PlanetMath: terminating reduction

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terminating reduction

(Definition)

Let be a set and $\to$ a reduction (binary relation) on . A chain with respect to $\to$ is a sequence of elements $x_1,x_2,x_3,\ldots$ in such that $x_1\to x_2$ , $x_2\to x_3$ , etc... A chain with respect to $\to$ is usually written

$\displaystyle x_1\to x_2 \to x_3 \to \cdots \to x_n \to \cdots.$

The length of a chain is the cardinality of its underlying sequence. A chain is finite if its length is finite. Otherwise, it is infinite.

Definition. A reduction $\to$ on a set is said to be terminating if it has no infinite chains. In other words, every chain terminates.

Examples.

If $\to$ is reflexive, or non-trivial symmetric, then it is never terminating.
Let be the set of all positive integers greater than . Define $\to$ on so that $a\to b$ means that for some $c\in X$ . Then $\to$ is a terminating reduction. By the way, $\to$ is also a normalizing reduction.
In fact, it is easy to see that a terminating reduction is normalizing: if has no normal form, then we may form an infinite chain starting from .
On the other hand, not all normalizing reduction is terminating. A canonical example is the set of all non-negative integers with the reduction $\to$ defined by $a\to b$ if and only if
- either $a,b\ne 0$ , $a\ne b$ , and ,
- or $a\ne 0$ and .
The infinite chain is given by $1\to 2\to 3\to \cdots$ , so that $\to$ is not terminating. However, $n\to 0$ for every positive integer . Thus every integer has 0 as its normal form, so that $\to$ is normalizing.

Remarks.

A reduction is said to be convergent if it is both terminating and confluent.
A relation is terminating iff the transitive closure of its inverse is well-founded.
To see this, first let be terminating on the set . And let be the transitive closure of $R^{-1}$ . Suppose is a non-empty subset of that contains no -minimal elements. Pick $x_0 \in A$ . Then we can find $x_1\in A$ with $x_1\ne x_0$ , such that . By the assumption on , this process can be iterated indefinitely. So we have a sequence $x_0, x_1, x_2, \ldots$ such that $x_{i+1} S x_i$ with $x_i\ne x_{i+1}$ . Since each pair $(x_i,x_{i+1})$ can be expanded into a finite chain with respect to , we have produced an infinite chain containing elements $x_0, x_1, x_2, \ldots$ , contradicting the assumption that is terminating.

On the other hand, suppose the transitive closure of $R^{-1}$ is well-founded. If the chain $x_0 R x_1 R x_2 R \cdots$ is infinite, then the set $\lbrace x_0, x_1, x_2, \ldots \rbrace$ has no -minimal elements, as whenever , and arbitrary.
The reflexive transitive closure of a terminating relation is a partial order.

A closely related concept is the descending chain condition (DCC). A reduction $\to$ on is said to satisfy the descending chain condition (DCC) if the only infinite chains on are those that are eventually constant. A chain $x_1\to x_2 \to x_3 \to \cdots$ is eventually constant if there is a positive integer such that for all $n\ge N$ , . Every terminating relation satisfies DCC. The converse is obviously not true, as a reflexive reduction illustrates.

Another related concept is acyclicity. Let $\to$ be a reduction on . A chain $x_0\to x_1 \to \cdots x_n$ is said to be cyclic if for some $0\le i < j\le n$ . This means that there is a “closed loop” in the chain. The reduction $\to$ is said to be acyclic if there are no cyclic chains with respect to $\to$ . Every terminating relation is acyclic, but not conversely. The usual strict inequality relation on the set of positive integers is an example of an acyclic but non-terminating relation.