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,...,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 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.
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
The multiset path ordering (>) can be defined as follows:[8]
s = f(s1,...,sm) > t = g(t1,...,tn) | if | ||||||||
f | .> | g | and | s | > | tj | for each | j∈{1,...,n}, | or |
si | ≥ | t | for some | i∈{1,...,m}, | or | ||||
f | = | g | and | { s1,...,sm } >> { t1,...,tn } |
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.[9]
More generally, an order functional is a function O mapping an ordering to another one, and satisfying the following properties:[10]
- 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
- Nachum Dershowitz, Jean-Pierre Jouannaud (1990). Jan van Leeuwen (ed.). Rewrite Systems. Handbook of Theoretical Computer Science. B. Elsevier. pp. 243–320. Here: sect.5.3, p.275
- Gerard Huet (May 1986). Formal Structures for Computation and Deduction. International Summer School on Logic of Programming and Calculi of Discrete Design. Archived from the original on 2014-07-14. Here: chapter 4, p.55-64
- N. Dershowitz (1982). "Orderings for Term-Rewriting Systems" (PDF). Theoret. Comput. Sci. 17 (3): 279–301. doi:10.1016/0304-3975(82)90026-3. S2CID 6070052.
- S. Kamin, J.-J. Levy (1980). Two Generalizations of the Recursive Path Ordering (Technical report). Univ. of Illinois, Urbana/IL.
- Kamin, Levy (1980)
- 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.
- Mitsuhiro Okada, Adam Steele (1988). "Ordering Structures and the Knuth–Bendix Completion Algorithm". Proc. of the Allerton Conf. on Communication, Control, and Computing.
- Huet (1986), sect.4.3, def.1, p.57
- Huet (1986), sect.4.1.3, p.56
- Huet (1986), sect.4.3, p. 58