Total order in computer science
In theoretical computer science, in particular in term rewriting, a path ordering is a well-founded strict total order ( ;) on the set of all terms such that 
"f"(...) ; "g"("s"1...,"s""n")   if   "f" . ; "g"   and   "f"(...) ; "s""i" for "i"=1...,"n",
where (. ;) is a user-given total precedence order on the set of all function symbols.
Intuitively, a term "f"(...) is bigger than any term "g"(...) built from terms "s""i" smaller than "f"(...) using a
lower-precedence root symbol "g".
In particular, by structural induction, a term "f"(...) is bigger than any term containing only symbols smaller than "f".
A path ordering is often used as reduction ordering in term rewriting, in particular in the Knuth–Bendix completion algorithm.
As an example, a term rewriting system for "multiplying out" mathematical expressions could contain a rule "x"*("y"+"z") → ("x"*"y") + ("x"*"z"). In order to prove termination, a reduction ordering ( ;) must be found with respect to which the term "x"*("y"+"z") is greater than the term ("x"*"y")+("x"*"z"). This is not trivial, since the former term contains both fewer function symbols and fewer variables than the latter. However, setting the precedence (*) . ; (+), a path ordering can be used, since both "x"*("y"+"z") ; "x"*"y" and "x"*("y"+"z") ; "x"*"z" is easy to achieve.
There may also be systems for certain general recursive functions, for example a system for the Ackermann function may contain the rule A("a"+, "b"+) → A("a", A("a"+, "b")), where "b"+ denotes the successor of "b".
Given two terms "s" and "t", with a root symbol "f" and "g", respectively, to decide their relation their root symbols are compared first.
The latter variations include:
Dershowitz, Okada (1988) list more variants, and relate them to Ackermann's system of ordinal notations. In particular, an upper bound given on the order types of recursive path orderings with "n" function symbols is φ("n",0), using Veblen's function for large countable ordinals.
Formal definitions.
The multiset path ordering ( ;) can be defined as follows:
where 
More generally, an order functional is a function "O" mapping an ordering to another one, and satisfying the following properties:
The multiset extension, mapping ( ;) above to (») above is one example of an order functional: (»)="O"( ;).
Another order functional is the lexicographic extension, leading to the lexicographic path ordering.
References.
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