terminating reduction

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

$$x_1\to x_2\to x_3\to \mathrm{\cdots }\to x_n\to \mathrm{\cdots }.$$

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 $X$ 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 $X$ be the set of all positive integers greater than $1$. Define $\to$ on $X$ so that $a\to b$ means that $a=bc$ 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 $a$ has no normal form, then we may form an infinite chain starting from $a$.

• 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 $b=0$.

  The infinite chain is given by $1\to 2\to 3\to \mathrm{\cdots }$, so that $\to$ is not terminating. However, $n\to 0$ for every positive integer $n$. 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 $R$ be terminating on the set $X$. And let $S$ be the transitive closure of $R^{-1}$. Suppose $A$ is a non-empty subset of $X$ that contains no $S$-minimal elements. Pick $x_0\in A$. Then we can find $x_1\in A$ with $x_1\ne x_0$, such that $x_{1}S{x}_0$. By the assumption on $A$, this process can be iterated indefinitely. So we have a sequence $x_0,x_1,x_2,\mathrm{\dots }$ 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 $R$, we have produced an infinite chain containing elements $x_0,x_1,x_2,\mathrm{\dots }$, contradicting the assumption that $R$ is terminating.

  On the other hand, suppose the transitive closure $S$ of $R^{-1}$ is well-founded. If the chain $x_{0}R{x}_{1}R{x}_{2}R\mathrm{\cdots }$ is infinite, then the set $\{x_0,x_1,x_2,\mathrm{\dots }\}$ has no $S$-minimal elements, as $x_{i}S{x}_j$ whenever $i>j$, and $j$ 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 $X$ is said to satisfy the descending chain condition (DCC) if the only infinite chains on $X$ are those that are eventually constant. A chain $x_1\to x_2\to x_3\to \mathrm{\cdots }$ is eventually constant if there is a positive integer $N$ such that for all $n\ge N$, $x_n=x_N$. 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 $X$. A chain $x_0\to x_1\to \mathrm{\cdots }{x}_n$ is said to be cyclic if $x_i=x_j$ for some . 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.

Title	terminating reduction
Canonical name	TerminatingReduction
Date of creation	2013-03-22 17:57:41
Last modified on	2013-03-22 17:57:41
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	11
Author	CWoo (3771)
Entry type	Definition
Classification	msc 68Q42
Related topic	NormalizingReduction
Related topic	Confluence
Related topic	DiamondLemma
Defines	terminating
Defines	descending chain condition
Defines	DCC
Defines	convergent reduction
Defines	acyclic