Path ordering

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In theoretical computer science, in particular in term rewriting, a path ordering is a strict 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.[1][2]

The latter variations include:

  • the multiset path ordering (mpo), originally called recursive path ordering (rpo)[3]
  • the lexicographic path ordering (lpo)[4]
  • a combination of mpo and lpo, called recursive path ordering by Dershowitz, Jouannaud (1990)[5][6][7]

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


Formal definitions

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As an auxiliary notion, define an order functional to be a function O mapping an ordering to another one, and satisfying the following properties:[8]

  • 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 most general rpo version in use is called recursive path ordering with status, due to Lescanne (1983).[9][10] It is parametrized by

  • a well-quasi-ordering (.>) on function symbols, and
  • an order functional Of for each function symbol f.

The recursive path ordering with status is defined recursively as follows: Define s = f(s1,...,sm) > t = g(t1,...,tn) if

  • f .> g and s > tj for each j∈{1,...,n}, or
  • si > t or si = t for some i∈{1,...,m}, or
  • f .= g and s O(>) t.

where

  • { 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.[11]


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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  1. ^ N. Dershowitz, J.-P. Jouannaud (1990). Jan van Leeuwen (ed.). Rewrite Systems. Handbook of Theoretical Computer Science. Vol. B. Elsevier. pp. 243–320. Here: sect.5.3, p.275
  2. ^ Gerard Huet (May 1986). Formal Structures for Computation and Deduction. International Summer School on Logic of Programming and Calculi of Discrete Design.{{cite book}}: CS1 maint: date and year (link) Here: chapter 4, p.55-64
  3. ^ N. Dershowitz (1982). "Orderings for Term-Rewriting Systems" (PDF). Theoret. Comput. Sci. 17 (3): 279–301. doi:10.1016/0304-3975(82)90026-3.
  4. ^ S. Kamin, J.-J. Levy (1980). Two Generalizations of the Recursive Path Ordering (Technical report). Univ. of Illinois, Urbana/IL.
  5. ^ Kamin, Levy (1980)
  6. ^ N. Dershowitz, M. Okada (1988). "Proof-Theoretic Techniques for Term Rewriting Theory". Proc. 3rd IEEE Symp. on Logic in Computer Science (PDF). pp. 104–111.
  7. ^ Mitsuhiro Okada, Adam Steele (1988). "Ordering Structures and the Knuth–Bendix Completion Algorithm". Proc. of the Allerton Conf. on Communication, Control, and Computing.
  8. ^ Huet (1986), sect.4.3, p. 58
  9. ^ Pierre Lescanne (1983). "Computer Experiments with the REVE Term Rewriting System Generator". Proc. 10th ACM Symp. on Principles of Programming Languages (POPL) (PDF). ACM. pp. 99–108.
  10. ^ Solange Coupet-Grimal, William Delobel (2006). A Constructive Axiomatization of the Recursive Path Ordering (PDF) (Technical report). Laboratoire d'Informatique Fondamentale de Marseille (LIF). p. 15. 28-2006. {{cite tech report}}: Unknown parameter |month= ignored (help)
  11. ^ Huet (1986), sect.4.1.3, p.56