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State Table Representation

A tabular method for defining the state machine, listing (Current State, Input) →\rightarrow (Next State, Output). It is a systematic way to implement the transition function δ\delta.
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The statement of the theorem

The state machine behavior is defined by the transition function δ:S×I→S\delta: S \times I \to S and the output function λ:S×I→O\lambda: S \times I \to O. The State Table Representation systematically maps the input pair (s,i)(s, i) to the resulting state s′s' and output oo. This mapping is formalized as: StateTable(s,i)=(s′,o)\text{StateTable}(s, i) = (s', o) where s′=δ(s,i)s' = \delta(s, i) and o=λ(s,i)o = \lambda(s, i). The table structure is a set of ordered quadruplets: Ttable={(s,i,s′,o)∣s∈S,i∈I,s′=δ(s,i),o=λ(s,i)}\mathcal{T}_{table} = \{(s, i, s', o) \mid s \in S, i \in I, s' = \delta(s, i), o = \lambda(s, i)\}.