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Next State Logic Equations

Boolean equations derived using Karnaugh maps or Quine-McCluskey method to define the next state variables SnextS_{next} based on current state SS and input II. Example: Snext,k=∑m(S,I)S_{next, k} = \sum m(S, I).
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The statement of the theorem

Let S={s1,s2,…,sn}S = \{s_1, s_2, \dots, s_n\} be the set of states, and I={i1,i2,…,im}I = \{i_1, i_2, \dots, i_m\} be the set of inputs. The next state variables Snext=⟨snext,1,snext,2,…,snext,n⟩S_{next} = \langle s_{next, 1}, s_{next, 2}, \dots, s_{next, n} \rangle are defined by a set of Boolean equations derived from the transition function δ:S×I→S\delta: S \times I \to S. For each state variable snext,ks_{next, k}, the equation is: snext,k=⋁j=1n⋁l=1mMk,j,l AND (sj AND il)s_{next, k} = \bigvee_{j=1}^{n} \bigvee_{l=1}^{m} \text{M}_{k, j, l} \text{ AND } (s_j \text{ AND } i_l) where Mk,j,l\text{M}_{k, j, l} is the Boolean value (1 or 0) indicating if the transition from state sjs_j to sks_k occurs upon input ili_l. This is often written compactly as snext,k=fk(s1,Input,s2,Input,...)s_{next, k} = f_k(s_1, \text{Input}, \text{s}_2, \text{Input}, \text{...}).
Source: Wikipedia