HOME

TheInfoList



OR:

The Knuth–Bendix completion algorithm (named after
Donald Knuth Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer ...
and Peter Bendix) is a semi-decision
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
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 Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
. 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 ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variable ...
s.


Introduction

For a set ''E'' of equations, its deductive closure () is the set of all equations that can be derived by applying equations from ''E'' in any order. Formally, ''E'' is considered a
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
, () is its
rewrite closure In theoretical computer science, in particular in automated reasoning about formal equations, reduction orderings are used to prevent endless loops. Rewrite orders, and, in turn, rewrite relations, are generalizations of this concept that have tur ...
, and () is the equivalence closure of (). For a set ''R'' of rewrite rules, its deductive closure ( ∘ ) is the set of all equations that can be confirmed by applying rules from ''R'' left-to-right to both sides until they are literally equal. Formally, ''R'' is again viewed as a binary relation, () is its rewrite closure, () is its converse, and ( ∘ ) is the relation composition of their reflexive transitive closures ( and ). For example, if are the group axioms, the derivation chain : demonstrates that ''a''−1⋅(''a''⋅''b'') ''b'' is a member of ''Es deductive closure. If is a "rewrite rule" version of ''E'', the derivation chains : demonstrate that (''a''−1⋅''a'')⋅''b'' ∘ ''b'' is a member of ''Rs deductive closure. However, there is no way to derive ''a''−1⋅(''a''⋅''b'') ∘ ''b'' similar to above, since a right-to-left application of the rule is not allowed. The Knuth–Bendix algorithm takes a set ''E'' of equations between terms, and a
reduction ordering Reduction, reduced, or reduce may refer to: Science and technology Chemistry * Reduction (chemistry), part of a reduction-oxidation (redox) reaction in which atoms have their oxidation state changed. ** Organic redox reaction, a redox react ...
(>) on the set of all terms, and attempts to construct a confluent and terminating term rewriting system ''R'' that has the same deductive closure as ''E''. While proving consequences from ''E'' often requires human intuition, proving consequences from ''R'' does not. For more details, see Confluence (abstract rewriting)#Motivating examples, which gives an example proof from group theory, performed both using ''E'' and using ''R''.


Rules

Given a set ''E'' of equations between terms, the following inference rules can be used to transform it into an equivalent convergent term rewrite system (if possible): They are based on a user-given
reduction ordering Reduction, reduced, or reduce may refer to: Science and technology Chemistry * Reduction (chemistry), part of a reduction-oxidation (redox) reaction in which atoms have their oxidation state changed. ** Organic redox reaction, a redox react ...
(>) on the set of all terms; it is lifted to a well-founded ordering (▻) on the set of rewrite rules by defining if * in the encompassment ordering, or * and are literally similar and .


Example

The following example run, obtained from the E theorem prover, computes a completion of the (additive) group axioms as in Knuth, Bendix (1970). It starts with the three initial equations for the group (neutral element 0, inverse elements, associativity), using f(X,Y) for ''X''+''Y'', and i(X) for −''X''. The 10 starred equations turn out to constitute the resulting convergent rewrite system. "pm" is short for "
paramodulation In mathematical logic and automated theorem proving, resolution is a rule of inference leading to a refutation complete theorem-proving technique for sentences in propositional logic and first-order logic. For propositional logic, systematically ...
", implementing ''deduce''. Critical pair computation is an instance of paramodulation for equational unit clauses. "rw" is rewriting, implementing ''compose'', ''collapse'', and ''simplify''. Orienting of equations is done implicitly and not recorded. See also Word problem (mathematics) for another presentation of this example.


String rewriting systems in group theory

An important case in
computational group theory In mathematics, computational group theory is the study of group (mathematics), groups by means of computers. It is concerned with designing and analysing algorithms and data structures to compute information about groups. The subject has attracted ...
are string rewriting systems which can be used to give canonical labels to elements or
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s of a finitely presented group as products of the generators. This special case is the focus of this section.


