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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6
votes
3answers
330 views

From lightface $\Sigma^1_1$ to boldface $\mathbf\Delta^1_1$

Fix some standard Polish space, e.g. Baire's space. It's a simple observation that every $\Delta^1_1$ is also $\mathbf\Delta^1_1$. It is the same observation that $\Sigma^1_1$ becomes ...
0
votes
1answer
51 views

Define the Complement of Factoring?

I just need some clarification as to what this terminology means in this situation. A decision problem for $FACTORING$ is as follows. INPUT: an integer $n$ and a integer $d$ QUESTION: does $n$ have a ...
2
votes
0answers
122 views

Unsolvability Degree in Turing's Proof 1

I have read that there is some debate over the exact origin of the Halting argument, which begins with Kleene and Davis in the 1950s [Copeland 2004]. Motivated by this I want to clarify the Degree of ...
-1
votes
1answer
165 views

Unbounded number of tapes of Turing Machine

Turing Machine with multiple tapes can be encoded such that its computational power is equivalent to Turing Machine with single tape. My question is if we have unbounded number of tapes, just like the ...
3
votes
1answer
203 views

Limits of Computable sequences

Turing introduced the fact that the limit of a computable sequence is not necessarily computable, and the Specker sequence is a specific example of such a number (with supremum not computable). My ...
4
votes
4answers
519 views

Examples of partial functions outside recursive function theory?

My math background is very narrow. I've mostly read logic, recursive function theory, and set theory. In recursive function theory one studies partial functions on the set of natural numbers. Are ...
2
votes
1answer
125 views

Showing a set of true sentences is recursive

Let's assume we are working in $(\mathbb{N}, +, \dot\ , 0,1)$. Let $T$ be a set of formulae that is closed under $\neg$ and such that the set of Godel numbers of formulae in $T$ is recursive. ...
12
votes
3answers
211 views

How does Borelness overlap with definability, computability, or constructiveness?

Background: I am writing a short paper aimed at math undergrads and focused as narrowly as possible on Borel equivalence relations. So, e.g., I am not assuming familiarity with recursion theory and am ...
5
votes
3answers
549 views

What does the concept of computation actually mean?

My question is very general, and the kind of answer I look for would be as low level as it could be. I think I may illustrate my query more succinctly with an example. In propositional logic, you ...
1
vote
2answers
243 views

Primitive recursive functions and mutual recursion

Let $g$ and $g'$ be primitive recursive, of arity 2, and let $a,a'\in\mathbb{N}$. Define $f$ and $f'$ by the following formulae: $f(0)=a$ $f'(0)=a'$ $f(n+1)=g(n,f'(n))$ $f'(n+1)=g'(n,f(n))$. How ...
0
votes
2answers
111 views

Total function and termination

If we have a total function, is it by default terminating function? How can we prove the termination for this total function?
2
votes
1answer
131 views

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 ...
4
votes
1answer
325 views

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 ...
3
votes
1answer
228 views

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 ...
3
votes
1answer
353 views

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 ...
3
votes
1answer
220 views

Computable function and decidable sets: For a computable $g$ and decidable set $A$ , Does $g(A), g^{-1}(A)$ necessarily decidable?

I'm trying to solve the following exercise from an old exam: For a computable function $G: \mathbb{N} \to \mathbb{N} $, and a set of numbers $A$, we define $g^{-1}(A)=\{x|g(x) \in A\} $ and ...
4
votes
3answers
264 views

Recursively enumerable set? Any hints?

I don't mean to be pulling answers out of you, but I'm stuck. Any advice on the right direction would be appreciated. I have the following set $X$ ={$n$ where $n$ is a number of a turing machine $M$ ...
2
votes
1answer
168 views

For a Turing machine $M$ can we decide whether there's an input which at $M(x)$ never moves left?

