Confluence

Remarks

Example: $A = \N$ with $n \to n - 1$ for $n \ge 1$; any two elements reachable from $a$ both reduce to $0$, so the system is confluent. Non-example: $A = \set{a, b, c}$ with $a \to b$ and $a \to c$; here $b$ and $c$ are distinct normal forms, so they are not joinable and the system is not even locally confluent. Every confluent system is locally confluent, but not conversely: with $b \to a$, $b \to c$, $c \to b$, $c \to d$, the system is locally confluent, yet $b \to^* a$ and $b \to^* d$ with $a$ and $d$ distinct normal forms. Confluence is equivalent to the Church–Rosser property: $a \leftrightarrow^* b$ implies that $a$ and $b$ are joinable. In a confluent system each element has at most one normal form.