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 In algebra, a presentation of a monoid (or a presentation of a semigroup) is a description of a monoid (or a semigroup) in terms of a set of generators and a set of relations on the free monoid (or the free semigroup ) generated by . The monoid ...
M = \langle X \mid R \rangle 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 ' it is desirable to choose a standard representative ''wk''. This representative is called the canonical or normal form for each word ''wk'' in the class. If there is a computable method to determine for each ''wk'' its normal form ''wi'' 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 Wi → Wi+1 it is possible to end up with two different irreducible reductions Wn ≠ 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 A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Present ...
\langle X \mid R \rangle , where X is a set of generators and R is a set of
relations Relation or relations may refer to: General uses *International relations, the study of interconnection of politics, economics, and law on a global level *Interpersonal relationship, association or acquaintance between two or more people *Public ...
giving the rewriting system. Suppose further that we have a reduction ordering < among the words generated by X (e.g.,
shortlex order In mathematics, and particularly in the theory of formal languages, shortlex is a total ordering for finite sequences of objects that can themselves be totally ordered. In the shortlex ordering, sequences are primarily sorted by cardinality (length) ...
). For each relation P_i = Q_i in R , suppose Q_i < P_i . Thus we begin with the set of reductions P_i \rightarrow 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 overlap. # 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 . #Case 2: either P_i is completely contained in (surrounded by) 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 \neq r_2 , then we have an instance where confluence could fail. Hence, add the reduction \max r_1, r_2 \rightarrow \min r_1, r_2 to R . After adding a rule to R , remove any rules in R that might have reducible left sides (after checking if such rules have critical pairs with other rules). Repeat the procedure until all overlapping left sides have been checked.


Examples


A terminating example

Consider the monoid: : \langle x, y \mid x^3 = y^3 = (xy)^3 = 1 \rangle . We use the
shortlex order In mathematics, and particularly in the theory of formal languages, shortlex is a total ordering for finite sequences of objects that can themselves be totally ordered. In the shortlex ordering, sequences are primarily sorted by cardinality (length) ...
. This is an infinite monoid but nevertheless, the Knuth–Bendix algorithm is able to solve the word problem. Our beginning three reductions are therefore A suffix of x^3 (namely x) is a prefix of (xy)^3=xyxyxy, so consider the word x^3yxyxy . Reducing using (), we get yxyxy . Reducing using (), we get x^2 . Hence, we get yxyxy = x^2 , giving the reduction rule Similarly, using xyxyxy^3 and reducing using () and (), we get xyxyx = y^2 . Hence the reduction Both of these rules obsolete (), so we remove it. Next, consider x^3yxyx by overlapping () and (). Reducing we get yxyx = x^2y^2 , so we add the rule Considering xyxyx^3 by overlapping () and (), we get xyxy = y^2x^2 , so we add the rule These obsolete rules () and (), so we remove them. Now, we are left with the rewriting system 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.


A non-terminating example

The order of the generators may crucially affect whether the Knuth–Bendix completion terminates. As an example, consider the
free Abelian group In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a su ...
by the monoid presentation: : \langle x, y, x^, y^\, , \, xy=yx, xx^ = x^x = yy^ = y^y = 1 \rangle . The Knuth–Bendix completion with respect to lexicographic order x < x^ < y < y^ finishes with a convergent system, however considering the length-lexicographic order x < y < x^ < y^ it does not finish for there are no finite convergent systems compatible with this latter order.


Generalizations

If Knuth–Bendix does not succeed, it will either run forever and produce successive approximations to an infinite complete system, or fail when it encounters an unorientable equation (i.e. an equation that it cannot turn into a rewrite rule). An enhanced version will not fail on unorientable equations and produces a ground confluent system, providing a
semi-algorithm In computability theory and computational complexity theory, RE (Recursively enumerable set, recursively enumerable) is the complexity class, class of decision problems for which a 'yes' answer can be verified by a Turing machine in a finite amount ...
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

* * * 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.


External links

*
Knuth-Bendix Completion Visualizer
{{DEFAULTSORT:Knuth-Bendix completion algorithm Computational group theory Donald Knuth Combinatorics on words Rewriting systems