After Reading again the answer to this question and the answer to this question, I am wondering if the language $L=\{\langle M \rangle | M $is a Turing machine and $\exists$ input $x$ such that in ...
2
votes
2answers
305 views

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: ...
2
votes
4answers
861 views

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 ...
2
votes
2answers
126 views

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 $n$. ...
3
votes
1answer
346 views

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 ...
2
votes
2answers
271 views

Why is the following language decidable? $L_{one\ right\ and\ never\ stops}$

I can't understand how the following language can ever be decidable: $L= \{ \langle M \rangle | M \ is \ a \ TM \ and\ there\ exists\ an\ input\ that\ in\ the\ computation\ $ $of\ M(w)\ the\ head\ ...
5
votes
1answer
192 views

The print problem: How to show it is not decidable?

I wonder the following reduction is correct. I'm trying to show that the following problem "PRINT_BLANK" is not decidable. Input: (a coding of) Turing machine M. Question: Does the machine never ...
1
vote
1answer
83 views

Showing that $\{0w:w \in A\} \cup \{1w: w \notin A \}$ computes $A$

I'm trying to construct a reduction from $A \in RE \setminus R$ under $\sum=\{0,1\}$ to $B$ which defined: $B=\{0w:w \in A\} \cup \{1w: w \notin A \}$. I need to show that $B\notin RE \cup co-RE$ ...
0
votes
1answer
114 views

For regular expression $E$ and a context free grammar $G$- why deciding if $L(G)\subseteq L(E)?$ is a recursive problem?

I'd love your help with understanding why does the following language is recursive: Input: a regular expression $E$ and a context free grammar $G$ question: $L(G)\subseteq L(E)?$ I tried to think ...
4
votes
1answer
136 views

An Obsessive Turing machine problem

Can you please help me understand whether or not the following the problem is recursive, recursively enumerable, or co-recursively enumerable? A Turing machine $M$ is said to be obsessive if on every ...
4
votes
1answer
145 views

Are there larger than countable chains in the join-semilattice of Turing degrees?

Recall that Turing degrees are equivalence classes of subsets of $\mathbb{N}$ under Turing equivalence (mutual Turing reducibility). They are partially ordered by Turing reducibility and form a ...
8
votes
1answer
269 views

Does there exist a universal pushdown automaton?

Let $\Sigma$ be a fixed alphabet and let $PDA(\Sigma)$ be the set of all Push-Down-Automata (PDA's) having input alphabet $\Sigma$. Is there an alphabet $S$ and a function $f:PDA(\Sigma) \to S^∗$ such ...
2
votes
1answer
159 views

Is the set of codes of Deterministic Finite-State Automata a regular language?

Let $\Sigma$ be a given alphabet. Is there a way to code up Deterministic Finite state Automata (DFA) over $\Sigma$ as strings of $\Sigma$ in such a way that the corresponding subset of $\Sigma^*$ is ...
2
votes
2answers
137 views

Complexity of subset relation on Borel Sets

Fix $A, B \in \mathbf{\Delta}_{1}^{1}$ (i.e. they're Borel). Is the statement $A \subseteq B$ generally only $\mathbf{\Pi}^{1}_{1}$ (at best)? Of course, it's $\mathbf{\Pi}^{1}_{1}$ via $\forall x[x ...
2
votes
2answers
148 views

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 ...
2
votes
3answers
232 views

Complexity of the computable entailment relation

The following definition comes from Richard Shore's 2010 paper 'Reverse Mathematics: The Playground of Logic'. Let $\varphi$ and $\psi$ be sentences in the language of second order arithmetic ...
1
vote
1answer
72 views

Satisfiability problem for FOL[<,R]

Let FOL[<,R] be the fragment of first-order logic enriched with two relational symbols < and R and the first-order axioms that say: < is a strict partial order and R is an irreflexive and ...
4
votes
0answers
516 views

algorithm for solving diagonal quadratic equations over real or complex numbers

I found the following statement in the paper http://www.math.uni-bonn.de/~saxena/papers/cubic-forms.pdf (page 22, in the middle): For $\mathbb F\in\{\mathbb R, \mathbb C\}$ and $b, a_i\in\mathbb ...
5
votes
1answer
318 views

Recursively enumerable languages are closed under the min(L) operation?

