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Rewrite order

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Rewrite order

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.

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,...,ln::=rn, ...} 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,...,ln::=rn, ...} is terminating if its rules are subset of the strict part of a simplification ordering.

    • References

    Rewrite order Wikipedia
    (Text) CC BY-SA


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