For any set $S$ of positive definite and integral quadratic forms with bounded rank, there is a finite subset $S_{0}$ of S such that any $S_{0}$-universal quadratic form is also $S$-universal. Such a set $S_{0}$ is called an $S$-universality criterion set. In this talk, we introduce various properties on minimal $S$-universality criterion sets. When $S$ is a subset of positive integers, we show that the minimal $S$-universality criterion set is unique. For higher rank cases, we prove that a minimal $S$-universality criterion set is not unique when $S$ is the set of all qudratic forms of rank $n$ with $n ≥ 9$. We say a ℤ-lattice $L$ is recoverable if there is a minimal $S_{L}$-universality criterion set other than ${L}$, where $S_{L}$ is the set of all sublattices of $L$ with same rank. We provide some necessary conditions, and some sufficient conditions for ℤ-lattices to be recoverable.