Decidability of Krohn-Rhodes complexity for all finite semigroups and automata

Anne Schilling; John Rhodes; Stuart Margolis

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Decidability of Krohn-Rhodes complexity for all finite semigroups and automata

Stuart Margolis, John Rhodes, Anne Schilling

Group Theory · Submitted 26 Jun 2024

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The main result of this paper proves that it is decidable whether a semigroup has complexity k for any k greater than or equal to 0.

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arXiv ID2406.18477 (version 1)

Pith ID5SDFAFER, Pith's permanent record of this paper and its review

CitationAnne Schilling, John Rhodes, Stuart Margolis. "Decidability of Krohn-Rhodes complexity for all finite semigroups and automata." arXiv:2406.18477 (2024). https://pith.science/paper/5SDFAFER

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What the abstract itself states, as probabilities: claims first 0.15 · states limitation 0.13 · human participants 0.02 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.66 · compares with prior results 0.94 · mentions code or data availability 0.02.

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The Krohn-Rhodes Theorem proves that a finite semigroup divides a wreath product of groups and aperiodic semigroups. Krohn-Rhodes complexity equals the minimal number of groups that are needed. Determining an algorithm to compute complexity has been an open problem for more than 50 years. The main result of this paper proves that it is decidable whether a semigroup has complexity k for any k greater than or equal to 0. This builds on our previous work for complexity 1. In that paper we proved using profinite methods and results on free Burnside semigroups by McCammond and others that the lower bound from a 2012 paper by Henckell, Rhodes and Steinberg is precise for complexity 1. In this paper we define an improved version of the lower bound from the 2012 paper and prove that it is exact for arbitrary complexity.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The Krohn-Rhodes Theorem proves that a finite semigroup divides a wreath product of groups and aperiodic semigroups.	background 1.00	0.04	0.02	asserted 1.00
s1	Krohn-Rhodes complexity equals the minimal number of groups that are needed.	background 1.00	0.07	0.02	asserted 1.00
s2	Determining an algorithm to compute complexity has been an open problem for more than 50 years.	gap 0.95	0.04	0.02	asserted 1.00
s3	The main result of this paper proves that it is decidable whether a semigroup has complexity k for any k greater than or equal to 0.	result 0.90	0.96	0.98	asserted 1.00
s4	This builds on our previous work for complexity 1.	background 0.93	0.51	0.02	asserted 0.97
s5	In that paper we proved using profinite methods and results on free Burnside semigroups by McCammond and others that the lower bound from a 2012 paper by Henckell, Rhodes and Steinberg is precise for complexity 1.	background 0.84	0.82	0.02	asserted 1.00
s6	In this paper we define an improved version of the lower bound from the 2012 paper and prove that it is exact for arbitrary complexity.	contribution 0.65	0.96	0.23	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v1

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full text
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P abstract

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S sections
T body sentences
C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

The Krohn-Rhodes Theorem proves that a finite semigroup divides a wreath product of groups and aperiodic semigroups. Krohn-Rhodes complexity equals the minimal number of groups that are needed. Determining an algorithm to compute complexity has been an open problem for more than 50 years. The main result of this paper proves that it is decidable whether a semigroup has complexity k for any k greater than or equal to 0. This builds on our previous work for complexity 1. In that paper we proved using profinite methods and results on free Burnside semigroups by McCammond and others that the lower bound from a 2012 paper by Henckell, Rhodes and Steinberg is precise for complexity 1. In this paper we define an improved version of the lower bound from the 2012 paper and prove that it is exact for arbitrary complexity.

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References

Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The Margolis-Rhodes Monoid of a Graph
math.GR 2026-09 conditional novelty 6.0 of 10

The monoid of continuous partial maps on a graph has regular J-classes matching induced subgraphs, and its Krohn-Rhodes complexity is exactly 2 for cycles of length at least 4 and for paths of length at least 14.
Open multiplication in relatively free profinite semigroupoids
math.GR 2025-07 accept novelty 6.0 of 10

Open multiplication in relatively free profinite semigroupoids over finite-vertex graphs is characterized by concatenation-closed pseudovarieties, yielding a profinite criterion for recurrent paths and new stabilizer results.
Master List of Examples in Complexity Theory of Finite Semigroup Theory
math.GR 2025-01 unverdicted novelty 4.0 of 10

A maintained list of finite semigroup examples with flow-based computations of their Krohn-Rhodes complexity.
Complexity of Finite Semigroups: History and Decidability
math.GR 2025-01 conditional novelty 2.0 of 10

A survey of Krohn-Rhodes complexity theory and an outline of the proof that the complexity of every finite semigroup is algorithmically computable.

How the literature uses it

5 arXiv papers cite this in 20 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

In recent papers, Margolis, Rhodes and Schilling proved that the complexity of a finite semigroup is computable.

Complexity of Finite Semigroups: History and Decidability · 0.89

Papers citing it · 15 sentences without a confident label

Uses its method or tool 1 paper

That is, [equation] The $GST$ expansion is used extensively in both [ref] and [ref] .

Complexity of Finite Semigroups: History and Decidability · More tools used in the proof of computability of complexity 1 · 0.92

Cites it as support 1 paper

The main theorems of [ref] and [ref] prove that the computability question has a positive solution.

Complexity of Finite Semigroups: History and Decidability · Introduction · 0.97

Cites it as background 3 papers

The purpose of this paper is to give the necessary background and an outline to read the papers [ref] .

Complexity of Finite Semigroups: History and Decidability · Introduction · 0.92
In the papers [ref] and [ref] Margolis, Rhodes and Schilling proved that it is decidable to compute the complexity of a finite semigroup.

Aperiodic Flows on Finite Semigroups: Foundations and First Examples · Introduction · 0.93
Important articles that give further background are [ref] .

Master List of Examples in Complexity Theory of Finite Semigroup Theory · Some background · 1.00

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