Confluence (term rewriting)

Confluence is a property of term rewriting systems, describing that terms in this system can be rewritten in more than one way, to yield the same result.

Motivating example

Consider the rules of regular arithmetic. We can think of these rules as forming a rewriting system. Suppose we are given the expression: $(11\; +\; 9)\; imes\; (2\; +\; 4)$We can rewrite this expression in two ways -- either simplifying the first bracket, or the second. Simplifying the first bracket, we have: $20\; imes\; (2\; +\; 4)\; =\; 20\; imes\; 6\; =\; 120$Simplifying the second, gives: $(11\; +\; 9)\; imes\; 6\; =\; 20\; imes\; 6\; =\; 120$We get the same result from rewriting the term in two different ways. This suggests that the rewriting system formed from regular arithmetic is a confluent rewriting system.

General case and theory

We can think of a rewriting system abstractly as being a set "S", together with a relation "R" which is a subset of "S"×"S". Instead of denoting relations as pairs ("s", "t"), with "s","t" ∈ "S", we write "s" →_"i" "t", where "i" is from some index set "I". If "u" ∈ "S", and "u" →_"j" "v" for some "j" ∈ "I", then we say "v" is a "reduct" of "u" (alternatively "v" is an "expansion" of "u", or "u" is "reduced to" "v"). We can also think of a chain of reductions of some element: if "w" ∈ "S", then we may have : $w\; ightarrow\_\{i\_1\}\; w\_1\; ightarrow\_\{i\_2\}\; w\_2\; ightarrow\_\{i\_3\}\; cdots$where "i"_"k" ∈ "I". We call this chain a "reduction sequence", or just a "reduction of" "w". If the reduction sequence terminates, say in "w"_"n", we write : $w\; ightarrow^*\_I\; w\_n$,reflecting that relations in "I" were chosen. If "I" is clear from context then we can omit this subscript.

With this established, confluence can be defined as follows. Let "a", "b", "c" ∈ "S", with "a" →* "b" and "a" →* "c". If "a" is confluent, 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.

Local confluence

An element "a" ∈ "S" is said to be locally (or weakly) confluent if for all "b", "c" ∈ "S" with "a" → "b" and "a" → "c" there exists "d" ∈ "S" with "b" →* "d" and "c" →* "d". If every "a" ∈ "S" is locally confluent, we say that → is locally (or weakly) confluent, or has the "weak Church-Rosser property". This is different from confluence in that "b" and "c" must be reduced from "a" in one step. In analogy with this, confluence is sometimes referred to as "global confluence".

We may view →*, which we introduced as a notation for reduction sequences, as a rewriting system in its own right, whose relation is the transitive closure of "R". Since a sequence of reduction sequences is again a reduction sequence (or, equivalently, since forming the transitive closure is idempotent), →** = →*. It follows that → is confluent if and only if →* is locally confluent.

A rewriting system may be locally confluent without being globally confluent. However, Newman's lemma states that if a locally confluent rewriting system has no infinite reduction sequences (in which case it is said to be "terminating" or "strongly normalizing"), then it is globally confluent.

Semi-confluence

The definition of local confluence differs from that of global confluence in that only elements reached from a given element in a single rewriting step are considered. By considering one element reached in a single step and another element reached by an arbitrary sequence, we arrive at the intermediate concept of semi-confluence: "a" ∈ "S" is said to be semi-confluent if for all "b", "c" ∈ "S" with "a" → "b" and "a" →* "c" there exists "d" ∈ "S" with "b" →* "d" and "c" →* "d"; if every "a" ∈ "S" is semi-confluent, we say that → is semi-confluent.

A semi-confluent element need not be confluent, but a semi-confluent rewriting system is necessarily confluent.

Strong confluence

Strong confluence is another variation on local confluence that allows us to conclude that a rewriting system is globally confluent. An element "a" ∈ "S" is said to be strongly confluent if for all "b", "c" ∈ "S" with "a" → "b" and "a" → "c" there exists "d" ∈ "S" with "b" →* "d" and either "c" → "d" or "c" = "d"; if every "a" ∈ "S" is strongly confluent, we say that → is strongly confluent.

A strongly confluent element need not be confluent, but a strongly confluent rewriting system is necessarily confluent.

See also

* critical pair
* Church–Rosser theorem

References

* "Term Rewriting Systems", Terese, Cambridge Tracts in Theoretical Computer Science, 2003
* "Term Rewriting and All That", Franz Baader and Tobias Nipkow, Cambridge University Press, 1998

External links

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