How can you modify a SUBSET-SUM instance so evaluating a set outputs either 0 or 1?


An SUBSET-SUM instance is a list of $ n$ integers $ \{ a_1, a_2,… a_n\}$ and a target $ W$ . To evaluate a set is to find the sum of a selection of numbers in the set. However, I want to know, is it possible to modify the instance, or SUBSET-SUM in general, where evaluating a set outputs $ 0$ if the sum equals $ W$ , and $ 1$ otherwise?

Bonus: Give a list of other NP-complete problems (other than 3SAT, where you evaluate a formula that either outputs $ 0$ or $ 1$ depending on the set of binary variables being passed into it), where evaluating an analogous instance outputs $ 0$ if it satisfies some objective related to the problem and $ 1$ otherwise.