We propose a new definition of the class of search problems that correspond to $\BPP$. Specifically, a problem in this class is specified by a polynomial-time approximable function $q:\bitset^*\times\bitset^*\to[0,1]$ that associates, with each possible solution $y$ to an instance $x$, a quality $q(x,y)$. Intuitively, quality 1 corresponds to perfectly valid solutions, quality 0 corresponds to perfectly invalid solutions, but the quality of other solutions can be anywhere in-between. The class of PPT-search problems contains $q$ if there exists a PPT algorithm that, on input $x$, finds a solution with value close to $\max_y\{q(x,y)\}$.
We relate this definition to previously studied definitions of ``BPP-search problems'' and articulate our preference for it. More importantly, we show that any PPT-search problem can be reduced in deterministic polynomial-time to a promise problem in $\prBPP$ (i.e., promise-BPP).
On the companion paper. This paper represents my perspective on a joint project conducted together with Roei Tell, whereas Roei's perspective is presented in [6]. The technical contents of both papers is similar and both papers present the foregoing definition of PPT-search problems (i.e., Definition 2.4), but the perspectives presented in the two papers differ.