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  "body": "\n## Definition\n\nLet $\\sigma$ be a signature and $X$ a set of variables.\nLet $T(X)$ be the set of terms of $\\sigma$ over $X$, as in the [[term algebra|3YYKGWJJND]].\nA *term rewriting system* is a set $R$ of pairs $(l, r)$ of terms, written $l \\to r$ and called rules.\nEach rule must satisfy two conditions: $l$ is not a variable, and every variable of $r$ occurs in $l$.\nFor terms $s, t$, one writes $s \\to_R t$ if there are a rule $l \\to r$ in $R$ and a [[substitution|NCDZ969V7X]] $\\tau$ with the following property.\nThe term $s$ contains $\\tau(l)$ as a subterm, and $t$ is obtained from $s$ by replacing one occurrence of it by $\\tau(r)$.\nLet $\\leftrightarrow_R^*$ be the smallest equivalence relation that contains $\\to_R$.\nThe system $R$ *presents* the [[equational theory|F1M51B4MDG]] $E$ if, for all terms $s, t$, one has $s \\leftrightarrow_R^* t$ if and only if $s \\approx t$ belongs to $E$.\nThe system $R$ is *convergent* if:\n\n1. it is terminating: there is no infinite chain $t_0 \\to_R t_1 \\to_R t_2 \\to_R \\cdots$;\n2. it is confluent: if $t \\to_R^* u$ and $t \\to_R^* v$, then $u \\to_R^* w$ and $v \\to_R^* w$ for some term $w$.\n\n## Remarks\n\nBy [[Birkhoff's completeness theorem|WDHWCQ6Y2Y]], a system $R$ presents the theory axiomatised by the [[identities|PKD440G2XW]] $\\set{l \\approx r : (l \\to r) \\in R}$.\nIf $R$ is convergent, each term $t$ has a unique normal form $\\operatorname{nf}(t)$, a term with no further rewrite step.\nThen $s \\approx t$ is in the theory if and only if $\\operatorname{nf}(s) = \\operatorname{nf}(t)$.\nFor a finite convergent system, this decides the equational theory.\nExample: $R = \\set{f(f(x)) \\to x}$ is convergent: each step shortens the term, and the only overlap, in $f(f(f(x)))$, rewrites to $f(x)$ in both ways.\nSo $f^m(x) \\approx f^n(x)$ holds in the theory of an involution exactly when $m$ and $n$ have the same parity.\nNon-example: $R = \\set{x \\cdot y \\to y \\cdot x}$ presents the theory of a commutative operation but is not terminating, since $a \\cdot b \\to b \\cdot a \\to a \\cdot b \\to \\cdots$.\nThe Knuth–Bendix completion procedure tries to turn a set of identities into an equivalent convergent system.\n",
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