[Paper Review] Termination of Original F5

Vasily Galkin|arXiv (Cornell University)|Mar 12, 2012

Polynomial and algebraic computation2 references9 citations

TL;DR

This paper provides a termination proof for the original F5 algorithm for any homogeneous polynomial input, without modifying the algorithm or relying on regularity assumptions. It establishes termination by showing that the existence of a non-terminating execution leads to a contradiction involving signature-safe reducers and F5 criteria, thereby proving the original F5 algorithm always terminates for homogeneous inputs.

ABSTRACT

The original F5 algorithm introduced by Faugère is formulated for any homogeneous polynomial set input. The correctness of output is shown for any input that terminates the algorithm, but the termination itself is proved only for the case of input being regular polynomial sequence. This article shows that algorithm correctly terminates for any homogeneous input without any reference to regularity. The scheme contains two steps: first it is shown that if the algorithm does not terminate it eventually generates two polynomials where first is a reductor for the second. But first step does not show that this reduction is permitted by criteria introduced in F5. The second step shows that if such pair exists then there exists another pair for which the reduction is permitted by all criteria. Existence of such pair leads to contradiction. Version v3 fixes the bibliography.

Motivation & Objective

• To establish a termination proof for the original F5 algorithm that does not depend on input regularity or algorithmic modifications.
• To address the gap in existing proofs, which only cover regular sequences or modified F5 variants.
• To provide a direct proof for the exact F5 algorithm as described in Faugère (2002), preserving its original behavior.
• To show that non-termination leads to a contradiction involving signature-safe reducers and F5 criteria.
• To strengthen the theoretical foundation of F5 for practical computer algebra systems by proving correctness and termination under general conditions.

Proposed method

• Introduces S-pair-chains as a structural tool to analyze infinite execution paths in F5.
• Uses a two-step contradiction argument: first, non-termination implies existence of a reductor pair; second, such a pair implies a valid pair satisfying all F5 criteria.
• Applies the representation rewriting technique from Eder & Perry (2009) to F5’s intermediate states, weakening hypotheses and strengthening conclusions.
• Relies on the signature-safe reductor property and the three F5 criteria (F5, Rewritten, and signature order) to derive contradictions.
• Uses the fact that if a polynomial is signature-safe reducible, then its reduction must satisfy all criteria, which fails under non-termination.
• Demonstrates that any non-terminating execution leads to a polynomial pair violating at least one criterion, contradicting the algorithm’s internal checks.

Experimental results

Research questions

• RQ1 Can the original F5 algorithm be proven to terminate for all homogeneous inputs without assuming regularity or modifying the algorithm?
• RQ2 What structural properties of F5’s execution path lead to non-termination, and how can they be exploited to derive a contradiction?
• RQ3 Is it possible to prove termination using only the original algorithm’s criteria and signature ordering, without introducing new checks or rules?
• RQ4 How do S-pair-chains and representation rewriting help in analyzing intermediate states of F5 during non-terminating runs?
• RQ5 Can the contradiction derived from a non-terminating execution be formally linked to the failure of F5’s reduction criteria?

Key findings

• The original F5 algorithm terminates for any homogeneous polynomial input, regardless of whether the input is regular or not.
• Non-termination leads to the existence of a pair of polynomials f′₁ and f where f′₁ is a signature-safe reductor for f but the reduction is not permitted by F5 or Rewritten criteria.
• The contradiction arises because such a pair must satisfy all criteria (a)–(d) in IsReducible, but the conditions are violated, contradicting the algorithm’s behavior.
• The proof relies on the equality of monomial order and signature order, which is essential for deriving the contradiction.
• The termination proof is valid only for homogeneous inputs and depends on the presence of the Rewritten criterion and signature-safe reduction.
• The result strengthens the theoretical foundation of F5, showing it is correct and terminating in its original form for homogeneous inputs.