Path ordering (term rewriting)

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(s₁,...,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)),^[1] 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.

• If f <^. g, then s can dominate t only if one of s's subterms does.
• If f ^.> g, then s dominates t if s dominates each of t's subterms.
• If f = g, then the immediate subterms of s and t need to be compared recursively. Depending on the particular method, different variations of path orderings exist.^[2][3]

The latter variations include:

• the multiset path ordering (mpo), originally called recursive path ordering (rpo)^[4]
• the lexicographic path ordering (lpo)^[5]
• a combination of mpo and lpo, called recursive path ordering by Dershowitz, Jouannaud (1990)^[6][7][8]

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.^[7]

The multiset path ordering (>) can be defined as follows:^[9]

where

• (≥) denotes the reflexive closure of the mpo (>),
• { s₁,...,s_m } denotes the multiset of s’s subterms, similar for t, and
• (>>) denotes the multiset extension of (>), defined by { s₁,...,s_m } >> { t₁,...,t_n } if { t₁,...,t_n } can be obtained from { s₁,...,s_m }
  • by deleting at least one element, or
  • by replacing an element by a multiset of strictly smaller (w.r.t. the mpo) elements.^[10]

More generally, an order functional is a function O mapping an ordering to another one, and satisfying the following properties:^[11]

• If (>) is transitive, then so is O(>).
• If (>) is irreflexive, then so is O(>).
• If s > t, then f(...,s,...) O(>) f(...,t,...).
• O is continuous on relations, i.e. if R₀, R₁, R₂, R₃, ... is an infinite sequence of relations, then O(∪^∞
  _i=0 R_i) = ∪^∞
  _i=0 O(R_i).

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.