Given a regular language L, prove or disprove L' is regular
$begingroup$
Given $NFA$ $N$ , $L(N)$ regular language and two words $w1$,$w2$ $in$ $sum^*$ such that $w1$ $neq$ $w2$.
I have to prove or disprove that
$L'=$ {$zin sum^*|exists$ $w1,w2$ :$w1z$ $in$ $L(N)$ $wedge$ $w2z$ $notin$ $L(N)$} is regular.
I believe this is correct but I'm having a hard time proving it.
Any help will do.
Thanks in advance.
computer-science formal-languages automata
asked Nov 30 '18 at 17:29
Avishai YanivAvishai Yaniv
73
$endgroup$
add a comment |
$begingroup$
Given $NFA$ $N$ , $L(N)$ regular language and two words $w1$,$w2$ $in$ $sum^*$ such that $w1$ $neq$ $w2$.
I have to prove or disprove that
$L'=$ {$zin sum^*|exists$ $w1,w2$ :$w1z$ $in$ $L(N)$ $wedge$ $w2z$ $notin$ $L(N)$} is regular.
I believe this is correct but I'm having a hard time proving it.
Any help will do.
Thanks in advance.
computer-science formal-languages automata
asked Nov 30 '18 at 17:29
Avishai YanivAvishai Yaniv
73
$endgroup$
$begingroup$
Are $w_1$ and $w_2$ fixed words? The description of $L’$ seems to use any string, presumably in $Sigma^*$.
$endgroup$
– Joey Kilpatrick
Nov 30 '18 at 18:02
$begingroup$
what do you mean by "fixed words"?
$endgroup$
– Avishai Yaniv
Nov 30 '18 at 20:13
$begingroup$
Are we to solve the problem for some given two words $w_1$ and $w_2$? Because the definition of $L’$ allows the words to change depending on the value of $z$. I’m asking if the words can vary based on $z$ or if they’re “fixed”.
$endgroup$
– Joey Kilpatrick
Nov 30 '18 at 20:18
add a comment |
$begingroup$
Given $NFA$ $N$ , $L(N)$ regular language and two words $w1$,$w2$ $in$ $sum^*$ such that $w1$ $neq$ $w2$.
I have to prove or disprove that
$L'=$ {$zin sum^*|exists$ $w1,w2$ :$w1z$ $in$ $L(N)$ $wedge$ $w2z$ $notin$ $L(N)$} is regular.
I believe this is correct but I'm having a hard time proving it.
Any help will do.
Thanks in advance.
computer-science formal-languages automata
asked Nov 30 '18 at 17:29
Avishai YanivAvishai Yaniv
73
$endgroup$
Given $NFA$ $N$ , $L(N)$ regular language and two words $w1$,$w2$ $in$ $sum^*$ such that $w1$ $neq$ $w2$.
I have to prove or disprove that
$L'=$ {$zin sum^*|exists$ $w1,w2$ :$w1z$ $in$ $L(N)$ $wedge$ $w2z$ $notin$ $L(N)$} is regular.
I believe this is correct but I'm having a hard time proving it.
Any help will do.
Thanks in advance.
computer-science formal-languages automata
computer-science formal-languages automata
asked Nov 30 '18 at 17:29
Avishai YanivAvishai Yaniv
73
asked Nov 30 '18 at 17:29
Avishai YanivAvishai Yaniv
73
asked Nov 30 '18 at 17:29
Avishai YanivAvishai Yaniv
73
asked Nov 30 '18 at 17:29
Avishai YanivAvishai Yaniv
73
asked Nov 30 '18 at 17:29
Avishai YanivAvishai Yaniv
73
73
$begingroup$
Are $w_1$ and $w_2$ fixed words? The description of $L’$ seems to use any string, presumably in $Sigma^*$.
$endgroup$
– Joey Kilpatrick
Nov 30 '18 at 18:02
$begingroup$
what do you mean by "fixed words"?
