Per this post
The hypothesis space used by a machine learning system is the set of all hypotheses that might possibly be returned by it.
Per this post, the Perceptron algorithm makes prediction
$$ \begin{equation} \hat y = \begin{cases} 1 & wx+b >= 0\\ 0 & wx+b<0 \end{cases} \end{equation} $$
we can conclude that the model to achieve an AND gate, using the Perceptron algorithm is
$x_1 + x_2 – 1.5$
In this case, what is the hypothesis space used by this AND gate Perceptron?
As far as I understand:
A hypothesis is a model which is capable of predicting outputs from inputs, hence the $x_1 + x_2 - 1.5$ is a hypothesis but not the only one. The whole models that have the same capability are regarded as hypothesis space.
We know that in AND gate:
x1 x2 output
|---------|--------|--------|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
|---------------------------|
and we have $w .\cdot x + b$, based on which, either $0$ or $1$ turns out as output. $$w \cdot x + b$$ $$w_1 \cdot x_1 + w_2 \cdot x_2 + b$$
trying all the inputs in this expression:
$w_1 \cdot 0 + w_2 \cdot 0 + b <= 0$ (because the output should be 0) $b < 0$
$w_1 \cdot 0 + w_2 \cdot 1 + b <= 0$ ---> $ w_2 + b <= 0$ So $w_1 < |b|$
$w_1 \cdot 1 + w_2 \cdot 0 + b <= 0$ ---> $ w_1 + b <= 0$ So $w_2 < |b|$
$w_1 \cdot 1 + w_2 \cdot 1 + b > 0$ ---> $ w_1 + w_2 + b > 0$
Firstly, we initialize the weights and bias parameters and then if needed, change them.
Here, since $b < 0$ we set it as $-1$
Since $w_1 < |b|$, $w_2 < |b|$ and weights are not negative, we set them as 1. So we would have:
$w_1 \cdot 0 + w_2 \cdot 0 + b = -1 < 0$ is right, returns 0 because is negative.
$w_1 \cdot 0 + w_2 \cdot 1 + b = w_2 + b = 1 - 1 = 0$ wrong, it returns 1 while it should return 0
$w_1 \cdot 1 + w_2 \cdot 0 + b <= 0$ ---> $ w_1 + b <= 0$ So $w_2 < |b|$ wrong, it returns 1 while it should return 0
$w_1 \cdot 1 + w_2 \cdot 1 + b > 0$ ---> $ w_1 + w_2 + b > 0$
So we set b for a smaller value like -1.5 (Note), then all the expressions would work appropriately. Hence $x_1 + x_2 - 1.5$ is a hypothesis for this problem.
Note: we know that in using perceptron algorithm, when reaching at any point that is not following the current model, weights and bias are updated as follows:
w = w + yx
b = b + y
Here, in the source that you referred to, maybe for simplicity they haven't done so and have just found a sample of a plausible model (hypothesis)
The other hypothesizes should follow the previously mentioned rules, thereby, $x_1 + x_2 - 2$ can also be another hypothesis for this problem, etc.
