No smoking is a formula used to indicate smoking is not allowed.
Why can't we use Yes smoking to indicate smoking is allowed? (Although, we might use humorously but I've never heard actually.)
Unlike yes, no can also be an adverb of degree and an adjective.
Here is the etymology of the adjective no from OED:
Variant of none adj. with loss of final -n, used originally before consonants except h, which ultimately became the standard form of the negative determiner except in archaic or poetic use.
Here is an explanation of the adjective (determiner) none from OED:
In later use the form none (or nane, etc.) occurs mainly before vowels and h, and after 1600 is almost entirely supplanted by the reduced form no (nae, etc.): exceptions are the phrase none effect and before other (see sense A. 2, where its survival may be due to interpreting other as postmodifying a pronominal none).
The earliest example where no denotes that something is forbidden or unwelcome is from a1625 according to OED:
Swet. Mercy yet. Bond. No talking: puff, there goes all your pitie.
J. Fletcher Bonduca iv. iv, in F. Beaumont & J. Fletcher Comedies & Trag. (1647) 65
No smoking is a phrase but can be used as an attributive adjective also.
The earliest example of No smoking as a phrase is from 1837 (OED):
No smoking aloud in the cabing [sic]!
W. G. Clark Ollapodiana xxi, in Knickerbocker Dec. 520
The earliest example of No smoking as an adjective is from 1944 (OED):
I wanted to smoke, but fell asleep before the No Smoking sign was switched off.
J. Gunther D Day i. 13,
Etymonline has the following for no-smoking (adj.):
1905; the sign wording itself is attested by 1817.
Smoking is a vice to [sic] -- and a national one, of such magnitude that railroad corporations throughout all their routes in the United States, have a special command in large letters, conspicuously placed at depots and inside of the cars -- "No smoking allowed here." ["The Sailor's Magazine," December 1840]
I've tried to answer my own question and it can be useful for other people but I would like to get some further information on this.
The original sense of the adjective no is not any. I wonder how it evolved to be used in the phrases like no smoking, no pets etc.
Also, "Yes smoking" could be a useful shortcut but didn't become a set phrase. It is related to yes not being an adjective but language evolves. It can be explained regarding evolutionary linguistics as well.
I think we will stick to "Smoking allowed" for now.