21
votes
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
18
votes
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
18
votes
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
17
votes
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
17
votes
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
16
votes
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
16
votes
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
15
votes
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
14
votes
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
13
votes
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 ...
- 131
13
votes
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
13
votes
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
13
votes
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 ...
- 541
13
votes
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
12
votes
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
11
votes
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
10
votes
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
10
votes
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
10
votes
Does every computable function have infinitely many "non-padded" representations?
Given any computable function $f : \mathbb{N} \to \mathbb{N}$ the set of its indices $S_f = \{n \in \mathbb{N} \mid \phi_n = f\}$ is for all practical purposes "the set of all programs computing $...
- 29.4k
9
votes
In the context of regular languages, must the alphabet be finite?
It makes sense in some contexts in mathematics to consider strings or languages over infinite alphabets. For instance, this concept is used in the strong version of Higman's lemma. But a finite ...
- 51.1k
9
votes
Impossibility in Computability and Complexity: always ultimately due to diagonal arguments?
The standard comparison-model lower bound for sorting, and most cell probe model lower bounds for data structures, are unconditional (for computing within the model but you could say the same about ...
- 51.1k
9
votes
Accepted
Enumerating decidable languages
You can enumerate exactly the decidable languages. I've given this question as a homework problem so I'll just give a hint here: You can modify a TM $M$ to a machine $M'$ such that if $M$ is total (...
- 8,741
9
votes
Accepted
a polynomial representation of boolean functions
Well done on your independent discovery.
This is a Hadamard matrix of Sylvester type, written in a different order. There is a massive literature on this topic. It is used in coding theory, ...
- 2,070
9
votes
Accepted
Non-comparable natural numbers
When you say "undecidable" I assume you mean it is independent of a theory such as ZFC. There will be statements like
$$B(m)>n$$
(for natural numbers $m$, $n$) that are not decided by ZFC, assuming ...
- 4,485
9
votes
Why/when do we ever need transfinite loop variants?
I would like to add the following to Andrej's response (not enough rep for a comment).
Indeed, we cannot avoid ordinals but we may hide them. One approach is to use some modal logic that takes ...
- 326
9
votes
Is there a difference between incompleteness and unknowable?
Yes, there is a difference.
Incompleteness is a property of a formal system logic. Given a formal system $L$, we say that it is incomplete if there is a sentence $S$ such that $L$ does not prove $S$ ...
- 29.4k
9
votes
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
9
votes
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
9
votes
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
8
votes
Accepted
Can all mathematical operations be encoded with a Turing Complete language?
But I've come away looking for proof. How do we know all this covers all computable functions? Is there an obvious branch of Mathematics for which this is not covered? Is there a shortcoming in Lambda ...
- 32.8k
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