# How would you define a superproblem?

Assuming that we have a problem, how would one classify its superproblem? There could be multiple ways to generalize a problem and for example two superproblems (although including the original as a subproblem) may not be compatible.

I am not sure whether there is a formal concept of a superproblem. But can you guide me to any formal source on this type of generalization?

• $A$ is a subproblem of $B$ if and only if $B$ is a superproblem of $A$? Commented Jan 9, 2019 at 19:25
• Perhaps, I used the word 'subproblem' in wrong context. What I meant was, has-a versus is-a distinction? A wider problem may contain the original problem but, there may not be an is-a relationship, only has-a? Is this possible?
– user
Commented Jan 9, 2019 at 19:29
• This seems analogous to $A$ being a subset of various sets. The sets can't be disjoint (since all have $A$ as a subset) but might have disjoint subsets. Each superset of $A$ corresponds to a different way of generalising the problem represented by set $A$. Commented Jan 9, 2019 at 20:30
• I don't think a problem is equivalent to a set though. At least there has to be an input-output relation, so more similar to a function maybe? So in the superproblem there maybe an additional hidden parameter that changes the output/solution, but at some value of this parameter this boils down to original problem.
– user
Commented Jan 9, 2019 at 21:32

I would say that A is a superproblem of B if B is a special case of A; in other words, A has a number of parameters such that, when some constant values are substituted for some of those parameters, the resulting problem is equivalent to B.

If you look at my questions and answers, you will see that many of them are generalizations of someone else's problem.

The way I put this internally is that I would rather solve an infinite number of problems than just one.

This is clearly an informal term, but my view is that for a problem $$A$$, a superproblem $$B$$ of problem $$A$$ has the following properties:

• Solving $$B$$ either solves $$A$$ directly or leads to a natural solution to $$A$$, but not vice versa.
• Solving $$B$$ leads to a natural solution of other problems ($$C, D, \ldots$$) but solving $$A$$ does not lead to a natural solution of those other problems.
• $$A$$ is a special case of $$B$$ but $$B$$ is not a special case of $$A$$.
• So this is analogous to, but the same as, $A$ being a subset of $B$? In which case we can maybe associate each problem with a set? Commented Jan 9, 2019 at 20:23
• @timtfj: Informally... I guess so. Commented Jan 9, 2019 at 20:30