# When is a stuck state not final and when is a maximal sequence not complete in transition systems?

Reading the book Practical Foundations for Programming Language.

In section 5.1 Transition Systems, the author said that

Whereas all final states are, by convention, stuck, there may be stuck states in a transition system that are not final.

Thus every complete transition sequence is maximal, but maximal sequences are not necessarily complete.

It seems that in practice a stuck state is always final, but I wonder when is a stuck state not final? (Could you give me an example?)

Also, if we think that the stuck state is always final, it seems no different between maximal and complete?

A transition sequence is maximal iff there is no $$s$$ such that $$s_n \mapsto s$$, and it is complete iff it is maximal and, in addition, $$s_n$$ final.

Also, do the states mentioned here refer to different states? If there is a fixed-point state in a transition system, the state is final or stuck?

Thanks.

• @JohnL. I am sorry for no further feedback on this issue (I forgot indeed...). Because later I seemed to have a better answer. A "final" state is something like final state in finite automata. They are marked by the creator of that system, whereas a "stuck" states is the state satisfy the "stuck" condition. So "final" state is included in "stuck" state. Nevertheless, your answer still gave me a lot of inspiration, thanks! (I have upvoted it.) Dec 22 '21 at 9:01