String rewriting and homology of monoids

1997 ◽  
Vol 7 (3) ◽  
pp. 207-240 ◽  
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.

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

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.

COMPLETE REWRITING SYSTEMS FOR CODIFIED SUBMONOIDS

2005 ◽  
Vol 15 (02) ◽  
pp. 207-216
Author(s):  
ANTÓNIO MALHEIRO
Keyword(s):  
The Other ◽  
Maximal Subgroups ◽  
Rewriting Systems ◽  
Group Of Units ◽  
Rewriting System ◽  
Other Hand

Given a complete rewriting system R on X and a subset X0 of X+ satisfying certain conditions, we present a complete rewriting system for the submonoid of M(X;R) generated by X0. The obtained result will be applied to the group of units of a monoid satisfying H1 = D1. On the other hand we prove that all maximal subgroups of a monoid defined by a special rewriting system are isomorphic.

scholarly journals An Automated Approach to the Collatz Conjecture

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.

scholarly journals String Rewriting Systems

2007 ◽  
Vol 15 (3) ◽  
Author(s):  
Michał Trybulec
Keyword(s):  
Rewriting Systems ◽  
String Rewriting

String-Rewriting Systems

1993 ◽  
pp. 35-64 ◽  
Author(s):  
Ronald V. Book ◽  
Friedrich Otto
Keyword(s):  
Rewriting Systems ◽  
String Rewriting

REWRITING SYSTEMS IN ALTERNATING KNOT GROUPS

2006 ◽  
Vol 16 (04) ◽  
pp. 749-769 ◽  
Author(s):  
FABIENNE CHOURAQUI
Keyword(s):  
Word Problem ◽  
Rewriting Systems ◽  
Rewriting System ◽  
Regular Position

Every tame, prime and alternating knot is equivalent to a tame, prime and alternating knot in regular position, with a common projection. In this work, we show that the augmented Dehn presentation of the knot group of a tame, prime, alternating knot in regular position, with a common projection has a finite and complete rewriting system. This provides an algorithm for solving the word problem with this presentation and we find an algorithm for solving the word problem with the Dehn presentation also.

scholarly journals Deleting string rewriting systems preserve regularity

2004 ◽  
Vol 327 (3) ◽  
pp. 301-317 ◽  
Author(s):  
Dieter Hofbauer ◽  
Johannes Waldmann
Keyword(s):  
Rewriting Systems ◽  
String Rewriting

String-rewriting systems

1993 ◽  
Vol 31 (02) ◽  
pp. 31-0990-31-0990
Keyword(s):  
Rewriting Systems ◽  
String Rewriting
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