Ackermann function
E208846
classical example in theoretical computer science
computable function
fast-growing function
recursive function
total function
The Ackermann function is a classic example of a computable function that grows faster than any primitive recursive function, often used in theoretical computer science to illustrate extreme computational complexity.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Ackermann function canonical | 4 |
| Ackermann–Péter function | 1 |
| three-argument Ackermann–Péter function | 1 |
| two-argument Ackermann–Péter function | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1859120 — 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: Ackermann function Context triple: [Wilhelm Ackermann, knownFor, Ackermann function]
-
A.
Knuth’s up-arrow notation
Knuth’s up-arrow notation is a mathematical notation introduced by Donald Knuth to concisely represent very large integers using iterated exponentiation and its higher-order generalizations.
-
B.
Conway chained arrow notation
Conway chained arrow notation is a mathematical system of hyper-operator-style notation introduced by John Horton Conway to concisely represent extremely large numbers.
-
C.
Gödel numbering
Gödel numbering is a method in mathematical logic that encodes symbols, formulas, and proofs as unique natural numbers, enabling arithmetic to represent and reason about syntactic statements.
-
D.
Look-and-say sequence
The look-and-say sequence is a famous integer sequence where each term is generated by verbally describing the digits of the previous term, studied for its surprising combinatorial and growth properties.
-
E.
Aitken
Aitken is a Scottish-origin surname notably borne by Max Aitken, 1st Baron Beaverbrook, a prominent Canadian-British newspaper magnate and politician.
- 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: Ackermann function Target entity description: The Ackermann function is a classic example of a computable function that grows faster than any primitive recursive function, often used in theoretical computer science to illustrate extreme computational complexity.
-
A.
Knuth’s up-arrow notation
Knuth’s up-arrow notation is a mathematical notation introduced by Donald Knuth to concisely represent very large integers using iterated exponentiation and its higher-order generalizations.
-
B.
Conway chained arrow notation
Conway chained arrow notation is a mathematical system of hyper-operator-style notation introduced by John Horton Conway to concisely represent extremely large numbers.
-
C.
Gödel numbering
Gödel numbering is a method in mathematical logic that encodes symbols, formulas, and proofs as unique natural numbers, enabling arithmetic to represent and reason about syntactic statements.
-
D.
Look-and-say sequence
The look-and-say sequence is a famous integer sequence where each term is generated by verbally describing the digits of the previous term, studied for its surprising combinatorial and growth properties.
-
E.
Aitken
Aitken is a Scottish-origin surname notably borne by Max Aitken, 1st Baron Beaverbrook, a prominent Canadian-British newspaper magnate and politician.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
classical example in theoretical computer science
ⓘ
computable function ⓘ fast-growing function ⓘ recursive function ⓘ total function ⓘ |
| alternativeName |
Ackermann–Péter function
ⓘ
linked to:
Ackermann function
|
| arity | two-argument function ⓘ |
| codomain | non-negative integers ⓘ |
| complexityRole |
basis for defining extremely slow-growing inverse functions used in amortized analysis
ⓘ
used to construct worst-case inputs for certain data structure algorithms ⓘ |
| definedBy |
multiple recursive clauses
ⓘ
nested recursion ⓘ |
| definedOn | non-negative integers ⓘ |
| domain | pairs of non-negative integers ⓘ |
| exampleValue |
A(0,n) = n + 1
ⓘ
A(1,n) = n + 2 ⓘ A(2,n) = 2n + 3 ⓘ A(3,n) = 2^(n+3) - 3 ⓘ A(4,2) is an enormous integer far beyond practical computation ⓘ |
| field |
complexity theory
ⓘ
computability theory ⓘ mathematical logic ⓘ theoretical computer science ⓘ |
| growthRate |
grows faster than any multiply-recursive function of fixed order
ⓘ
grows faster than any primitive recursive function ⓘ |
| hasVariant |
three-argument Ackermann–Péter function
ⓘ
linked to:
Ackermann function
two-argument Ackermann–Péter function ⓘ
linked to:
Ackermann function
|
| historicalNote | one of the earliest-discovered examples of a total computable non–primitive recursive function ⓘ |
| namedAfter | Wilhelm Ackermann ⓘ |
| property |
computable by a Turing machine
ⓘ
extremely rapidly growing ⓘ not primitive recursive ⓘ total but not primitive recursive ⓘ well-defined for all non-negative integer arguments ⓘ |
| relatedConcept |
Turing computability
ⓘ
linked to:
Computability Theory
fast-growing hierarchy ⓘ inverse Ackermann function ⓘ primitive recursive functions ⓘ μ-recursive functions ⓘ |
| teachingRole |
illustrates difference between primitive recursive and general recursive functions
ⓘ
illustrates extreme computational complexity growth ⓘ |
| usedIn |
complexity analysis of algorithms
ⓘ
defining the inverse Ackermann function ⓘ demonstrating limits of primitive recursion ⓘ examples of computable but not primitive recursive functions ⓘ hierarchies of fast-growing functions ⓘ proofs in recursion theory ⓘ |
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: Ackermann function Description of subject: The Ackermann function is a classic example of a computable function that grows faster than any primitive recursive function, often used in theoretical computer science to illustrate extreme computational complexity.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Ackermann function
linked to: Ackermann function
linked to: Ackermann function