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For all transitions of $\delta$ that are not yet specified, they are the identity transition, as in a transition that maps a state to itself. - We added the transition $\delta(\bot, \tau)=A_1$, but it is important to note $\bot$, being the initial state, does not have any attached timer within ZKE, and will not be triggered automatically. This transition simply represents the upper protocol initiating a session through ZKE. Most upper protocols will attempt to restart sessions on a timer, which is why we chose to represent this unique transition with the $\tau$ symbol. + $\beta$ contains the transition $\delta(\bot, \tau)=A_1$, but it is important to note $\bot$, being the initial state, does not have any attached timer within ZKE, and will not be triggered automatically. This transition simply represents the upper protocol initiating a session through ZKE. Most upper protocols will attempt to restart sessions on a timer, which is why we chose to represent this unique transition with the $\tau$ symbol. Let $\beta=(Q, \Sigma, \delta, \bot, \{S_2\})$ be the ZKE Automata. \end{definition}