If I pay for a program can I install it on the three computers I use on a daily basis?
How is licensing decided or are there any restrictions on licenses used?
Where are these kinds of questions documented?
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Currently there's no DRM-like application security system. Developers have to rely on the good will of the users to appreciate the fact that there's no DRM and show that they're responsible enough to not need any DRM.
If you want to install your previous purchages after reinstalling Ubuntu, just click on the File -> Reinstall Previous Purchases menu item in Software Center. You'll be prompted for your login details when you click it. And from here it all depends on what happens then: I expect you can download and install the app (not tested ;) ) but technically the website you download from could block you and make you pay again. Save the .deb if you want to be sure.
This really depends on the publisher of the application.
There is a page with information on how licenses work for paid applications in the Ubuntu Software Center:
Once you’ve enabled licence keys, Ubuntu Software Centre will fetch a licence key when it completes a purchase, and copy that into the right file for your app to check. It is up to you, the developer, to provide keys that are hard to guess, check if the key is there when your application starts and behave accordingly.
You will receive a notification by email when the number of available licence keys goes below a certain configurable threshold, and you will see notices when you visit the website telling you about this also. If we run out of license keys we will necessarily stop selling copies of your app.
The new website has a FAQ page at http://developer.ubuntu.com/publish/commercial-software-faqs/ that answers these questions.