# Tag Info

## Hot answers tagged computability

### Importance of irrational numbers in computer science

The question displays a fundamental misunderstanding about the nature of what can be done with computers. To correct this misunderstanding, I suggest getting a copy of the computer algebra software ...
• 6,985

### Importance of irrational numbers in computer science

You're assuming that numbers can only be represented as fractions (either literally, by having a datatype storing integer numerator and denominator, or implicitly by using some kind of floating point ...
• 2,805
Accepted

### Is Hartmanis-Stearns conjecture settled by this article?

First, the name of the conjecture is "Hartmanis-Stearns", not "Hartmanis-Stearn". Second, the Hartmanis-Stearns conjecture concerns those real numbers computable by a multi-tape Turing machine in ...
• 6,985
Accepted

### Is the Kolmogorov complexity of the truth tables of the halting problem known asymptotically?

Hmm, turns out there's actually an matching upper bound that isn't too hard: To produce the truth table $HALT_n$ in a finite amount of time, the only information that is needed is the number of ...
• 371
Accepted

### Entscheidungsproblem vs. Unvollständigkeitssatz (soft question)

The two words do not refer to the same thing. Hilbert's Entscheidungsproblem was the question whether there is an algorithm that decides the universal truth of first-order logical sentences, which was ...
• 4,640

### Importance of irrational numbers in computer science

I can see why the question is being downvoted, but this is too good not to post. Suppose you have a coin with bias $p\in[0,1]$, which might be rational or irrational. Question: using only finite ...
• 10.6k
Accepted

### A definition of computable numbers that requires to "wait an infinite amount of time" to get the correct result; how to make this precise

This is not a research-level question, but since the general level of interest seems high, here is an answer. I cannot guess from your question whether you're shooting for something that will result ...
• 29.4k

### Is there a notion of computability on sets other than the natural numbers?

This question is not research-level, but since it is receiving answers, I would like to offer an answer that may actually clear things up a bit, and provide references. There is an entire area of ...
• 29.4k

### Does the uncomputability of Kolmogorov complexity follow from Lawvere's Fixed Point Theorem?

EDIT: Adding the caveat that Roger's fixed-point theorem may not be a special case of Lawvere's. Here is a proof that may be "close"... It uses Roger's fixed-point theorem instead of Lawvere's ...
• 10.9k

### How is proving a context free language to be ambiguous undecidable?

The answer by apolge presents the proof that it is undecidable whether an arbitrary context free grammar is ambiguous. The question of whether a context free grammar defines an inherently ambiguous ...

### Is there a result in computability theory that does not relativize?

Higman's Embedding Theorem: The finitely generated computably presented groups are precisely the finitely generated subgroups of finitely presented groups. Furthermore, every computably presented ...
• 37.8k

### Why/when do we ever need transfinite loop variants?

You are correct when you observe that for any particular terminating loop $L$ we may simply define the invariant "we're getting one step closer to termination". But proving that this is indeed a valid ...
• 29.4k
Accepted

### Is there a language of first-order logic such that every r.e. set is Turing-equivalent to some finitely axiomatizable theory in that language?

The answer is yes. This was proved by Hanf (Model-theoretic methods in the study of elementary logic, in the Theory of models volume). A "uniform" version of this result was conjectured by ...
Accepted

### Computability and continuity

Let $L_1 = L_2 = \mathbb{N}$ and let $M \subseteq \mathbb{N}$ be a maximal set and let $L = \mathbb{N} \setminus M$ be its complement. Recall that $L$ is infinite, and that every computably enumerable ...
• 29.4k
Accepted

### Is there a useful notion of being “approximately computable”

If the family function $f(x,n)=f_n(x)$ is computable then these are exactly the $\Delta^0_2$ functions, or equivalently, the functions that are Turing reducible to the halting set $0'$, which are very ...
• 4,485
Accepted

### Equilibrium in a Halting Game

Even if you have a one-player game there is no computable equilibrium. Consider nature putting probability $1/2^i$ on program $i$. Any computable strategy will achieve some value strictly less than ...
• 8,741

### Importance of irrational numbers in computer science

Even if you're not trying to compute them, irrational numbers arise throughout computer science, particularly in the analysis of the complexity of algorithms. For example, if you want to know how ...
• 239

### Does the physical Church-Turing thesis imply that all physical constants are computable?

You appear to be positing a universe where (a) the fine-structure constant has an exact value and (b) we can measure as many digits of it as we want. Thus, if a Turing machine cannot compute the exact ...
• 24.9k

• 29.4k
Accepted

### Proof and computational complexity

Maybe the keyword you are looking for is "Implicit Complexity". It is more general than Curry-Howard correspondence, but several lines of research investigate along the axis you are ...
• 8,923
Accepted

### What is the complexity class of higher-order primitive recursion?

If I understand correctly, the primitive recursive functionals defined in the Wikipedia page linked in the question coincide with Gödel's system T, which is well-known to correspond to the class of ...
• 5,598

### What are the computations model with a constant slowdown ? (and why do we care about Turing machines)

For any fixed k > 1, the k-tape Turing machine model has a universal Turing machine with only linear slow down. This is a result due to Martin Fürer: The Tight Deterministic Time Hierarchy. STOC ...
• 2,271