# Tagged Questions

This tag is suited for questions involving Turing machines. Not to be confused with finite state machines and finite automata.

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### Are there any Turing-undecidable problems whose undecidability is independent of the Halting problem?

To be more specific, does there exist a decision problem $P$ such that given an oracle machine solving $P$, the Halting problem remains undecidable, and given an oracle machine solving the Halting ...
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### Why do we believe the Church-Turing Thesis?

The Church-Turing Thesis, which says that the Turing Machine model is at least as powerful as any computer that can be built in practice, seems to be pretty unquestioningly accepted in my exposure to ...
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### How can Busy beaver($10 \uparrow \uparrow 10$) have no provable upper bound?

This wikipedia article claims that the number of steps for a $10 \uparrow \uparrow 10$ state (halting) Turing Machine to halt has no provable upper bound: "... in the context of ordinary ...
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### How large is the set of all Turing machines?

How large is the set of all Turing machines? I am confident it is infinitely large, but what kind of infinitely large is its size?
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### What turmite runs the longest before becoming predictable?

When looking at 2D Turing machines, many of them eventually become predictable. For example, Langton's Ant, the champion 2-color 1-state turmite, develops a highway after 10,000 steps. Predictable ...
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### Show that the question “Is there life beyond earth?” is decidable

I was given a question to prove that there exists a turing machine that solves the question Is there life beyond earth? and is decidable. I actually don't understand how to show a turing ...
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### The mother of all undecidable problems

It is usual to show that a problem P is undecidable by showing that the halting problem reduces to P. Is it the case that the halting problem is the mother of all undecidable problems in the sense ...
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### Is the set of PA theorems the same as the set of solvable halting problems?

I am not sure if this is a trivial question. By Post's theorem we know that every PA (first order logic) theorem is equivalent to stating that a given input C in a given Turing machine halts or doesn'...
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### Are there known natural problems of intermediate degrees of unsolvability?

I know there exist intermediate degrees of unsolvability, i.e. there are undecidable problems which can be reduced to the Halting Problem, but not vice versa. Are there any "natural" problems known or ...
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### Why is propositional logic not Turing complete?

According to 1 (probably not the most relevant source), propositional logic is not Turing complete. Aren't all computations in computers performed using logic gates, which can be represented as ...
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### Lower bounds for bb(7) and bb(8) wanted

The busy beaver function $bb(n)$ is not known for $n \geq 5$. Does Anyone know suitable lower bounds for $bb(7)$ and $bb(8)$? Remark: $bb(6)$ as a trivial lower bound does not count as a suitable ...
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### An Undecidable but not Universal Turing Machine?

I have seen many examples of universal Turing machines, all of which are undecidable due to the undecidability of the halting problem. I have also seen proofs that certain really small Turing ...
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### Show that minimal CFG is undecidable (Sipser 5.36)

Question: Say that a CFG (context-free grammar) is minimal if none of its rules can be removed without changing the language generated. Let $MIN_{\text{CFG}}$ = $\{\, \langle G \rangle$ | $G$ is a ...
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### Milton Green's lower bounds of the busy beaver function

Wikipedia states that Milton Green demonstrated in 1964, that the busy beaver function $\Sigma(n)$ has the lower bound $$\Sigma(2k)>3\uparrow^{k-2}3$$ I read the talk about the busy beaver ...
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### Is there a turing machine for which halting is equivalent to the Axiom of Choice or its negation?

As seen in "A Turing machine for which halting is outside ZFC", Gödel's incompletness theorem can that there a turing machines for which halting can not be decided. My question is, is there a turing ...
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### Doubt about neural networks that allegedly compute beyond the Turing limit

The question is about a new result by Cabessa & Siegelmann. http://binds.cs.umass.edu/papers/2014_cabessa.pdf : "In this context, we show that the so-called plastic recurrent neural networks (RNNs)...
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### Designing a turing machine for primality check.

I am designing some turing machines, so far I have made Binary Addition and subtraction. Now I've been thinking that what if turing machine can check if the number is prime or not. Lets suppose we ...
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### What Constitutes a Pattern

Mathematics is often referred to as the "study of patterns." What I'm wondering is whether there is somehow a technical way to describe a pattern. For the length of this question let's assume that we'...
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### Where does this argument showing there are uncountably many TMs fail?

This argument comes up once every while on Lambda the Ultimate. I want to know where the flaw is. Take a countable number of TMs all generating different bitstreams. Construct a Cantor TM which runs ...
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### Why is it so hard to translate some proves into machine-readable form?

I have just read a topic on mathoverflow about man vs. machine in mathematics. The topic was inspired by the recent victory of Alpha Go over the World Go Champion, Lee Sedol. It reminded me of an ...
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### What breaks the Turing Completeness of simply typed lambda calculus?

On the Wikipedia page about Turing Completeness, we can read that: Although (untyped) lambda calculus is Turing-complete, simply typed lambda calculus is not. I am curious as to what exactly ...
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### How many digits of Chaitin's $\Omega$ constant would we know if we had a $\Sigma_1$-Oracle?

According to Wikipedia (and it seems intuitive from the definition itself), $\Omega$ is Turing equivalent to the halting problem and thus at level $\Delta_2^0$ of the arithmetical hierarchy. Do this ...
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### Build a deterministic turing machine to decide L = { ww }

As the title says. w is in {a, b}^*.Note that I am not looking for the non-deterministic one. Use a Turing machine of one tape and "pointer". An idea: I thought that I would do something like ...
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### Turing Machine Decidability

I have been working on this problem for few hours, but haven't been able to come up with a solution : Is the following problem decidable? Given a TM M, whether there is a w such that M enters each of ...
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### Baker-Gill-Solovay theorem

I have been trying to understand the proof of Baker-Gill-Solovay theorem as described in Complexity Theory: Modern Approach. I think I do understand most of it, but what troubles me is that let's say ...
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### Designing a Turing machine for Binary Multiplication

I need help designing a turing machine that will compute the following $$f(x,y) = x\times y$$ How to approach this problem in binary base? This is a assignment so I don't want anyone to solve it ...