Greedy Alphabet Reducing (GAR)

\[ \begin{aligned} &\text{begin} \\ &\quad \Gamma \leftarrow \Sigma \\ &\quad \psi(a) \leftarrow a \text{ for each } a \in \Sigma \\ &\quad \text{repeat} \\ &\qquad \text{for each } (a, b) \in \Gamma \times \Gamma \text{ with } a \neq b \\ &\qquad\qquad \text{let } \phi_{a\to b} \text{ be a mapping obtained by letting } \psi(\sigma) = b, \\ &\qquad\qquad\qquad \text{for all } \sigma \in \Sigma \text{ with } \psi(\sigma) = a \\ &\qquad\qquad \text{choose } \phi_{a\to b} \text{ satisfying } \tilde{\phi}_{a\to b}(P) \cap \tilde{\phi}_{a\to b}(N) = \varnothing \\ &\qquad\qquad\qquad \text{and minimizing } \left|\tilde{\phi}_{a\to b}(P) \cup \tilde{\phi}_{a\to b}(N)\right| \\ &\qquad\qquad \text{(Note: } \tilde{\phi}_{a\to b} \text{ denotes the natural homomorphism associated with } \phi_{a\to b}\text{.)} \\ &\qquad\qquad \text{remove } a \text{ from } \Gamma \\ &\quad \text{until no more replacements are found} \\ &\text{end.} \end{aligned} \]