Causal or anti-causal?

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Joined Feb 15, 2011
Consider a system function (discrete time system) H(z)=1/(1-a*z^-1);
We can define both a causal and anti-causal unit sample response with different regions of convergence. So the system being a causal or an anti-causal system depends on the ROC. Now a different representation of the system behavior is in terms of difference equation which for this system is y[n] = x[n] + a*y[n-1]; Now, from this difference equation we tend to infer that the system is always a causal system. But as pointed out before, we can define an anti-causal h[n] and so it can be viewed as an anti-causal system. So what is the point i am missing in this argument? Given a difference equation, is it possible that the system can be both causal and anti-causal?
Please help me out...