Girish Mahajan (Editor)

Path ordering (term rewriting)

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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,...,sn)   if   f .> g   and   f(...) > si 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 si 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 less function symbols and less 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.

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.

    • The latter variations include:

  • the multiset path ordering (mpo), originally called recursive path ordering (rpo)
  • the lexicographic path ordering (lpo)
  • a combination of mpo and lpo, called recursive path ordering by Dershowitz, Jouannaud (1990)

    • Dershowitz, Okada (1988) list more variants, and relate them to Ackermann's system of ordinals.

    Formal definitions

    The multiset path ordering (>) can be defined as follows:

    where

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

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

  • 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     Ri) = ∪∞
    i=0     O(Ri).

    • 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

    Path ordering (term rewriting) Wikipedia
    (Text) CC BY-SA