<p>No, it is not, although I have admittedly never heard of either theorem before so I'm just going off the definitions you posted, of which the one for Gödel's theorem is rather lackluster.</p>
<p>Gödel's theorem, as stated on the linked Wiki page, seems to talk about specific theorems having shorter proofs. There are actual examples stated on the page.</p>
<p>On the other hand, Blum's theorem just states that there exists some function which can be sped up as much as you want. In no way is it guaranteed that that function relates to your statement or proof.</p>

CC BY-SA 4.0
2023-10-02T06:24:04.570Z
2023-10-02T06:24:04.570Z
Andreas T
<p><a href="https://en.wikipedia.org/wiki/Blum%27s_speedup_theorem" rel="nofollow noreferrer">Blum's speedup theorem</a> is a statement about a certain class of computable functions for which it is always possible to find a program of lower complexity.
<a href="https://en.wikipedia.org/wiki/G%C3%B6del%27s_speed-up_theorem" rel="nofollow noreferrer">Gödel's speed-up theorem</a> is a statement about the possibility of drastically reducing the size of a theorem proof, by going to a more powerful axiomatic system.</p>
<p>Is Gödel's speed-up theorem an instance of Blum's speedup theorem, in which the computable function takes as input the axioms and outputs the theorem?
Such a function could concretely be built in theorem proving programming languages, say for instance with <a href="https://en.wikipedia.org/wiki/Coq" rel="nofollow noreferrer">Coq</a>.</p>
<p>Additional info:
I found <a href="https://onlinelibrary.wiley.com/doi/abs/10.1002/malq.19870330505" rel="nofollow noreferrer">this</a> that looks related, but it doesn't clearly answer my question.</p>

CC BY-SA 4.0
2023-10-03T21:25:32.650Z
There is a difference between static complexity measures (like program size or proof length) and dynamic complexity measures (like running time); the latter depend on the input as well. Blum axiomatized both; the speedup theorem you cite refers to dynamic measures. There is a theorem to the effect of: if a programming language L can diagonalize over K, then L has shorter programs than K even for K-computable functions. This resembles Gödel's speedup theorem and I wouldn't be surprised if there was a formal connection. Odifreddi's Classical Recursion Theory vol. 2 is an excellent reference.
Siddharth
CC BY-SA 4.0
2023-10-04T09:18:34.290Z
On History of Science and Mathematics, there is a related question whose answer elaborates a bit on the resemblance mentioned by @Siddharth: https://hsm.stackexchange.com/questions/12317/does-blums-speedup-theorem-have-any-conceptual-predecessors
Hermann Gruber
<ds.algorithms><speed-up>
Is Gödel's speed-up theorem an instance of Blum's speedup theorem?
Rexcirus
# Is Gödel's speed-up theorem an instance of Blum's speedup theorem?

(Asked by: username_0 on 2020-02-10)

[Blum's speedup theorem](https://en.wikipedia.org/wiki/Blum%27s_speedup_theorem) is a statement about a certain class of computable functions for which it is always possible to find a program of lower complexity.
[Gödel's speed-up theorem](https://en.wikipedia.org/wiki/G%C3%B6del%27s_speed-up_theorem) is a statement about the possibility of drastically reducing the size of a theorem proof, by going to a more powerful axiomatic system.

Is Gödel's speed-up theorem an instance of Blum's speedup theorem, in which the computable function takes as input the axioms and outputs the theorem?
Such a function could concretely be built in theorem proving programming languages, say for instance with [Coq](https://en.wikipedia.org/wiki/Coq).

Additional info:
I found [this](https://onlinelibrary.wiley.com/doi/abs/10.1002/malq.19870330505) that looks related, but it doesn't clearly answer my question.

**Tags:** ds.algorithms, speed-up

> **Comments:**
> **username_1**: There is a difference between static complexity measures (like program size or proof length) and dynamic complexity measures (like running time); the latter depend on the input as well. Blum axiomatized both; the speedup theorem you cite refers to dynamic measures. There is a theorem to the effect of: if a programming language L can diagonalize over K, then L has shorter programs than K even for K-computable functions. This resembles Gödel's speedup theorem and I wouldn't be surprised if there was a formal connection. Odifreddi's Classical Recursion Theory vol. 2 is an excellent reference.  
> **username_2**: On History of Science and Mathematics, there is a related question whose answer elaborates a bit on the resemblance mentioned by @username_1: https://hsm.stackexchange.com/questions/12317/does-blums-speedup-theorem-have-any-conceptual-predecessors  

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## 1 answers

### Answer by username_3 on 2023-10-02. Score: -1

No, it is not, although I have admittedly never heard of either theorem before so I'm just going off the definitions you posted, of which the one for Gödel's theorem is rather lackluster.

Gödel's theorem, as stated on the linked Wiki page, seems to talk about specific theorems having shorter proofs. There are actual examples stated on the page.

On the other hand, Blum's theorem just states that there exists some function which can be sped up as much as you want. In no way is it guaranteed that that function relates to your statement or proof.


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