Completion [in multiway systems]

If one has a multiway system that terminates but is not confluent then it turns out often to be possible to make it confluent by adding a finite set of new rules. Given a string p which gets transformed either to q or r by the original rules, one can always imagine adding a new rule q r or r q that makes the paths from p immediately converge. To do this explicitly for all possible p that can occur would however entail having infinitely many new rules. But as noted by Donald Knuth and Peter Bendix in 1970 it turns out often to be sufficient just iteratively to add new rules only for each so-called critical pair q, r that is obtained from strings p that represent minimal overlaps in the left-hand sides of the rules one has. To decide whether to add q r or r q in each case one can have some kind of ordering on strings. For the procedure to work this ordering must be such that the strings generated on successive steps in every possible evolution of the multiway system follow the ordering. A number of variations of the basic procedure—using different orderings and with different schemes for dropping redundant rules—have been proposed for systems arising in different kinds of applications. The original Knuth-Bendix procedure was for equations (of the form a ↔ b) had the feature that it could terminate yet not give a confluent multiway system. But in the 1980s so-called unfailing completion algorithms (see page 1158) were developed that—if they terminate—guarantee to give confluent systems. (The question of whether any procedure of this type will terminate in a particular case is nevertheless in general undecidable.)

The basic idea of so-called critical pair completion procedures has arisen several times—notably in the Gröbner basis approach of Bruno Buchberger from 1965 to finding canonical forms for systems of polynomials.