Define $\min(L)$, an operation over a language, as follows: $$ min(L) = \{ w \mid \nexists x \in L, y \in \Sigma^+ , w=xy \} $$ In words: all strings in language L that don't have a proper prefix in ...
3
votes
3answers
199 views

A non-arithmetical set?

A set is called arithmetical if it can be defined by a first-order formula in Peano arithmetic. I first encountered these sets when exploring the arithmetical hierarchy in the context of ...
11
votes
3answers
210 views

Must a function that 'preserves r.e.-ness' be computable itself?

Does there exist a non-recursive function (say, from naturals to naturals) such that the inverse of every r.e. set is r.e.? If yes, how to construct one? If no, how to prove that? Any References?
1
vote
1answer
500 views

Computable functions' set is countable

I have to prove that computable functions (by computable we mean recursive functions or functions calculated by a program with a register machine) are countable. Let $\mathcal{C}$ be the set of ...
0
votes
1answer
376 views

Definition of effective enumerability and empty set

Let $S$ be a set. We say that $S$ is effectively enumerable iff (by definition) there exists a function $f \colon N \to N$ which has $S$ as codomain. My question is: is the empty set an effectively ...
9
votes
2answers
715 views

Why is it undecidable whether two finite-state transducers are equivalent?

According to the Wikipedia page on finite-state transducers, it is undecidable whether two finite-state transducers are equivalent. I find this result striking, since it is decidable whether two ...
7
votes
3answers
371 views

Reductions for regular languages?

To reason about whether a language is R, RE, or co-RE, we can use many-one reductions to show how the difficulty (R, RE, or co-RE-ness) of one language influences the difficulty of another. To reason ...
6
votes
2answers
155 views

Effective cardinality

Consider $X,Y \subseteq \mathbb{N}$. We say that $X \equiv Y$ iff there exists a bijection between $X$ and $Y$. We say that $X \equiv_c Y$ iff there exist a bijective computable function between ...
11
votes
3answers
2k views

Proving that the halting problem is undecidable without reductions or diagonalization?

I'm currently teaching a class on computability and recently covered the proof that the halting problem is undecidable. I know of three major proof avenues that can be used here: Diagonalization - ...
2
votes
3answers
757 views

Is the set of total recursive functions countable?

There are many reasons to hold that the set of total recursive functions is countable, and among them the two following seem to me to be very powerful and sound: The set of total recursive functions ...
4
votes
3answers
756 views

Primitive recursive functions, Recursive functions and recursive set

I'm trying to understand basic computability notions, and I'm a bit confused concerning the following questions : Is the set of (Gödel numbers of) partial recursive functions recursive ? Is the set of ...
0
votes
2answers
112 views

A problem that is Turing degree greater than $0'$ and not co-re

I know that halting problem is Turing degree $0'$. So, what degree would Co-RE complete problems be in? And is there any problem we can formulate (<- this might be too general, but let us say in ...
4
votes
1answer
147 views

expressiveness of computable infinitary logic

A language $L_{\omega_1\omega}$ generalizes an ordinary first-order language by allowing countably long disjunctions. If we take its nonlogical vocabulary to contain just a predicate for the ...
2
votes
2answers
171 views

Turing reduction

I'm learning algorithm theory. Homework question is: Are $A$ and $B$ possible so that $A\not\le_{tt}B$ (impossible to reduce using tt), but $A\le_T B$. But I can't think of any example..
4
votes
1answer
162 views

Projecting onto (lightface) Borel sets

Suppose $A \subseteq \omega^{\omega} \times \omega^{\omega}$ is Borel. If we project $A$ onto $\omega^\omega$, we get a $\mathbf{\Sigma^{1}_{1}}$ set $\{y: \exists x (y,x) \in A\}$. What if we project ...