semi-knowledge proof vs zero-knowledge proof

[wang...@yahoo.com.cn]

Hi,all, maybe I post a non-sense topic.But when I learned zero-
knowledge proof, I wonder why there is nothing about semi-knowledge
proof,or maybe quantitive-knowledge proof? Then I came to the
literature,I only found something about knowledge complextiy of proof
writen by Bellare and Goldreich.
In an interactive proof system,a zero -knowledge proof system require
the prover give nothing about witness via proving something
validly .Maybe we can define semi-knowlege proof system as
following :When the prover proves something, We allow the prover leak
some knowledge to the verifier, but not the hardcore knowledge.Maybe
this form of semi-knowledge proof is equal to zero-knowledge proof
with auxiliary input.But in some situations ,
these two notions are not same.Maybe we can improve the effiency of
some zero-knowlege proof system via this way! Thanks for your
attention.Please give some hints on this.