Mfcttrf

How to convert a file with several spaces into a tab-delimited file?

Referring to different instances of the same character in time travel

Misspelling my name on my mathematical publications

Are neural networks prone to catastrophic forgetting?

Why does the U.S. tolerate foreign influence from Saudi Arabia and Israel on its domestic policies while not tolerating that from China or Russia?

How can one write good dialogue in a story without sounding wooden?

During copyediting, journal disagrees about spelling of paper's main topic

Credit score and financing new car

Reverse dots and boxes, swastika edition

Why are they 'nude photos'?

How many hours would it take to watch all of Doctor Who?

Simple interepretation problem regarding Polynomial Hierarchy?

Machine learning and operations research projects

C program to parse source code of another language

Why was hardware diversification an asset for the IBM PC ecosystem?

Can the Mage Hand cantrip be used to trip an enemy who is running away?

For a hashing function like MD5, how similar can two plaintext strings be and still generate the same hash?

Why do players in the past play much longer tournaments than today's top players?

How would vampires avoid contracting diseases?

If a non-friend comes across my Steam Wishlist, how easily can he gift me one of the games?

Are randomly-generated passwords starting with "a" less secure?

Why isn't pressure filtration popular compared to vacuum filtration?

Why were Er and Onan punished if they were under 20?

What is the job of the acoustic cavities inside the main combustion chamber?

Simple interepretation problem regarding Polynomial Hierarchy?

Why isn't this undecidable problem in NP?Do any decision problems exist outside NP and NP-Hard?P is contained in NP ∩ Co-NP?Constructing a promise problem equivalent to XSAT from subset sumIs a problem in NP if it is decided by some non-deterministic, polynomial time turing machine?Is there a task that is solvable in polynomial time but not verifiable in polynomial time?A clarification on $NP=coNP$?NP-complete problem with a polynomial number of yes-instances?Is This A Solution for does P=NP problem?How does a co-NP problem differ from an NP (its complement) one?

.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;

6

$begingroup$

So $NP$ stands for problems where we have small verifiable witnesses for $YES$ instances and $coNP$ for small verifiable witnesses for $NO$ instances. How does this work for

1. $P^NP$




2. $NP^NP$




3. $coNP^NP$




4. and so on?

complexity-theory np

share|cite|improve this question

asked 9 hours ago

TurboTurbo

1,27266 silver badges1818 bronze badges

$endgroup$

add a comment | 

6

$begingroup$

So $NP$ stands for problems where we have small verifiable witnesses for $YES$ instances and $coNP$ for small verifiable witnesses for $NO$ instances. How does this work for

1. $P^NP$




2. $NP^NP$




3. $coNP^NP$




4. and so on?

complexity-theory np

share|cite|improve this question

asked 9 hours ago

TurboTurbo

1,27266 silver badges1818 bronze badges

$endgroup$

add a comment | 

$begingroup$

So $NP$ stands for problems where we have small verifiable witnesses for $YES$ instances and $coNP$ for small verifiable witnesses for $NO$ instances. How does this work for

1. $P^NP$




2. $NP^NP$




3. $coNP^NP$




4. and so on?

complexity-theory np

share|cite|improve this question

asked 9 hours ago

TurboTurbo

1,27266 silver badges1818 bronze badges

$endgroup$

share|cite|improve this question

asked 9 hours ago

TurboTurbo

1,27266 silver badges1818 bronze badges

share|cite|improve this question

asked 9 hours ago

TurboTurbo

1,27266 silver badges1818 bronze badges

asked 9 hours ago

TurboTurbo

1,27266 silver badges1818 bronze badges

asked 9 hours ago

TurboTurbo

1,27266 silver badges1818 bronze badges

1 Answer
1

active

oldest

votes

5

$begingroup$

There is a logical interpretation of the various levels of the polynomial hierarchy, which extends the witness characterization of $mathsfNP$ and $mathsfcoNP$.

A language $L$ is in $Sigma_k^P$ if there is a polytime predicate $f$ and a polynomial $ell$ such that
$$
x in L Leftrightarrow exists |y_1| le ell(|x|) ; forall |y_2| le ell(|x|) ; cdots Q |y_k| le ell(|x|) ; f(x,y_1,ldots,y_k).
$$
Here:

• 
  $exists |y| le ell(|x|)$ means that there exists a number $y$ whose length is at most $ell(|x|)$ such that ...


• 
  $forall |y| le ell(|x|)$ mean that for all $y$ whose length is at most $ell(|x|)$, the following holds ...


• 
  $Q$ is $exists$ if $k$ is odd and $forall$ if $k$ is even.

Similarly, $L$ is in $Pi_k^P$ if it can be written in a similar way, only starting with $forall$.

As an example, $mathsfNP^mathsfNP$ is $Sigma_2^P$, and consists of all languages such that
$$
x in L Leftrightarrow exists |y_1| leq ell(|x|) ; forall |y_2| leq ell(|x|) ; f(x,y_1,ldots,y_k).
$$
As another example, $mathsfcoNP^mathsfNP$ is $Pi_2^P$.

Your third example is $mathsfP^mathsfNP$, which is $Delta_2^P$. I'm not sure what is the logical characterization.

share|cite|improve this answer

answered 8 hours ago

Yuval FilmusYuval Filmus

204k1515 gold badges198198 silver badges360360 bronze badges

$endgroup$

add a comment | 

Your Answer

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "419"
;
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: false,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: null,
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
,
onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

draft saved

draft discarded

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

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fcs.stackexchange.com%2fquestions%2f111699%2fsimple-interepretation-problem-regarding-polynomial-hierarchy%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

5

$begingroup$

There is a logical interpretation of the various levels of the polynomial hierarchy, which extends the witness characterization of $mathsfNP$ and $mathsfcoNP$.

