If Anti-causal systems are defined as those whose output depends solely upon future inputs.(Is this definition correct as I understand)
So i see that $y[n] = x[n+2]$ ; is anticausal system
How is a time reversing system such as:
$$y(t) = x(-t) $$
anticausal?
e.g. $y(2) = x(-2)$ it infact depends upon past.
Looking for some explanation.
$y(t) = x(-t)$ is not anticausal, but it is acausal. For negative values of $t$, $y(t)$ is not causal, while for positive $t$, it is causal, since, for example, $y(2) = x(-2)$, but $y(-2) = x(2)$.
If Anti-causal systems are defined as those whose output depends solely upon future inputs. (Is this definition correct as I understand)
How particular are you about the words solely and future in your definition?
A causal system is one with the property that the output $y$ at every time $t$ depends only on the current and past inputs, that is, the values of the input $x$ at times in $(-\infty, t]$, or more formally,
For each $t$, $-\infty < t < \infty$, $y(t)$ is a function of $\{x(\tau) \colon \tau \in (-\infty, t]\}$ only and does not depend on any $x(\tau^\prime), \tau^\prime > t$.
If anticausal is taken to mean not causal, then the complement of the definition of causal is not what you have written. In this sense of "non-causal = not causal",
a non-causal system is defined as one for which there is at least one time instant $t_0$ (and a $\tau > t_0$) such that the output $y(t_0)$ depends on the value of $x(\tau)$, a future input.
Note that $y(t_0)$ might depend on past inputs as well as on the future input $x(\tau)$.
Note that there may be other time instants $t_1$ for which $y(t_1)$ depends only on past and curent $\{x(t) \colon t \in (-\infty, t_1]\}$ and on no future inputs. All it takes for a system to be bad-mouthed as non-causal is one bad apple time instant $t_0$ for which causality is violated.
On the other hand, your definition of anti-causal is that for every instant $t$, $y(t)$ depends solely on future inputs $\{x(\tau) \colon \tau \in (t,\infty)\}$; even the current input is excluded. So, the time-reversal system $y(t) = x(-t)$ is not an anti-causal system by your definition. Trivially, $y(0)$ equals current input $x(0)$, and if you chill a bit and amend your definition to say "current and future inputs", then, as Jason R has already pointed out, $y(2)$ depends on a past input $x(-2)$ and so the system is not anti-causal as per your amended definition either. In fact, there is a huge class of systems that would be classified as non-causal as per the definition of non-causal given here that do not meet your definition of anti-causality; the time-reversal system is just one example of a system that is not anti-causal.
