Knuth-Bendix completion algorithm

The Knuth-Bendix completion algorithm is an algorithm for transforming a set of equations (over terms) into a confluent term rewriting system. When the algorithm succeeds, it has effectively solved the word problem for the specified algebra.

An important case in computational group theory are string rewriting systems which can be used to give canonical labels to elements or cosets of a finitely presented group as products of the generators. This special case is the current focus of this article.

Motivation in group theory

The critical pair lemma states that a term rewriting system is weakly confluent if and only if the critical pairs are convergent. Furthermore, we have Newman's lemma which states that if an (abstract) rewriting system is strongly normalizing and weakly confluent, then the rewriting system is confluent. So, if we can add rules to the term rewriting system in order to force all critical pairs to be convergent while maintaining the strong normalizing property, then this will force the resultant rewriting system to be confluent.

Consider a finitely presented monoid where X is a finite set of generators and R is a set of defining relations on X. Let X^* be the set of all words in X (i.e. the free monoid generated by X). Since the relations R define an equivalence relation on X*, one can consider elements of M to be the equivalence classes of X^* under R. For each class "{w₁, w₂, ... }" it is desirable to choose a standard representative "w_k". This representative is called the canonical or normal form for each word "w_k" in the class. If there is a computable method to determine for each "w_k" its normal form "w_i" then the word problem is easily solved. A confluent rewriting system allows one to do precisely this.

Although the choice of a canonical form can theoretically be made in an arbitrary fashion this approach is generally not computable. (Consider that an equivalence relation on a language can produce an infinite number of infinite classes.) If the language is well-ordered then the order < gives a consistent method for defining minimal representatives, however computing these representatives may still not be possible. In particular, if a rewriting system is used to calculate minimal representatives then the order < should also have the property:

: A < B -> XAY < XBY for all words A,B,X,Y

This property is called translation invariance. An order that is both translation-invariant and a well-order is called a reduction order.

From the presentation of the monoid it is possible to define a rewriting system given by the relations R. If A x B is in R then either A < B in which case B -> A is a rule in the rewriting system, otherwise A > B and A -> B. Since < is a reduction order a given word W can be reduced W > W_1 > ... > W_n where W_n is irreducible under the rewriting system. However, depending on the rules that are applied at each W_i -> W_i+1 it is possible to end up with two different irreducible reductions W_n $e$ W'_m of W. However, if the rewriting system given by the relations is converted to a confluent rewriting system via the Knuth-Bendix algorithm, then all reductions are guaranteed to produce the same irreducible word, namely the normal form for that word.

Description of the algorithm for finitely presented monoids

Suppose we are given a presentation $langle\; X\; mid\; R\; angle$, where $X$ is a set of generators and $R$ is a set of relations giving the rewriting system. Suppose further that we have a reduction ordering among the words generated by $X$. For each relation $P\_i\; =\; Q\_i$ in $R$, suppose . Thus we begin with the set of reductions $P\_i\; ightarrow\; Q\_i$.

First, if any relation $P\_i\; =\; Q\_i$ can be reduced, replace $P\_i$ and $Q\_i$ with the reductions.

Next, we add more reductions (that is, rewriting rules) to eliminate possible exceptions of confluence. Suppose that $P\_i$ and $P\_j$, where $i\; eq\; j$, overlap. That is, either the prefix of $P\_i$ equals the suffix of $P\_j$, or vice versa. In the former case, we can write $P\_i\; =\; BC,\; P\_j\; =\; AB$; in the latter case, $P\_i\; =\; AB,\; P\_j\; =\; BC$.

Reduce the word $ABC$ using $P\_i$ first, then using $P\_j$ first. Call the results $r\_1,\; r\_2$, respectively. If $r\_1\; eq\; r\_2$, then we have an instance where confluence could fail. Hence, add the reduction $max\; r\_1,\; r\_2\; ightarrow\; min\; r\_1,\; r\_2$ to $R$.

After adding a rule to $R$, remove any rules in $R$ that might have reducible left sides.

Repeat the procedure until all overlapping left sides have been checked.

Example

Consider the presentation $\{\; x,\; y\; mid\; x^3\; =\; y^3\; =\; (xy)^3\; =\; 1\; \}$. We use the shortlex order. In fact, this is an infinite group. Nevertheless, the Knuth-Bendix algorithm is able to solve the word problem.

Our beginning three reductions are therefore (1) $x^3\; ightarrow\; 1$, (2) $y^3\; ightarrow\; 1$, and (3) $(xy)^3\; ightarrow\; 1$.

First, we see an overlap of $x$ in (1) and (3). Consider the word $x^3yxyxy$. Reducing using (1), we get $yxyxy$. Reducing using (3), we get $x^2$. Hence, we get $yxyxy\; =\; x^2$, giving the reduction rule (4) $yxyxy\; ightarrow\; x^2$.

Similarly, using the overlap of $y$ in (2) and (3), we get the reduction (5) $xyxyx\; ightarrow\; y^2$.

Both of these rules obsolete (3), so we remove it.

Next, consider the overlap of $x$ of (1) and (5). Considering $x^3yxyx$ we get $yxyx\; =\; x^2y^2$, so we add the rule (6) $yxyx\; ightarrow\; x^2y^2$. This obsoletes rules (4) and (5), so we remove them. Considering $xyxyx^3$, we get $xyxy\; =\; y^2x^2$, so we add the rule (7) $y^2x^2\; ightarrow\; xyxy$.

Now, we are left with the rewriting system
* (1) $x^3\; ightarrow\; 1$
* (2) $y^3\; ightarrow\; 1$
* (6) $yxyx\; ightarrow\; x^2y^2$
* (7) $y^2\; x^2\; ightarrow\; xyxy$Checking the overlaps of these rules, we find no potential failures of confluence. Therefore, we have a confluent rewriting system, and the algorithm terminates successfully.

Generalizations

If Knuth-Bendix does not succeed, it will either run forever, or fail when it encounters an unorientable equation (i.e. an equation that it cannot turn into a rewrite rule). The enhanced completion without failure will not fail on unorientable equations and provides a semi-decision procedure for the word problem.

References

* D. Knuth and P. Bendix. "Simple word problems in universal algebras." "Computational Problems in Abstract Algebra" (Ed. J. Leech) pages 263--297, 1970.
* C. Sims. 'Computations with finitely presented groups.' Cambridge, 1994.

External links

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