Fragment answers may convey a complete thought, hence being complete sentences on their own. They've been analysed to involve either sluicing or the elision of a copular construction (pseudosluicing).
What did Mary eat?
Cake1
Mary ate t1Cake1
it was t1
An imperative may convey a complete thought and is regarded as a complete sentence even if it is only a VP, because it actually is underlyingly a complete sentence - the subject is omitted.
Youstop!
Has anyone ever regarded isolated NPs like the ones below as complete sentences and extended the pseudosluice analysis to such isolated NPs?
Look! A bird.
What a pretty flower!
It does seem that pseudosluicing is involved.
[A bird]1
it is t1[What a pretty flower]1
it is t1
This could be supported by data from Japanese where such phrases are ended with the copula だ(da)/です(desu).
見て! 鳥 だ.
Mite! Tori da.
Look! Bird COP
Look! It's a bird.
なんて きれい-な 花 だ!
Nante kirei-na hana da!
EMPH pretty flower COP
What a pretty flower it is.