I have problems creating a timed Kripke structure $ mathcal {TK} = langle S, mathcal {T}, rightarrow, L rangle $ for a system that I have.

My system, a timed automaton (time base $ mathcal {T} = mathbb {R} ^ + $) has a clock $ c $.

I want to define a single atomic proposition $ p $ which marks the states where the clock is not 0 ($ c neq $ 0).

However, I struggle to define states and transitions.

My intuition is to create two states $ s $, such as $ p notin L (s) $ and $ p in L (s) $.

The two states are then connected using a transition of $ (s, epsilon, s) in rightarrow $, such as $ epsilon <r, forall r in mathbb {R} ^ {+} $.

However, when reading publications such as Lepri et al., They do not seem to need to $ epsilon $.

Can any one explain why they do not need such transitions or is it impossible for them to express such structures?