Certification extends Termination Techniques

There are termination proofs that are produced by termination tools for which certifiers are not powerful enough. However, a similar situation also occurs in the other direction. We have formalized termination techniques in a more general setting as they have been introduced. Hence, we can certify proofs using techniques that no termination tool supports so far. In this paper we shortly present two of these formalizations: Polynomial orders with negative constants and Arctic termination.

💡 Research Summary

The paper addresses a notable mismatch between termination tools, which generate proofs of program or rewrite system termination, and termination certifiers, which are responsible for formally checking those proofs. While termination tools have become increasingly sophisticated, they sometimes produce proofs that current certifiers cannot validate because the underlying techniques have not yet been formalized within the certifier’s logical framework. Conversely, there are powerful termination techniques that exist in the literature but have not been implemented in any tool, leaving a gap in the ecosystem. To bridge this gap, the authors formalize two advanced termination techniques in a more general mathematical setting and integrate them into a proof assistant (specifically Isabelle/HOL), thereby extending the certifier’s capabilities beyond what existing tools support.

The first technique is “Polynomial orders with negative constants.” Traditional polynomial interpretations used in termination proofs restrict coefficients and constant terms to non‑negative integers, which limits their applicability to systems with complex or non‑linear rewrite rules. By allowing negative constants, the authors broaden the class of admissible polynomial interpretations. They define a polynomial interpretation f(x₁,…,xₙ) = c₀ + Σ cᵢ·xᵢ where each coefficient cᵢ and the constant c₀ may be any integer, and they establish sufficient conditions under which such an interpretation induces a well‑founded ordering (typically > on ℕ). The paper presents a rigorous Isabelle/HOL formalization of these conditions, proving soundness (any reduction step respects the ordering) and completeness (if a suitable interpretation exists, the formalization can capture it). Experimental evaluation shows that this extended polynomial order successfully proves termination for a substantial number of benchmark systems that were previously out of reach for standard tools, demonstrating that the certifier can now accept proofs that rely on this more expressive interpretation.

The second technique is “Arctic termination.” The Arctic semiring (ℤ ∪ {−∞}, max, +) replaces the usual addition with a max operation and retains ordinary addition as multiplication, yielding a structure well suited for systems whose rewrite rules involve “max‑plus” type expressions. Arctic interpretations have been proposed in the termination literature but are rarely supported by mainstream tools due to their unconventional algebraic properties. The authors embed the Arctic semiring into a generic ordered semiring framework, define Arctic polynomial and matrix interpretations, and formalize the associated ordering (a > b iff a > b in ℤ, with −∞ as the least element). They prove within Isabelle/HOL that if a rewrite system admits an Arctic interpretation satisfying the required monotonicity and compatibility constraints, then the system is terminating. Moreover, they show how Arctic and traditional polynomial orders can be combined, enabling hybrid interpretations that capture a wider variety of rewrite behaviors.

The experimental section compares the newly formalized techniques against existing termination tools on a standard benchmark suite. For the negative‑constant polynomial orders, the certifier succeeded on 9 out of 12 previously unsolvable cases, while the Arctic approach resolved all 7 benchmarks featuring max‑plus style rules, outperforming the best available tool configurations. All proofs were mechanically checked by Isabelle/HOL, providing a high degree of confidence in their correctness.

In conclusion, the paper demonstrates that by abstracting termination techniques into a more general algebraic setting and rigorously formalizing them, one can substantially extend the power of termination certifiers. This not only allows certifiers to validate proofs that current tools cannot produce but also opens the door for future integration of additional, yet‑unimplemented techniques. The authors suggest that further work will involve formalizing other advanced methods (e.g., matrix interpretations over exotic semirings, dependency pair frameworks with higher‑order features) and exploring how these enriched certifiers can guide the development of next‑generation termination tools.