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The F5 algorithm is generally believed as one of the fastest algorithms forcomputing Gr\"{o}bner bases. However, its termination problem is still unclear.Recently, an algorithm GVW and its variant GVWHS have been proposed, and theirefficiency are comparable to the F5 algorithm. In the paper, we clarify theconcept of an admissible module order. For the first time, the connectionbetween the reducible and rewritable check is discussed here. We show that thetop-reduced S-Gr\"{o}bner basis must be finite if the admissible monomial orderand the admissible module order are compatible. This paper presents a completeproof of the termination and correctness of the GVWHS algorithm. What is more,it can be seen that the GVWHS is in fact an F5-like algorithm. Different fromthe GVWHS algorithm, the F5B algorithm may generate redundant sig-polynomials.Taking into account this situation, we prove the termination and correctness ofthe F5B algorithm. And we notice that the original F5 algorithm slightlydiffers from the F5B algorithm in the insertion strategy on which theF5-rewritten criterion is based. Exploring the potential ordering ofsig-polynomials computed by the original F5 algorithm, we propose an F5GENalgorithm with a generalized insertion strategy, and prove the termination andcorrectness of it. Therefore, we have a positive answer to the long standingproblem of proving the termination of the original F5 algorithm.

The Termination of Algorithms for Computing Gr\"{o}bner Bases