# Why aren't promise problems just decision problems; can't we encode the promised inputs in the alphabet?

I don't really understand why promise problems are classified differently than decision problems.

Consider this problem as an example. Given some real number between $$0$$ and $$1$$, determine if it greater than $$\frac{1}{2}$$ or not.

In this problem, that the real number is between $$0$$ and $$1$$. Let's say our alphabet is $$\{0, 1, 2\}$$. Then we can say that our input is encoded like so:

$$0$$ represents $$0$$, $$1$$ represents $$\frac{1}{2}$$, and $$2$$ represents $$1$$

From these 3 cases, we can define any input to represent

$$\frac{1}{n} \sum_{i=1}^{n} R(x_i)$$

where $$x_i$$ represents the letter at index $$i$$ of the input, and $$R$$ is the transformation that takes $$0$$ to $$0$$, $$1$$ to $$\frac{1}{2}$$, and $$2$$ to $$1$$.

So what is the need to define "Promise-Problems?"

If there is some encoding for that promise which has polynomial length, can produce all possible inputs for the problem, and can be parsed in polynomial time, then for a given SAT instance, you could add a single trivial solution to it, and an encoding which produces that would be a coNP certificate for the SAT instance, which is impossible assuming $$\text{NP}\neq\text{coNP}$$.