Please ponder this question: What part of speech is "cannot" ? You can't make it past tense. I have always written cannot as 2 words not one because I think it's a helping verb + "not" as an adverb written before an action verb.

There is a lot of discussion about the word "cannot" around there.

1) Is "cannot" a contraction of "can not"? (formal aspect)

We can see that a space between the two words has been suppressed, so it is obviously a "contraction" in the sense "A reversible reduction in size."
http://en.wiktionary.org/wiki/contraction

For instance, Wiktionary defines "cannot" as a contraction:
"Contraction
cannot
1. can not ( = am/is/are unable to); contracted into can't (see the usage notes)
I cannot open the window. It is stuck.
2. am/are/is forbidden, not permitted to
You cannot enter the hall without a ticket."
Source and further information:
http://en.wiktionary.org/wiki/cannot

The Oxford dictionaries also define "cannot" as a contraction:
"contraction
can not
Usage
Both the one-word form cannot and the two-word form can not are acceptable, but cannot is more common (in the Oxford English Corpus, three times as common). The two-word form is better only in a construction in which not is part of a set phrase, such as‘ not only ... but (also)’: Paul can not only sing well, he also paints brilliantly"
Source and further information:
http://www.oxforddictionaries.com/view/entry/m_en_gb0120300#m_en_gb0120300

However, various grammatical definitions of a contraction would not apply to "cannot":
- "A contraction is the shortening of a word, syllable, or word group by omission of internal letters. In traditional grammar, contraction can denote the formation of a new word from one word or a group of words, for example, by elision."
http://en.wikipedia.org/wiki/Contraction_(grammar)
Does not apply: a space is not a letter.
- "(grammar) A shortened word or phrase, with the missing letters represented by an apostrophe."
http://en.wiktionary.org/wiki/contraction
Does not apply: no apostrophe.
- "A contraction is two words combined with an apostrophe such as can't, wouldn't, I'd, won't."
http://www.tooter4kids.com/classroom/contractions.htm
Does not apply: no apostrophe.

Notice, by the way, that "can't" would be a contraction of "can not", but not of "cannot", according to the last definition ("cannot" is not two words).


2) Does "cannot" mean the same as "can not"? (semantic aspect).
Usage seems different.

I'll just point at various references discussing those aspects:
http://www.wisegeek.com/what-is-the-difference-between-cannot-and-can-not.htm
http://www.semanticargument.com/2010/01/06/is-cannot-a-word-or-a-contraction/
http://alexfiles.com/cannot.shtml
http://en.wikipedia.org/wiki/English_modal_verb#Can_and_could
http://eslblogs.englishclub.com/english/vocabulary/cannot/
http://boards.straightdope.com/sdmb/showthread.php?t=4103
http://www.wsu.edu/~brians/errors/cannot.html
http://www.usingenglish.com/forum/ask-teacher/1617-cannot-vs-can-not.html
http://www.dailywritingtips.com/cannot-or-can-not/
http://www.educationbug.org/a/can---t--cant--cannot.html

3) Former, similar questions:
http://www.answerbag.com/q_view/634634
http://www.answerbag.com/q_view/634623

Cannot is a contraction i thought.

Example:

Cannot - - > Can't
Do not - - > Don't