General Case and Theory
A rewriting system can be expressed as a directed graph in which nodes represent expressions and edges represent rewrites. So, for example, if the expression can be rewritten into, then we say that is a reduct of (alternatively, reduces to, or is an expansion of ). This is represented using arrow notation; indicates that reduces to . Intuitively, this means that the corresponding graph has a directed edge from to .
If there is a path between two graph nodes (let's call them and ), then the intermediate nodes form a reduction sequence. So, for instance, if, then we can write, indicating the existence of a reduction sequence from to .
With this established, confluence can be defined as follows. Let a, b, c ∈ S, with a →* b and a →* c. a is deemed confluent if there exists a d ∈ S with b →* d and c →* d. If every a ∈ S is confluent, we say that → is confluent, or has the Church-Rosser property. This property is also sometimes called the diamond property, after the shape of the diagram shown on the right. Caution: other presentations reserve the term diamond property for a variant of the diagram with single reductions everywhere; that is, whenever a → b and a → c, there must exist a d such that b → d and c → d. The single-reduction variant is strictly stronger than the multi-reduction one.
Read more about this topic: Confluence (abstract Rewriting)
Famous quotes containing the words general, case and/or theory:
“A writer who writes, “I am alone” ... can be considered rather comical. It is comical for a man to recognize his solitude by addressing a reader and by using methods that prevent the individual from being alone. The word alone is just as general as the word bread. To pronounce it is to summon to oneself the presence of everything the word excludes.”
—Maurice Blanchot (b. 1907)
“I say you must not win an unjust case by oaths.”
—Aeschylus (525–456 B.C.)
“The whole theory of modern education is radically unsound. Fortunately in England, at any rate, education produces no effect whatsoever. If it did, it would prove a serious danger to the upper classes, and probably lead to acts of violence in Grosvenor Square.”
—Oscar Wilde (1854–1900)