9
votes
Accepted
For a specific unbounded Turing machine, is its Halting problem undecidable?
It depends in which sense you mean "undecidable".
If you evaluate $M$ on the empty input, and want only to find a yes/no answer, then the algorithmic problem is trivially decidable, as answered by ...
- 8,598
7
votes
Can we not output the Kolmogorov complexity?
The question can be rephrased as whether or not $\lim \inf_{\vert x \vert \rightarrow \infty}{\vert T(x) - K(x) \vert} = 0$, and as Denis points out in the comments this is false for some encodings. ...
- 548
6
votes
For a specific unbounded Turing machine, is its Halting problem undecidable?
For every concrete Turing machine $M$, the halting problem (Problem $P_M$ without input: "Does the Turing machine $M$ halt on the empty input $\varepsilon$?") is decidable.
The corresponding decision ...
- 5,772
5
votes
Accepted
Equivalence of a physical computer and Turing machine
You certainly are right that if someone says "a physical computer is just a Turing machine" then they are telling a bit of a stretcher. More accurate would be something like "anything ...
- 213
3
votes
Accepted
Multi Head universal Turing machine
If we have a fixed number of tapes then yes we can simulate them without the logarithmic overhead. E.g simulation of two-tape (and in general $k$-tape) TMs on a two-tape machine can be done without ...
- 21.5k
3
votes
Accepted
Fastest Turing Machine
Probably the best one can say at this level of generality is that $T_U(L,n)$ and $T_V(L,n)$ are computably related (if $U$ and $V$ are both universal), i.e. there are computable functions $f,g$ such ...
- 36.9k
3
votes
Fastest Turing Machine
The statement $T_U(L,n) \le c_{UV} \cdot T_V(L,n)$ is not true for all choices of $U$. It's easy to think of a Universal Turing Machine that is simply inefficient. For example choose $U$ as the ...
- 131
3
votes
Accepted
Is my language Turing-complete?
Here's a proof that the language is not Turing-complete: its Halting problem can be solved for all programs in your language. This language contains only single loops in sequence, and each loop ...
- 1,934
3
votes
Can we not output the Kolmogorov complexity?
I think the following works. I'll use $C(x)$ for the Kolmogorov complexity
Give $U$ a time bound $t$ (say, some exponential function of the length of the input program), and call the result $U^t$. ...
- 459
3
votes
Can Pattern Recognition algorithms be considered Oracle Machines (in the Turing sense)?
There is currently zero evidence contrary to the Church-Turing thesis -- namely that the Turing machine is the strongest physically realizable computational paradigm. In the case of AI systems, the ...
- 10.3k
3
votes
Accepted
How fast is an equivalent 2-tape TM compared to a $O(n^2)$ 1-tape TM?
Just an extended comment to underline how the question is (up to my knowledge) far from being solved (and easy).
First of all there are no "natural" quadratic lower bounds with respect to ...
- 22.6k
3
votes
Accepted
Can a normal form term be extensionally equivalent to a term with no WHNF?
S(C(KM)M)I ~ MM suffices
The reduction is as follows:
S(C(KM)M)Ix
C(KM)Mx(Ix)
C(KM)Mxx
KMxMx
MMx
- 916
2
votes
Accepted
What is the VC dimension of Turing machines with specified maximum size?
The exact VC bounds will depend on the alphabet size and the exact specification of the transition function (must it always move left or right, or can it stay put, etc). For fixed alphabet size, say 2,...
- 10.3k
2
votes
Formalization of simulation for Turing machines
First, let's clearly settle what it means for a Turing Machine to compute a function.
Any specification of a deterministic TM $M$ implicitly defines, for each word $w \in \Sigma^*$, a (possibly ...
1
vote
Cook's theorem and universal machine
what Papadimitriou and Yannakakis mean is something along the following lines. Consider the language L consisting of all triples <M,x,t> where M is a nondeterministic Turing machine, x is a ...
- 11
1
vote
Asymptotic time required to simulate a Turing machine M for k steps
This problem has given me a headache too so I'll post for future reference. The way I understand it the two wikipedia articles you're referencing aren't about the same problem.
Problem number one is : ...
- 149
1
vote
Accepted
About reduction relation between $HP$ and $\mathcal{E}\mbox{*}$
Note that any computable language $L$ can be turing-reduced to any other language $L'$ which contains at least one yes-instance and one no-instance (i.e. we have $L \leq L'$) by letting the reduction ...
- 257
1
vote
Accepted
Where can I find a Turing machine evaluating arithmetic expressions?
Probably the main reason why you don't find such a machine is its complexity - most textbooks try to not overwhelm the reader with details. We can give a sketch of how to construct it, however. Note ...
- 2,500
Only top scored, non community-wiki answers of a minimum length are eligible