Calculus of constructions

The Knuth–Bendix completion algorithm (named after Donald Knuth and Peter Bendix) is an algorithm for transforming a set of equations (over terms) into a confluent term rewriting system. When the algorithm succeeds, it effectively solves 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.

Buchberger's algorithm for computing Gröbner bases is a very similar algorithm. Although developed independently, it may also be seen as the instantiation of Knuth-Bendix algorithm in the theory of polynomial rings.

Motivation in group theory

The critical pair lemma states that a term rewriting system is locally confluent (or weakly confluent) if and only if all its 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 $M=⟨X\mid R⟩$ 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 generate 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 ≠ 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 $⟨X\mid R⟩$, 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 $X$(e.g., shortlex order). For each relation $P_i=Q_i$ in $R$, suppose $Q_i<P_i$. Thus we begin with the set of reductions $P_i\to Q_i$.

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 $i\ne j$, overlap.

1. Case 1: either the prefix of $P_i$ equals the suffix of $P_j$, or vice versa. In the former case, we can write $P_i=BC$ and $P_j=AB$; in the latter case, $P_i=AB$ and $P_j=BC$.
2. Case 2: either $P_i$ is completely contained (surrounded) in $P_j$, or vice versa. In the former case, we can write $P_i=B$ and $P_j=ABC$; in the latter case, $P_i=ABC$ and $P_j=B$.

Reduce the word $ABC$ using $P_i$ first, then using $P_j$ first. Call the results $r_1,r_2$, respectively. If $r_1\ne r_2$, then we have an instance where confluence could fail. Hence, add the reduction $\max r_1,r_2\to \min r_1,r_2$ 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 monoid: $⟨x,y\mid x^3=y^3=(xy{)}^3=1⟩$. We use the shortlex order. This is an infinite monoid but nevertheless, the Knuth–Bendix algorithm is able to solve the word problem.

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

Consider the word $x^{3}yxyxy$. Reducing using (1), we get $yxyxy$. Reducing using (3), we get $x^2$. Hence, we get $yxyxy=x^2$, giving the reduction rule (4) $yxyxy\to x^2$.

Similarly, using $xyxyx{y}^3$ and reducing using (2) and (3), we get $xyxyx=y^2$. Hence the reduction (5) $xyxyx\to y^2$.

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

Next, consider $x^{3}yxyx$ by overlapping (1) and (5). Reducing we get $yxyx=x^2{y}^2$, so we add the rule (6) $yxyx\to x^2{y}^2$. Considering $xyxy{x}^3$ by overlapping (1) and (5), we get $xyxy=y^2{x}^2$, so we add the rule (7) $y^2{x}^2\to xyxy$. These obsolete rules (4) and (5), so we remove them.

Now, we are left with the rewriting system

• (1) $x^3\to 1$
• (2) $y^3\to 1$
• (6) $yxyx\to x^2{y}^2$
• (7) $y^2{x}^2\to xyxy$

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.

The notion of logged rewriting discussed in the paper by Heyworth and Wensley listed below allows some recording or logging of the rewriting process as it proceeds. This is useful for computing identities among relations for presentations of groups.

References

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• C. Sims. 'Computations with finitely presented groups.' Cambridge, 1994.
• Anne Heyworth and C.D. Wensley. "Logged rewriting and identities among relators." Groups St. Andrews 2001 in Oxford. Vol. I, 256–276, London Math. Soc. Lecture Note Ser., 304, Cambridge Univ. Press, Cambridge, 2003.

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