## what is the function of a turing machine

The main question asked me to build a certain turing machine such that given a word w over {0,1}* the turing machine accepts all such words and ends in accept state with the tape string = the word apended with the number of zeros in it. I built the state diagram of this turing machine..

But now there is a question next to it which say something like –

1. What is the function the machine you build computes ??

I dont understand what this means ?? What i can think of is that the function the question talks about is the transition function… and that just copies 0’s from w to the end of it… it doesnt computes as such anything..

Any lights on the above ??

Thanks

## Universal Turing Machine algorithm

First, I learned this based on these facts:

1. Turing machine (TM) will be define with 7-tuple Notation, M=<Q,G,b,S,d,q0,F>.
2. Any computation rules that can use to simulate any possible TM is called Turing-Complete.
3. Universal TM (UTM) is TM that is Turing-Complete.

Then, the question begins:

• If we have 7-tuple Notation of any U that is UTM, Is there an algorithm to find initial tape content P that U to simulate any TM T with any I input(s)? If it exists, Does it based on each U or pattern of U? If it does, give me some example(s)? If it does, explain the algorithm?
• Since all possible computations can be done with TM, Is there an algorithm to make TM simulate any algorithm P written in any language? If it exists, give me some example(s)?
• If both questions above exist the algorithm, Why don’t we just make a single UTM U and use it to program itself then do every possible computation?

## Can partial Turing completeness be quantified as a subset of Turing-computable functions?

Can partial Turing completeness be coherently defined this way:
An an abstract machine or programming language can be construed as Turing complete on its computable subset of Turing-computable functions.

In computability theory, several closely related terms are used to describe the computational power of a computational system (such as an abstract machine or programming language):

Turing completeness A computational system that can compute every Turing-computable function is called Turing-complete (or Turing-powerful). https://en.wikipedia.org/wiki/Turing_completeness

## Undecidability of the language of all Turing Machines with repeat strings as their language

Show that the language consisting of all Turing machines whose language consists of strings that can be broken up into two consecutive and equal strings is undecidable.

I would prefer if reduction was used and not Rice’s theorem.

## Transforming multi-tape Turing machines to equivalent single-tape Turing machines

We know that multi-tape Turing machines have the same computational power as single-tape ones. So every $$k$$-tape Turing machine has an equivalent single-tape Turing machine.

About the computability and complexity analysis of such a transformation:

Is there a computable function that receives as input an arbitrary multi-tape Turing machine and returns an equivalent single-tape Turing machine in polynomial time and polynomial space?

## Existence of a loop in a Turing machine?

Consider a Turing Machine which (1) reads all its input and (2) accepts inputs arbitrarily large. Can we conclude that there must be a loop in the finite-state control as its inputs get larger?

## Is there a query language variant of Turing Completeness?

By this I mean a theory where you can say Language X is Query Complete so that you know that language is able to do any sort of query? I’m guessing not because some queries would run into things that a language would have to be Turing complete to work?

Why do I wonder – well you might have a relational database and a graph database and someone might say anything that can be done in the relational database can be done in the graph database (albeit at different speeds) so I would like if there were some terms DB A and DB B are both Query Complete – or if not that if there was a way to categorize levels of "query completeness" (I’m just going to assume my ill defined concept is totally understandable to everyone) so one can say stuff like "DB A is query level 4 but DB B is at query level 3, but of course much faster because of those limitations."

I sure hope (so as to not feel like a bigger idiot than normal) that the answer to this question isn’t just a flat No.

## How to prove the language of all Turing Machines that accept an undecidable language is undecidable?

I want to prove that $$L=\{\langle M \rangle |L(M)\text{ is undecidable}\}$$ is undecidable

Suppose L is decidable. Let $$E$$ be the decider from $$L$$. Let $$A$$ be a TM which is recognizing $$A_{TM}$$. Let $$S$$ be a TM which works on input $$\langle M,w \rangle$$ in the following way:

1. Construct a TM $$N$$ which works on Input $$x$$ as follows: Run $$M$$ on $$w$$. If $$M$$ $$accepts$$ run $$A$$ on $$x$$ and accept $$x$$ if $$A$$ accepts.(In this case is $$L(N)=A_{TM}$$). If $$M$$ $$rejects$$ $$w$$, $$accept$$ $$x$$.(In this case is $$L(N)=\Sigma^*$$)
2. Run $$E$$ on $$N$$ and accept if N accepts. Otherwise reject

I am not sure if my reduction is the right way or not. Maybe someone can help to finish the reduction 🙂

## Rice’s Theorem for Turing machine with fixed output

So I was supposed to prove with the help of Rice’s Theorem whether the language: $$L_{5} = \{w \in \{0,1\}^{*}|\forall x \in \{0,1\}^{*}, M_{w}(w) =x\}$$ is decidable.

First of all: I don’t understand, why we can use Rice’s Theorem in the first place: If I would chose two Turingmachines $$M_{w}$$ and $$M_{w’}$$ with $$w \neq w’$$ then I would get $$M_{w}(w) = M_{w’}(w) = x$$. But this would lead to $$w’$$ not being in $$L_{5}$$ and $$w \in L_{5}$$. Or am I misunderstanding something?

Second: The solution says, that the Language $$L_{5}$$ is decidable as $$L_{5} = \emptyset$$ because the output is clearly determined with a fixed input. But why is that so? I thought that $$L_{5}$$ is not empty because there are TM which output x on their own input and there are some which do not.

## Limiting where a Turing Machine can write

Suppose I have a Turing Machine A, generated by somebody else, trying to break my system. To break my system they would have to write to the tape at some point X (this can be a range of the tape).

If I can force the machine A to be in some state q1 at some point, can I make a Turing Machine B such that for any Machine A it will never write in X?