About the behaviour of multi-layer perceptrons I have a multilayer perceptron. It has an input layer with two neurons, a hidden layer with an arbitrary number of neurons, and an output layer with two neurons.

Given that randomboolean and targetboolean are random boolean values, and the network operates as such:

input(randomboolean); //Set the input neurons to reflect the random boolean
propagateforwards(); //Perform standard forward propagation
outputboolean = output(); //To get the networks output
ideal(targetboolean); //Performs connection updating via back-prop


Is it possible to get the network to map the randomboolean value to the targetboolean value in such a way as the the outputboolean value will correctly match the targetboolean while running in an 'on-line' (where prediction occurs along with continued learning) mode after some arbitrary number of training cycles. 

I hear that the network needs to be recurrent to process this as it may be temporal behaviour, however the MLP is a universal computing platform and I assume it should be able to approximate the temporal behaviour needed for this task.
The answer is no. What you want to do is to predict randomness. The perceptron network takes randomboolean(true/false) and it outputs outputboolean(not random!!). The random generation of targetboolean is independent from the generation of outputboolean.

Perceptrons generally learn functions. If you have $f(A)=B$ and $f(A)=C$ and $B\neq C$, then $f$ is not a function.

EDIT: To predict temporal behavior you should add some time dependent variable in the input of the network.