$endgroup$
– Avishai Yaniv
Nov 30 '18 at 20:13
$begingroup$
Are we to solve the problem for some given two words $w_1$ and $w_2$? Because the definition of $L’$ allows the words to change depending on the value of $z$. I’m asking if the words can vary based on $z$ or if they’re “fixed”.
$endgroup$
– Joey Kilpatrick
Nov 30 '18 at 20:18
add a comment |
$begingroup$
Are $w_1$ and $w_2$ fixed words? The description of $L’$ seems to use any string, presumably in $Sigma^*$.
$endgroup$
– Joey Kilpatrick
Nov 30 '18 at 18:02
$begingroup$
what do you mean by "fixed words"?
$endgroup$
– Avishai Yaniv
Nov 30 '18 at 20:13
$begingroup$
Are we to solve the problem for some given two words $w_1$ and $w_2$? Because the definition of $L’$ allows the words to change depending on the value of $z$. I’m asking if the words can vary based on $z$ or if they’re “fixed”.
$endgroup$
– Joey Kilpatrick
Nov 30 '18 at 20:18
$begingroup$
Are $w_1$ and $w_2$ fixed words? The description of $L’$ seems to use any string, presumably in $Sigma^*$.
$endgroup$
– Joey Kilpatrick
Nov 30 '18 at 18:02
$begingroup$
Are $w_1$ and $w_2$ fixed words? The description of $L’$ seems to use any string, presumably in $Sigma^*$.
$endgroup$
– Joey Kilpatrick
Nov 30 '18 at 18:02
$begingroup$
what do you mean by "fixed words"?
$endgroup$
– Avishai Yaniv
Nov 30 '18 at 20:13
$begingroup$
what do you mean by "fixed words"?
$endgroup$
– Avishai Yaniv
Nov 30 '18 at 20:13
$begingroup$
Are we to solve the problem for some given two words $w_1$ and $w_2$? Because the definition of $L’$ allows the words to change depending on the value of $z$. I’m asking if the words can vary based on $z$ or if they’re “fixed”.
$endgroup$
– Joey Kilpatrick
Nov 30 '18 at 20:18
$begingroup$
Are we to solve the problem for some given two words $w_1$ and $w_2$? Because the definition of $L’$ allows the words to change depending on the value of $z$. I’m asking if the words can vary based on $z$ or if they’re “fixed”.
$endgroup$
– Joey Kilpatrick
Nov 30 '18 at 20:18
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The automaton $A$ for $L'$ can work like this:
- Without reading any input, it guesses a $w_1$ and simulates $N$ on this string. The guessing is done letter by letter with $lambda$-transitions. The states in which $N$ ends after reading $w_1$ is remembered.
$A$ guesses a $w_2$ and simulates $N$ on this string. The guessing is again done letter by letter with $lambda$-transitions. The states in which $N$ ends after reading $w_2$ is remembered.- Now $A$ reads the input $z$ and simulates two cmputations of $N$ in parallel: that of $w_1z$ and that of $w_2z$. If the former ends in some final state while the latter does not, $A$ accepts.
For fixed $w_1$ and $w_2$ you could do the same without the guessing. However, it would be more efficient to do the simulations of $N$ once on paper and let $A$ start from there directly.
answered Dec 1 '18 at 18:00
Peter LeupoldPeter Leupold
56826
$endgroup$
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%2f3020373%2fgiven-a-regular-language-l-prove-or-disprove-l-is-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
$begingroup$
The automaton $A$ for $L'$ can work like this:
- Without reading any input, it guesses a $w_1$ and simulates $N$ on this string. The guessing is done letter by letter with $lambda$-transitions. The states in which $N$ ends after reading $w_1$ is remembered.
$A$ guesses a $w_2$ and simulates $N$ on this string. The guessing is again done letter by letter with $lambda$-transitions. The states in which $N$ ends after reading $w_2$ is remembered.- Now $A$ reads the input $z$ and simulates two cmputations of $N$ in parallel: that of $w_1z$ and that of $w_2z$. If the former ends in some final state while the latter does not, $A$ accepts.
