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State Transition Diagram (STD)

A graphical representation showing all possible states and the transitions between them, triggered by specific inputs. It defines the system's finite set of operational modes.
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The statement of the theorem

Let SS be a finite set of states, II be a finite set of inputs, and Transition:S×I→S\text{Transition}: S \times I \to S be the transition function. The State Transition Diagram (STD) is formally defined by the tuple ⟨S,I,Transition⟩\langle S, I, \text{Transition} \rangle. A transition from state si∈Ss_i \in S to state sj∈Ss_j \in S upon input i∈Ii \in I is represented by the directed edge (si,i,sj)(s_i, i, s_j), such that sj=Transition(si,i)s_j = \text{Transition}(s_i, i). The set of all possible transitions is T={(si,i,sj)∣si,sj∈S,i∈I,sj=Transition(si,i)}\mathcal{T} = \{(s_i, i, s_j) \mid s_i, s_j \in S, i \in I, s_j = \text{Transition}(s_i, i)\}.