Can neural nets have State?  Can they learn to create and use state
without prior knowledge?  Neural nets are good at recognizing but to
perform more complex tasks like planning they must have state, like a
world model.  The neural net may perform acts like creating,
modifying, and acting on state in an internal model.  Even more
promising would be the ability to learn these states without a priori
knowledge.

I have previously asked if neural nets can perform logic. 
http://answers.google.com/answers/threadview?id=289529  But I think a
better question is if they hold and manipulate state.

An acceptable answer will contain at least one reference to a paper or
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seeks to answer this exact question.  Five stars if more than one
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and the research is available to be read for free, without paying the
publisher.

Yes, they can. Nets with states (memories) are called Hopfield Nets. These
are recurrent networks whose outputs are fed back into inputs. These
incessant self-reference forms basis for memory retention (the same is
with digital circuits, when connecting outputs to inputs (caen) lead
to inner state retention - memory. Have a look at Kevin Gurney's
online book on Neural Nets,

http://www.shef.ac.uk/psychology/gurney/notes/download.html

especially chapter 5 on Hopfield Nets.