For fixed $w_1$ and $w_2$ you could do the same without the guessing. However, it would be more efficient to do the simulations of $N$ once on paper and let $A$ start from there directly.
answered Dec 1 '18 at 18:00
Peter LeupoldPeter Leupold
56826
$endgroup$
add a comment |
$begingroup$
The automaton $A$ for $L'$ can work like this:
- Without reading any input, it guesses a $w_1$ and simulates $N$ on this string. The guessing is done letter by letter with $lambda$-transitions. The states in which $N$ ends after reading $w_1$ is remembered.
$A$ guesses a $w_2$ and simulates $N$ on this string. The guessing is again done letter by letter with $lambda$-transitions. The states in which $N$ ends after reading $w_2$ is remembered.- Now $A$ reads the input $z$ and simulates two cmputations of $N$ in parallel: that of $w_1z$ and that of $w_2z$. If the former ends in some final state while the latter does not, $A$ accepts.
For fixed $w_1$ and $w_2$ you could do the same without the guessing. However, it would be more efficient to do the simulations of $N$ once on paper and let $A$ start from there directly.
answered Dec 1 '18 at 18:00
Peter LeupoldPeter Leupold
56826
$endgroup$
add a comment |
$begingroup$
The automaton $A$ for $L'$ can work like this:
- Without reading any input, it guesses a $w_1$ and simulates $N$ on this string. The guessing is done letter by letter with $lambda$-transitions. The states in which $N$ ends after reading $w_1$ is remembered.
$A$ guesses a $w_2$ and simulates $N$ on this string. The guessing is again done letter by letter with $lambda$-transitions. The states in which $N$ ends after reading $w_2$ is remembered.- Now $A$ reads the input $z$ and simulates two cmputations of $N$ in parallel: that of $w_1z$ and that of $w_2z$. If the former ends in some final state while the latter does not, $A$ accepts.
For fixed $w_1$ and $w_2$ you could do the same without the guessing. However, it would be more efficient to do the simulations of $N$ once on paper and let $A$ start from there directly.
answered Dec 1 '18 at 18:00
Peter LeupoldPeter Leupold
56826
$endgroup$
The automaton $A$ for $L'$ can work like this:
- Without reading any input, it guesses a $w_1$ and simulates $N$ on this string. The guessing is done letter by letter with $lambda$-transitions. The states in which $N$ ends after reading $w_1$ is remembered.
$A$ guesses a $w_2$ and simulates $N$ on this string. The guessing is again done letter by letter with $lambda$-transitions. The states in which $N$ ends after reading $w_2$ is remembered.- Now $A$ reads the input $z$ and simulates two cmputations of $N$ in parallel: that of $w_1z$ and that of $w_2z$. If the former ends in some final state while the latter does not, $A$ accepts.
For fixed $w_1$ and $w_2$ you could do the same without the guessing. However, it would be more efficient to do the simulations of $N$ once on paper and let $A$ start from there directly.
answered Dec 1 '18 at 18:00
Peter LeupoldPeter Leupold
56826
answered Dec 1 '18 at 18:00
Peter LeupoldPeter Leupold
56826
answered Dec 1 '18 at 18:00
Peter LeupoldPeter Leupold
56826
answered Dec 1 '18 at 18:00
Peter LeupoldPeter Leupold
56826
56826
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.
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%2f3020373%2fgiven-a-regular-language-l-prove-or-disprove-l-is-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
$begingroup$
Are $w_1$ and $w_2$ fixed words? The description of $L’$ seems to use any string, presumably in $Sigma^*$.
$endgroup$
– Joey Kilpatrick
Nov 30 '18 at 18:02
$begingroup$
what do you mean by "fixed words"?
$endgroup$
– Avishai Yaniv
Nov 30 '18 at 20:13
$begingroup$
Are we to solve the problem for some given two words $w_1$ and $w_2$? Because the definition of $L’$ allows the words to change depending on the value of $z$. I’m asking if the words can vary based on $z$ or if they’re “fixed”.
$endgroup$
– Joey Kilpatrick
Nov 30 '18 at 20:18