# Tagged Questions

Questions about Turing computability and recursion theory, including the halting problem and other unsolvable problems. Questions about the resources required to solving particular problems should be tagged (computational-complexity).

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### On the Decision Problem for Two-variable First-Order Logic

I have a question concerning the model construction of the $\forall \forall \land \forall \exists$ - Scott sentence on page 6 in this paper: www.cs.rice.edu/~vardi/papers/basl96.ps.gz Why do we ...
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### Show $f$ is primitive recursive, where $f(n) = 1$ if the decimal expansion of $\pi$ contains $n$ consecutive $5$'s

Let $f:\mathbb{N}\to\mathbb{N}$ be given by $f(n)=1$ if the decimal expansion of $\pi$ contains $n$ consecutive $5$'s, and $f(n)=0$ otherwise. How would you go about showing such a function is ...
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### A qualitative, yet precise statement of Godel's incompleteness theorem?

I read online a statement to the effect that (I'm paraphrasing): Goedel's incompleteness theorem shows that we cannot even have a complete and consistent theory for the natural numbers. I am ...
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### The emptiness problem for “lunatic” and “crazy” Turing machines

Crazy Turing Machine is the same as Turing machine with one stripe , except of the fact that after each ten steps the head jumps back to the beginning of the stripe. Lunatic Turing Machine is the ...
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### For a Turing machine and input $w$- Is “Does M stop or visit the same configuration twice” a decidable question?

I have the following question out of an old exam that I'm solving: ...
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### Halting problem on finite set of programs

As I understand the halting problem, it imply the fact that there doesn't exist one program which can answer the halting problem for every computable program and it rely on Cantor diagonalization to ...
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### Solovay Randomness

Say that an $x\in 2^{\omega}$ is Solovay random if for all computably enumerable collections of intervals $\{I_n\}$ such that $\sum_n\mu(I_n)<\infty$, then $x\in I_n$ for at most finitely many $n$. ...
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### Proving Turing Completeness by Simulating Rule 110

Something I've heard often is that Rule 110 is the `simplest' Turing-complete formalism. As a programming exercise in a language I am new to, I implemented a function that computes from an initial ...
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### Were PR functions considered to be the class of total recursive functions?

At some point of history, were the class of primitive recursive functions considered (or even conjectured) to be the class of total recursive functions? I think I faced this claim sometime ago, but ...
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### Equivalence of sequences and subsets of natural numbers

For me, facts like the independence of the continuum hypotheses from ZFC cast a doubt on the "law of the excluded middle". (In this context, the doubt is that there might be no "final set theory" such ...
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### Creating ways to encode recursive function.

This is from an exercise in Boolos' Computability text. My problem is as follows: I am looking for a method that encode numbers for recursive functions. Then given such an encoding for recursive ...
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### there is no partial recursive function f s.t. whenever N-W_e has one element, f converges and N-W_e = f(e)

question is as written in the title: show that there is no partial recursive function f s.t. whenever N-W_e has one element, f converges and N-W_e = {f(e)}. W_e is the domain of the program coded by ...
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### Proof of a Theorem in Gao's 'Invariant Descriptive Set Theory'

Theorem 1.7.5 on p.35 of Gao's Invariant Descriptive Set Theory reads Theorem 1.7.5 (Kleene) If $A\subseteq X \times \omega^{\omega}$ is $\Pi^{1}_{1}$ and x \in B \Longleftrightarrow \exists y ...
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### Notation in Sacks' 'Higher Recursion Theory'

I'm having trouble with the notation in Sacks' Higher Recursion Theory. I've asked specific questions from page 4. Instead of reading my question in detail and trying to understand my confusion (which ...
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