Rewrite order

File:Triangle diagram of rewrite rule application.pdf

Rewriting s to t by a rule l::=r. If l and r are related by a rewrite relation, so are s and t. A simplifcation ordering always relates l and s, and similarly r and t.

In theoretical computer science, in particular in automated theorem proving and term rewriting, a binary relation (→) on the set of terms is called a rewrite relation if it is closed under contextual embedding and under instantiation; formally: if l→r implies u[lσ]_p→u[rσ]_p for all terms l, r, u, each path p of u, and each substitution σ. If (→) is also irreflexive and transitive, then it is called a rewrite ordering, or rewrite preorder. If the latter (→) is moreover well-founded, it is called a reduction ordering, or a reduction preorder. Given a binary relation R, its rewrite closure is the smallest rewrite relation containing R. A transitive and reflexive rewrite relation that contains the subterm ordering is called a simplification ordering.

Overview of rewrite relations
	rewrite relation	rewrite order	reduction order	simplification order
closed under context x R y implies u[x]p R u[y]p	Yes	Yes	Yes	Yes
closed under instantiation x R y implies xσ R yσ	Yes	Yes	Yes	Yes
contains subterm relation y subterm of x implies x R y				Yes
reflexive always x R x		(No)	(No)	Yes
irreflexive never x R x		Yes	Yes	(No)
transitive x R y and y R z implies x R z		Yes	Yes	Yes
well-founded no infinite chain x1 R x2 R x3 R ...			Yes	(Yes)

Properties

• The converse, the symmetric closure, the reflexive closure, and the transitive closure of a rewrite relation is again a rewrite relation, as are the union and the intersection of two rewrite relations.
• The converse of a rewrite order is again a rewrite order.
• While rewrite orders exist that are total on the set of ground terms ("ground-total" for short), no rewrite order can be total on the set of all terms.
• A term rewriting system {l₁::=r₁,...,l_n::=r_n, ...} is terminating if its rules are subset of a reduction ordering.
• Conversely, for every terminating term rewriting system, the transitive closure of (::=) is a reduction ordering, which needn't be extendable to a ground-total one, however. For example, the ground term rewriting system { f(a)::=f(b), g(b)::=g(a) } is terminating, but can be shown so using a reduction ordering only if the constants a and b are incomparable.
• A ground-total and well-founded rewrite ordering necessarily contains the proper subterm relation on ground terms.
• Conversely, a rewrite ordering that contains the subterm relation is necessarily well-founded, when the set of function symbols is finite.
• A finite term rewriting system {l₁::=r₁,...,l_n::=r_n, ...} is terminating if its rules are subset of the strict part of a simplification ordering.