Question about function φ

[slmjlf]

I am still not clear about the function φ in Algorithm 1. It is obvious from the paper that by using the function φ the input to Q-network is clipped into a 84×84×4 image. But how did it do that?

In Algorithm 1 we found that

and

This makes me confused. What on earth is s_t+1? Does that mean:

s1 = x1
s2 = s1,a1,x2 = x1,a1,x2
s3 = s2,a2,x3 = x1,a1,x2,a2,x3
s4 = s3,a3,x4 = x1,a1,x2,a3,x3,a3,x4
......

So how did φ process s3, for instance? φ(3) should equal to φ(s3) = φ(x1,a1,x2,a2,x3)? I feel hard to understand this.

I would appreciate if anyone could help.