"not found" or "is not found"
Solution 1:
These are status messages similar to traffic signs like No Parking, One Way etc.
Even many news headlines in newspaper are very short and are not full sentences like "Man shot dead", "Man shot at", "Gold up" etc.
Statuses, titles etc need not be full sentences.
Solution 2:
I think the main reason terse language, such as this, is used for error messages is simple: people don't want to read horrendously long error messages.
Case in point:
The computer searched the file system, unfortunately however, the file was not found.
Versus
file not found.
It's simple and yet it conveys all of the necessary information.
To answer your question, the shortened messages would not be appropriate anywhere that formal language is used. However, "the access is denied" is incorrect. You would not use "the" to refer to access, in this case. To be formal, you would say something like "Access to the resource was denied."
Solution 3:
"The access" is not a proper construct. "Your access" is fine, but I don't think it's necessary. I'd say "Access was denied" if you want to spell it out more than tradition.
"The file was not found" is grammatically proper.
But "File not found" and "Access denied" get your point across just as well
Solution 4:
As a programmer, I suspect that the message is coming from an underlying error class. The application is merely reporting the name of the type of error that occurred; it's not trying to have an intelligent conversation with the user.
This answer applies to programming in general, but here is an example of a list of status codes in the HTTP protocol. See http://en.wikipedia.org/wiki/List_of_HTTP_status_codes#4xx_Client_Error .
You will recognize some of them as messages you have received in your web browser at one time or another.