Dehn algorithm
E265416
The Dehn algorithm is a decision procedure in combinatorial group theory that solves the word problem for certain groups by systematically reducing words using defining relations.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Dehn algorithm canonical | 2 |
| Dehn presentations | 1 |
| Dehn’s algorithm | 1 |
| Dehn’s word problem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2416884 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Dehn algorithm Context triple: [Max Dehn, notableConcept, Dehn algorithm]
-
A.
Knuth–Bendix completion algorithm
The Knuth–Bendix completion algorithm is a procedure in term rewriting and automated theorem proving that transforms a set of equations into a confluent rewriting system, enabling decision of word problems in algebraic structures.
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B.
Dowker–Thistlethwaite notation
Dowker–Thistlethwaite notation is a numerical encoding system used in knot theory to uniquely represent knot diagrams and facilitate their classification and study.
-
C.
Euler’s polyhedron formula
Euler’s polyhedron formula is a fundamental result in topology and geometry that relates the numbers of vertices, edges, and faces of a convex polyhedron through the equation V − E + F = 2.
-
D.
geometrization conjecture
The geometrization conjecture is a fundamental statement in 3-dimensional topology that classifies all closed 3-manifolds into pieces each admitting one of eight canonical geometric structures, a result proven by Grigori Perelman.
-
E.
Alexander–Briggs notation
Alexander–Briggs notation is a classical system for naming and classifying knots in knot theory, assigning each distinct knot a unique label based on its crossing number and order in knot tables.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Dehn algorithm Target entity description: The Dehn algorithm is a decision procedure in combinatorial group theory that solves the word problem for certain groups by systematically reducing words using defining relations.
-
A.
Knuth–Bendix completion algorithm
The Knuth–Bendix completion algorithm is a procedure in term rewriting and automated theorem proving that transforms a set of equations into a confluent rewriting system, enabling decision of word problems in algebraic structures.
-
B.
Dowker–Thistlethwaite notation
Dowker–Thistlethwaite notation is a numerical encoding system used in knot theory to uniquely represent knot diagrams and facilitate their classification and study.
-
C.
Euler’s polyhedron formula
Euler’s polyhedron formula is a fundamental result in topology and geometry that relates the numbers of vertices, edges, and faces of a convex polyhedron through the equation V − E + F = 2.
-
D.
geometrization conjecture
The geometrization conjecture is a fundamental statement in 3-dimensional topology that classifies all closed 3-manifolds into pieces each admitting one of eight canonical geometric structures, a result proven by Grigori Perelman.
-
E.
Alexander–Briggs notation
Alexander–Briggs notation is a classical system for naming and classifying knots in knot theory, assigning each distinct knot a unique label based on its crossing number and order in knot tables.
- F. None of above. chosen
Statements (34)
| Predicate | Object |
|---|---|
| instanceOf |
algorithm
ⓘ
decision procedure ⓘ word problem algorithm ⓘ |
| appliesTo |
Dehn presentations
ⓘ
linked to:
Dehn algorithm
word-hyperbolic groups with suitable presentations ⓘ |
| assumes |
finite generating set
ⓘ
finite set of defining relations ⓘ |
| basedOn | systematic reduction of words using defining relations ⓘ |
| category |
decision problems in algebra
ⓘ
group theory algorithms ⓘ |
| characteristicProperty |
reduces word length at each step when applicable
ⓘ
terminates in finitely many steps for groups admitting a Dehn presentation ⓘ |
| complexity | linear time for groups with a Dehn presentation ⓘ |
| concludesIdentityIf | reduced word is empty ⓘ |
| concludesNonIdentityIf | reduced word is nonempty ⓘ |
| contrastsWith | general undecidability of the word problem for finitely presented groups ⓘ |
| field |
combinatorial group theory
ⓘ
geometric group theory ⓘ |
| formalizedAs | rewriting system on words in group generators ⓘ |
| guarantees | decidability of the word problem for groups with a Dehn presentation ⓘ |
| historicalContext | early work on decision problems in group theory ⓘ |
| input | word in the generators of a group ⓘ |
| inspired | later linear-time algorithms for the word problem in hyperbolic groups ⓘ |
| method | searches for relator subwords whose replacement strictly shortens the word ⓘ |
| namedAfter | Max Dehn ⓘ |
| output | decision whether the word represents the identity element ⓘ |
| relatedTo |
automatic groups
ⓘ
hyperbolic groups ⓘ isoperimetric inequality in groups ⓘ small cancellation theory ⓘ |
| solves | word problem for certain groups ⓘ |
| stopsWhen | no further length-reducing relator replacements are possible ⓘ |
| uses | finite presentation of a group ⓘ |
| yearProposed | 1911 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Dehn algorithm Description of subject: The Dehn algorithm is a decision procedure in combinatorial group theory that solves the word problem for certain groups by systematically reducing words using defining relations.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Dehn algorithm
linked to: Dehn algorithm
linked to: Dehn algorithm