Formalizing Monotone Algebras for Certification of Termination and Complexity Proofs

Sternagel, Christian; Thiemann, René

doi:10.1007/978-3-319-08918-8_30

Cited by 6 publications

(9 citation statements)

References 25 publications

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“…We state (at a high level) the key results on matrix/arctic interpretations that we use in our implementation. For more details we refer the reader to existing work [2,6,10,15,26].…”

Section: Interpretation Methodsmentioning

confidence: 99%

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An Automated Approach to the Collatz Conjecture

Yolcu

Aaronson

Heule

2021

Automated Deduction – CADE 28

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We explore the Collatz conjecture and its variants through the lens of termination of string rewriting. We construct a rewriting system that simulates the iterated application of the Collatz function on strings corresponding to mixed binary–ternary representations of positive integers. Termination of this rewriting system is equivalent to the Collatz conjecture. To show the feasibility of our approach in proving mathematically interesting statements, we implement a minimal termination prover that uses the automated method of matrix/arctic interpretations and we perform experiments where we obtain proofs of nontrivial weakenings of the Collatz conjecture. Finally, we adapt our rewriting system to show that other open problems in mathematics can also be approached as termination problems for relatively small rewriting systems. Although we do not succeed in proving the Collatz conjecture, we believe that the ideas here represent an interesting new approach.

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“…We state (at a high level) the key results on matrix/arctic interpretations that we use in our implementation. For more details we refer the reader to existing work [2,6,10,15,26].…”

Section: Interpretation Methodsmentioning

confidence: 99%

“…Additionally setting (M σ ) 1,1 = 1 satisfies the requirements for an extended monotone algebra. Matrix interpretations can also be adapted to the max-plus algebra of arctic numbers A := N∪{−∞} as coefficients with different arithmetic operations and order relations [15,26]. Example 1.…”

Section: Interpretation Methodsmentioning

confidence: 99%

An Automated Approach to the Collatz Conjecture

Yolcu

Aaronson

Heule

2021

Automated Deduction – CADE 28

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“…Although trying to prove the Collatz conjecture via automated deduction is clearly a moonshot goal, there are two recent technological advances that provide reasons for optimism that at least some interesting variants of the problem might be solvable. First, the invention of the method of matrix interpretations [9,10] and its variants such as arctic interpretations [11,12] turns the quest of finding a ranking function to witness termination into a problem that is suitable for systematic search. Second, the progress in satisfiability (SAT) solving makes it possible to solve many seemingly difficult combinatorial problems efficiently in practice.…”

Section: Introductionmentioning

confidence: 99%

An Automated Approach to the Collatz Conjecture

2023

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We explore the Collatz conjecture and its variants through the lens of termination of string rewriting. We construct a rewriting system that simulates the iterated application of the Collatz function on strings corresponding to mixed binary–ternary representations of positive integers. We prove that the termination of this rewriting system is equivalent to the Collatz conjecture. We also prove that a previously studied rewriting system that simulates the Collatz function using unary representations does not admit termination proofs via natural matrix interpretations, even when used in conjunction with dependency pairs. To show the feasibility of our approach in proving mathematically interesting statements, we implement a minimal termination prover that uses natural/arctic matrix interpretations and we find automated proofs of nontrivial weakenings of the Collatz conjecture. Although we do not succeed in proving the Collatz conjecture, we believe that the ideas here represent an interesting new approach.

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“…Also, to ultimately trust termination tools, one needs to formalize proof methods using proof assistants and obtain trusted certifier that validates outputs of termination tools, see, e.g., IsaFoR/CeTA [31] or CoLoR/Rainbow [4] frameworks. Although some interpretation methods have already been formalized [28,30], adding missing variants one by one would cost a significant effort.…”

Section: Introductionmentioning

confidence: 99%

Multi-Dimensional Interpretations for Termination of Term Rewriting

Yamada

2021

Automated Deduction – CADE 28

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Interpretation methods constitute a foundation of termination analysis for term rewriting. From time to time remarkable instances of interpretation methods appeared, such as polynomial interpretations, matrix interpretations, arctic interpretations, and their variants. In this paper we introduce a general framework, the multi-dimensional interpretation method, that subsumes these variants as well as many previously unknown interpretation methods as instances. Employing the notion of derivers, we prove the soundness of the proposed method in an elegant way. We implement the proposed method in the termination prover and verify its significance through experiments.

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Formalizing Monotone Algebras for Certification of Termination and Complexity Proofs

Cited by 6 publications

References 25 publications

An Automated Approach to the Collatz Conjecture

An Automated Approach to the Collatz Conjecture

An Automated Approach to the Collatz Conjecture

Multi-Dimensional Interpretations for Termination of Term Rewriting

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