A problem such that all NP problems can be reduced in polynomial time to it.
Interesting because of the Cook-Levin theorem: if only a single NP-complete problem were in P, then all NP-complete problems would also be P!
We all know the answer for this: either false or independent.
Strictly speaking, only defined for decision problems: cs.stackexchange.com/questions/9664/is-it-necessary-for-np-problems-to-be-decision-problems/128702#128702
To get an intuition for it, see the sample computation at: en.wikipedia.org/w/index.php?title=Ackermann_function&oldid=1170238965#TRS,_based_on_2-ary_function where in this context. From this, we immediately get the intuition that these functions are recursive somehow.
Non-primitive total recursive function by 
Ciro Santilli 35 Updated 2025-03-28 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-03-28 +Created 1970-01-01In intuitive terms it consists of all integer functions, possibly with multiple input arguments, that can be written only with a sequence of:and such that
- variable assignments
- addition and subtraction
- integer comparisons and if/else
- for loops
for (i = 0; i < n; i++)n does not change inside the loop body, i.e. no while loops with arbitrary conditions.n does not have to be a constant, it may come from previous calculations. But it must not change inside the loop body.Primitive recursive functions basically include every integer function that comes up in practice. Primitive recursive functions can have huge complexity, and it strictly contains EXPTIME. As such, they mostly only come up in foundation of mathematics contexts.
The cool thing about primitive recursive functions is that the number of iterations is always bound, so we are certain that they terminate and are therefore computable.
This also means that there are necessarily functions which are not primitive recursive, as we know that there must exist uncomputable functions, e.g. the busy beaver function.
Adding unbounded while loops of course enables us to simulate arbitrary Turing machines, and therefore increases the complexity class.
More finely, there are non-primitive total recursive functions, e.g. most famously the Ackermann function.
Stronger version of the big O notation, basically means that ratio goes to zero. In big O notation, the ratio does not need to go to zero.
E.g.:
- K does not tend to zero
Governments should provide basic Internet infrastructure by Ciro Santilli 35 Updated 2025-03-28 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-03-28 +Created 1970-01-01Companies are getting too much power to distort regulations and destroy privacy.
Taxes pay for the physical car roads, so why shouldn't they also pay for the "online roads" of today?
The following services are obvious picks because they are so simple:
Other less simple ones that might also be feasible:
- geographic information system. Notable anti-example: United Kingdom's Ordnance Survey's apparently non-free-data
- App stores
All of them should have strong privacy enabled by default: end-to-end encryption, logless, etc. Governments are not going to like this part.
And then if you ever forget a password or lose a multi-factor authentication token, you can just go to an ID center with your ID to recover it.
Pinned article: ourbigbook/introduction-to-the-ourbigbook-project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivative - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. - Internal cross file references done right:
- Infinitely deep tables of contents:
Figure 6. Dynamic article tree with infinitely deep table of contents.Live URL: ourbigbook.com/cirosantilli/chordateDescendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact