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Figure 1: Cyclic, locally-confluent, but not globally confluent rewrite system [4] Figure 2: Infinite non-cyclic, locally-confluent, but not globally confluent rewrite system [4] An element a ∈ S is said to be locally confluent (or weakly confluent [5]) if for all b, c ∈ S with a → b and a → c there exists d ∈ S with b d and c d.
A critical pair arises in a term rewriting system when two rewrite rules overlap to yield two different terms. In more detail, (t 1, t 2) is a critical pair if there is a term t for which two different applications of a rewrite rule (either the same rule applied differently, or two different rules) yield the terms t 1 and t 2.
Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines, [1] [2] or reduction systems). In their most basic form, they consist of a set of objects, plus relations on how to transform those objects. Rewriting can be non-deterministic. One rule to rewrite a term could be applied in many different ways to ...
Given a set E of equations between terms, the following inference rules can be used to transform it into an equivalent convergent term rewrite system (if possible): [4] [5] They are based on a user-given reduction ordering (>) on the set of all terms; it is lifted to a well-founded ordering ( ) on the set of rewrite rules by defining (s → t) (l → r) if
Suppose the set of objects is T = {a, b, c} and the binary relation is given by the rules a → b, b → a, a → c, and b → c. Observe that these rules can be applied to both a and b to get c. Furthermore, nothing can be applied to c to transform it any further. Such a property is clearly an important one.
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This leads to the idea of rewriting "modulo commutativity" where a term is in normal form if no rules but commutativity apply. [8] Weakly but not strongly normalizing rewrite system [9] The system {b → a, b → c, c → b, c → d} (pictured) is an example of a weakly normalizing but not strongly normalizing system.
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