A language $L$ is in $Sigma_k^P$ if there is a polytime predicate $f$ and a polynomial $ell$ such that
$$
x in L Leftrightarrow exists |y_1| le ell(|x|) ; forall |y_2| le ell(|x|) ; cdots Q |y_k| le ell(|x|) ; f(x,y_1,ldots,y_k).
$$
Here:

• 
  $exists |y| le ell(|x|)$ means that there exists a number $y$ whose length is at most $ell(|x|)$ such that ...


• 
  $forall |y| le ell(|x|)$ mean that for all $y$ whose length is at most $ell(|x|)$, the following holds ...


• 
  $Q$ is $exists$ if $k$ is odd and $forall$ if $k$ is even.

Similarly, $L$ is in $Pi_k^P$ if it can be written in a similar way, only starting with $forall$.

As an example, $mathsfNP^mathsfNP$ is $Sigma_2^P$, and consists of all languages such that
$$
x in L Leftrightarrow exists |y_1| leq ell(|x|) ; forall |y_2| leq ell(|x|) ; f(x,y_1,ldots,y_k).
$$
As another example, $mathsfcoNP^mathsfNP$ is $Pi_2^P$.

Your third example is $mathsfP^mathsfNP$, which is $Delta_2^P$. I'm not sure what is the logical characterization.

share|cite|improve this answer

answered 8 hours ago

Yuval FilmusYuval Filmus

204k1515 gold badges198198 silver badges360360 bronze badges

$endgroup$

add a comment | 

5

$begingroup$

There is a logical interpretation of the various levels of the polynomial hierarchy, which extends the witness characterization of $mathsfNP$ and $mathsfcoNP$.

A language $L$ is in $Sigma_k^P$ if there is a polytime predicate $f$ and a polynomial $ell$ such that
$$
x in L Leftrightarrow exists |y_1| le ell(|x|) ; forall |y_2| le ell(|x|) ; cdots Q |y_k| le ell(|x|) ; f(x,y_1,ldots,y_k).
$$
Here:

• 
  $exists |y| le ell(|x|)$ means that there exists a number $y$ whose length is at most $ell(|x|)$ such that ...


• 
  $forall |y| le ell(|x|)$ mean that for all $y$ whose length is at most $ell(|x|)$, the following holds ...


• 
  $Q$ is $exists$ if $k$ is odd and $forall$ if $k$ is even.

Similarly, $L$ is in $Pi_k^P$ if it can be written in a similar way, only starting with $forall$.

As an example, $mathsfNP^mathsfNP$ is $Sigma_2^P$, and consists of all languages such that
$$
x in L Leftrightarrow exists |y_1| leq ell(|x|) ; forall |y_2| leq ell(|x|) ; f(x,y_1,ldots,y_k).
$$
As another example, $mathsfcoNP^mathsfNP$ is $Pi_2^P$.

Your third example is $mathsfP^mathsfNP$, which is $Delta_2^P$. I'm not sure what is the logical characterization.

share|cite|improve this answer

answered 8 hours ago

Yuval FilmusYuval Filmus

204k1515 gold badges198198 silver badges360360 bronze badges

$endgroup$

add a comment | 

$begingroup$

There is a logical interpretation of the various levels of the polynomial hierarchy, which extends the witness characterization of $mathsfNP$ and $mathsfcoNP$.

A language $L$ is in $Sigma_k^P$ if there is a polytime predicate $f$ and a polynomial $ell$ such that
$$
x in L Leftrightarrow exists |y_1| le ell(|x|) ; forall |y_2| le ell(|x|) ; cdots Q |y_k| le ell(|x|) ; f(x,y_1,ldots,y_k).
$$
Here:

• 
  $exists |y| le ell(|x|)$ means that there exists a number $y$ whose length is at most $ell(|x|)$ such that ...


• 
  $forall |y| le ell(|x|)$ mean that for all $y$ whose length is at most $ell(|x|)$, the following holds ...


• 
  $Q$ is $exists$ if $k$ is odd and $forall$ if $k$ is even.

Similarly, $L$ is in $Pi_k^P$ if it can be written in a similar way, only starting with $forall$.

As an example, $mathsfNP^mathsfNP$ is $Sigma_2^P$, and consists of all languages such that
$$
x in L Leftrightarrow exists |y_1| leq ell(|x|) ; forall |y_2| leq ell(|x|) ; f(x,y_1,ldots,y_k).
$$
As another example, $mathsfcoNP^mathsfNP$ is $Pi_2^P$.

Your third example is $mathsfP^mathsfNP$, which is $Delta_2^P$. I'm not sure what is the logical characterization.

share|cite|improve this answer

answered 8 hours ago

Yuval FilmusYuval Filmus

204k1515 gold badges198198 silver badges360360 bronze badges

$endgroup$

share|cite|improve this answer

answered 8 hours ago

Yuval FilmusYuval Filmus

204k1515 gold badges198198 silver badges360360 bronze badges

answered 8 hours ago

Yuval FilmusYuval Filmus

204k1515 gold badges198198 silver badges360360 bronze badges

answered 8 hours ago

Yuval FilmusYuval Filmus

204k1515 gold badges198198 silver badges360360 bronze badges