# Question about IO automaton

I'm currently reading Nancy Lynch book about distributed systems, chapter about IO automaton. And I have following questions related to book exercise 8.13(c).

We are given some automaton A with sig(A) is empty. Traces(P) is the set of sequences over {1,2} in wich every occurance of 1 is immediatly followed by a 2. I need to show that P is neither a safety property nor a liveness property. and show explicitly that P could be expressed as intersections of S and L.

Here is my problem: I can show that P is not safety because it breaks prefix-closed property, e.g. {2,1,2} has no prefix belonging to P (it should be of the form {...,1} which is impossible for P). But I don't know how to deal with L property -- either it is empty or include trace(P) -- trace(P) $\subset$ trace(L). If it is empty then traces(P) is empty because traces(P)= traces(S) $\cap$ traces(L) which is wrong. So I think that traces(P) is subset of traces(L).

Is my conclusion related to traces(L) is right?

How can I explicitly express traces(P)=traces(S) $\cap$ traces(L) for this problem?

Let S be the set of all traces in which "11" does not appear (a safety property) and L be the liveness property defined such that if 1 appears then eventually 2 will appear after (in temporal logic: $\square( 1 \rightarrow \lozenge 2)$). Then trace(P)=trace(S)$\cap$trace(L).