History and Basic Features of the Critical-Pair/Completion Procedure

A synopsis of the critical-pair/completion approach for solving algorithmic problems in theorem proving, finitely generated algebras and rewrite systems is given. The emphasis is on tracing the main intuition behind the approach, which consists in considering ''critical pairs''. These are the ''first possible'' situations where ''superpositions'' may occur. Extensive references to the original literature are provided. Some directions for future research are outlined. The presentation is biased towards the development of the approach in algorithmic polynomial ideal theory.

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