notin L$, would very appreciate an explanation or example of how you found the term so it can be concluded that $L$ is not regular using the pumping lemma.
Regarding 2)
Assuming that $L$ is regular, it means that $a^ib^jcd^k$ is also regular. So it should be closed under intersection, however, the form of ${a^ib^jcd^k mid igeq 0;k>j>0}$ is not regular, and it can be achieved using completion, so contradiction and because of that $L$ is not regular.
Hoping to learn from my mistakes and improve.
Thank you very much for your aid.
computer-science automata regular-language
computer-science automata regular-language
edited Dec 4 at 6:20
J.-E. Pin
18.3k21754
asked Dec 2 at 11:27
mathnoobie
93
edited Dec 4 at 6:20
J.-E. Pin
18.3k21754
asked Dec 2 at 11:27
mathnoobie
93
edited Dec 4 at 6:20
J.-E. Pin
18.3k21754
edited Dec 4 at 6:20
J.-E. Pin
18.3k21754
edited Dec 4 at 6:20
J.-E. Pin
18.3k21754
18.3k21754
asked Dec 2 at 11:27
mathnoobie
93
asked Dec 2 at 11:27
mathnoobie
93
asked Dec 2 at 11:27
mathnoobie
93
93
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
For (1): You assume the language satisfies the pumping lemma for some $n$.
You want to choose a word that would have at least one of its letters related to $n$.
A good start would be taking one of your letters $lin w$ to be $l^n$, and using that to show that for some $i$, $xy^iznotin L$
For (2): You want to show that if $L$ is regular, then some other language $L_{not}$ which you know is not regular, is the intersection of another regular language $L_{reg}cap L=L_{not}$.
answered Dec 3 at 4:34
NL1992
8311
add a comment |
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%2f3022519%2fhow-can-i-prove-given-language-is-not-regular%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
For (1): You assume the language satisfies the pumping lemma for some $n$.
You want to choose a word that would have at least one of its letters related to $n$.
A good start would be taking one of your letters $lin w$ to be $l^n$, and using that to show that for some $i$, $xy^iznotin L$
For (2): You want to show that if $L$ is regular, then some other language $L_{not}$ which you know is not regular, is the intersection of another regular language $L_{reg}cap L=L_{not}$.
answered Dec 3 at 4:34
NL1992
8311
add a comment |
For (1): You assume the language satisfies the pumping lemma for some $n$.
You want to choose a word that would have at least one of its letters related to $n$.
A good start would be taking one of your letters $lin w$ to be $l^n$, and using that to show that for some $i$, $xy^iznotin L$
For (2): You want to show that if $L$ is regular, then some other language $L_{not}$ which you know is not regular, is the intersection of another regular language $L_{reg}cap L=L_{not}$.
answered Dec 3 at 4:34
NL1992
8311
add a comment |
For (1): You assume the language satisfies the pumping lemma for some $n$.
You want to choose a word that would have at least one of its letters related to $n$.
A good start would be taking one of your letters $lin w$ to be $l^n$, and using that to show that for some $i$, $xy^iznotin L$
For (2): You want to show that if $L$ is regular, then some other language $L_{not}$ which you know is not regular, is the intersection of another regular language $L_{reg}cap L=L_{not}$.
answered Dec 3 at 4:34
NL1992
8311
For (1): You assume the language satisfies the pumping lemma for some $n$.
You want to choose a word that would have at least one of its letters related to $n$.
A good start would be taking one of your letters $lin w$ to be $l^n$, and using that to show that for some $i$, $xy^iznotin L$
For (2): You want to show that if $L$ is regular, then some other language $L_{not}$ which you know is not regular, is the intersection of another regular language $L_{reg}cap L=L_{not}$.
answered Dec 3 at 4:34
NL1992
8311
answered Dec 3 at 4:34
NL1992
8311
answered Dec 3 at 4:34
NL1992
8311
answered Dec 3 at 4:34
NL1992
8311
8311
add a comment |
add a comment |
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3022519%2fhow-can-i-prove-given-language-is-not-regular%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
