I'm modelling a real-world problem as having instances $i$ in a set $P$. As a very simple artificial example, consider the problem of choosing a meeting room given a certain number of people. Then $i = 2$ represents a problem instance of finding a meeting room for 2 people. Let's put $P = \mathbb{Z}$ for the sake of example, even though we have better options like $\mathbb{Z_0^+}$.
I define certain restrictions $A$ on $P$ using logical formulae over structure of $i$. e.g. I can say $\forall i \in P,~i > -10$ , i.e. any problem in the problem domain, if represented using $i \in P$ would satisfy $A$. Let $P$ under $A$ be $P' = \{p : p \in P \land p~\text{satisfies}~A\}$.
But it's possible that $\exists i' \in P'$ such that $i'$ is a valid mathematical structure but doesn't actually correspond to valid any real-world problem (e.g. $-5 \in P'$).
What is the standard terminology to say
- All valid problem instances are a part of $P'$ (I'm informally calling this "sufficiency")
- All problem instances which are a part of $P'$ are valid (I'm informally calling this "completeness")
I may then use this terminology to say that in my example, $P'$ is sufficient but incomplete (replacing these two words with the actual terminology)
Because I'm looking for terminology which is domain-independent, I'll give another example. Let's model valid sudoku problem instances (where I define "valid" as having a unique solution) using $P = \{\_, 1, \ldots, 9\}^{9*9}$ where "_" represents a blank and numbers start from top left cell, go towards right and then down.
Let $A$ be the condition that all rows, columns, and boxes have unique numbers (disregarding the blanks).
Here also you can specify all valid problem instances as members of $P$, but $P$ might include some invalid problem instances too (those with multiple or no solutions). It's possible to make the condition $A$ stronger to get rid of invalid instances, and that's precisely one of the reasons I need terminology to distinguish between that and the $A$ I've currently given.