Diamond Subgraphs in the Reduction Graph of a One-Rule String Rewriting System

Arthur Adinayev; Itamar Stein

doi:10.3233/fi-2021-2002

Diamond Subgraphs in the Reduction Graph of a One-Rule String Rewriting System

Fundamenta Informaticae ◽

10.3233/fi-2021-2002 ◽

2021 ◽

Vol 178 (3) ◽

pp. 173-185

Author(s):

Arthur Adinayev ◽

Itamar Stein

Keyword(s):

Isomorphism Problem ◽

Complete Characterization ◽

Hasse Diagram ◽

Subgraph Isomorphism ◽

Rewriting Systems ◽

Rewriting System ◽

String Rewriting ◽

Reduction Graph

In this paper, we study a certain case of a subgraph isomorphism problem. We consider the Hasse diagram of the lattice Mk (the unique lattice with k + 2 elements and one anti-chain of length k) and find the maximal k for which it is isomorphic to a subgraph of the reduction graph of a given one-rule string rewriting system. We obtain a complete characterization for this problem and show that there is a dichotomy. There are one-rule string rewriting systems for which the maximal such k is 2 and there are cases where there is no maximum. No other intermediate option is possible.

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String rewriting and homology of monoids

Mathematical Structures in Computer Science ◽

10.1017/s0960129596002149 ◽

1997 ◽

Vol 7 (3) ◽

pp. 207-240 ◽

Cited By ~ 12

Author(s):

DANIEL E. COHEN

Keyword(s):

The Other ◽

Rewriting Systems ◽

Rewriting System ◽

String Rewriting

Results of Anick (1986), Squier (1987), Kobayashi (1990), Brown (1992b), and others, show that a monoid with a finite convergent rewriting system satisfies a homological condition known as FP∞.In this paper we give a simplified version of Brown's proof, which is conceptual, in contrast with the other proofs, which are computational.We also collect together a large number of results and examples of monoids and groups that satisfy FP∞ and others that do not. These may provide techniques for showing that various monoids do not have finite convergent rewriting systems, as well as explicit examples with which methods can be tested.

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An Automated Approach to the Collatz Conjecture

Automated Deduction – CADE 28 - Lecture Notes in Computer Science ◽

10.1007/978-3-030-79876-5_27 ◽

2021 ◽

pp. 468-484

Author(s):

Emre Yolcu ◽

Scott Aaronson ◽

Marijn J. H. Heule

Keyword(s):

Open Problems ◽

New Approach ◽

Collatz Conjecture ◽

Rewriting Systems ◽

Automated Method ◽

Rewriting System ◽

Positive Integers ◽

String Rewriting

AbstractWe 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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