abstract reduction system
Using this abstraction, it is easy to define a range of basic notions that play a role in computing with rules.
(1) An element a ∈ A is a normal form for →R if there does not exist b, different from a, such that a →R b.
(2) The reduction system →R is Church–Rosser (or confluent) if for any a ∈ A if there are b1,b2 ∈ A such that a →R b1 and a →R b2 then there exists c ∈ A such that b1 →R c and b2 →R c.
(3) The reduction system →R is weakly terminating (or weakly normalizing) if for each a ∈ A there is some normal form b ∈ A so that a →R b.
(4) The reduction system →R is strongly terminating (or strongly normalizing or Noetherian) if there does not exist an infinite chain a0 →R a1 →R …→R an →R …
of reductions in A wherein ai ≠ ai+1 for i = 0, 1, 2, …
(5) The reduction system →R is complete if it is Church–Rosser and strongly terminating.
(6) A reduction system is Church–Rosser and weakly terminating if, and only if, every element reduces to a unique normal form. Let ≡R denote the smallest equivalence relation on A containing →R. If →R is a Church–Rosser weakly terminating reduction system then the set NF(→R) of normal forms is a transversal for ≡R, i.e. a set that contains one and only one representative of each equivalence class.
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