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).

learn more… | top users | synonyms (1)

3
votes
1answer
202 views

Recursion schema and the arithmetical hierarchy

In computability we define the following basic functions, the zero function, the successor function, and the functions $I_{n,k}(x_1,\ldots,x_n)=x_k$ for $k\leq n$. Next we define three schemata for ...
4
votes
1answer
88 views

A paradox related to computable reals?

Let O be a computable ordering of all computable reals in ⟨0,1) (eg. first by length of programs computing them and then lexicographically). (it does not matter that they appear there more than one ...
6
votes
1answer
144 views

A topological example from Church's undecidability paper

A. Church, in his classical paper An unsolvable problem in elementary number theory in American Journal of Mathematics Vol. 58 No. 2. (1936), pp. 345-363, (available here), wrote: There is a class ...
3
votes
1answer
111 views

Is my logic here correct? Primitive recursion and diagonalization proof

I'm trying to understand diagonalization as it applies to proving that not all total functions are primitive recursive functions. For example, say that we enumerate all primitive recursive functions ...
2
votes
1answer
72 views

Proving a class of relations is closed under an operation

Given a class of functions $\mathcal{A}$ is closed under substitution and the operation $(\mu y)_{\leq z}$, where $$ (\mu y)_{\leq z}f(y, \vec x) = \begin{cases} \min\{y : y \leq z \land f(y, ...
3
votes
2answers
96 views

If $L\in REG$ then $M$ has a finite number of distinct rows

Let $L \subseteq \Sigma^{\star}$ and let $M^{\Sigma^{\star} \times \Sigma^{\star}}(\{0,1\})$ an infinite matrix such that for each $x,y\in \Sigma^\star$: $$ m_{x,y}=\begin{cases} 1 & x y\in L\\ 0 ...
0
votes
1answer
108 views

Computability function - how to express it in set theory/arithmetic hierarchy

Let's say that $f$ is computable function such that for particular inputs $x$ and $y$, $f(x) = 0$ and $f(y) = 0$. If we want to express this in logical form (arithmetic hierarchy formula), what would ...
0
votes
1answer
126 views

recursively enumerable of Godel numbering

There are statements for natural number x, like followings m: "x is even natural number" n: "x+1 is odd number and x>1" l: "x is positive integer multiple of two" m, n, l has same boolean value ...
2
votes
1answer
85 views

What's the error in this argument that Fin$\le_m$Inf

There must be an error in the following argument since Fin is not many-one reducible to Inf, I can't seem to find it. Here it is informally (I hope it's straightforward and not confusing): Take any ...
1
vote
1answer
83 views

For a one-to-one function is the pullback of a recursively enumerable set $f^{-1}[W_e] = W_{g(e)}$ where $g$ is one-to-one?

Let $\phi_e : \mathbb{N} \to \mathbb{N}$ be the recursive function coded by $e$ and $W_e = \{ x : \phi_e(x) \text{ is convergent}\}$. A set $A$ is recursively enumerable if $A = W_e$ for some $e$. ...
0
votes
1answer
127 views

completeness and creative

I'm trying to show that any complete $\Sigma_1^0$ set is creative. The definition of creative I understand is: if there is a total recurvise function f s.t. f(e) is an element of A iff f(e) is an ...
0
votes
1answer
69 views

recursive maps and decimal representation

I'm trying to think of an example of a real number $R$ such that the map $n \mapsto R[n]$, where $R[n]$ is the $n$th digit of the decimal representation of $R$, is not recursive. So I was thinking to ...
1
vote
1answer
157 views

recursive and creative theorem

How can we show that if A is creative, then A is not recursive. Only thing I can get out is the fact that if A is creative, if it is rec. enumerable and the complement(A) is productive. Thanks
1
vote
1answer
178 views

How does one prove that 1-generic set is not computable?

Without resorting to diagonalization proof of halting problem, how does one prove that 1-generic set is not computable?
2
votes
0answers
163 views

How to show Simp. and Creat. are $\Sigma^0_2$-Hard

Let Simp={$e:W_e$ is simple} and Creat={$e:W_e$ is creative} I'm having troubles showing these sets are $\Sigma^0_2$-Hard, ie that any $\Sigma^0_2$ set can be many-one reduced to them. I've already ...
0
votes
1answer
72 views

Identifying a pattern in an array

Is there a way to identifying a pattern and/or recursive function for an array? If yes, how can I do this. Could anyone please help me with some information and/or resource for this? Any help is ...
2
votes
1answer
195 views

Proof of Kleene's T predicate being primitive recursive

As I am looking over Kleene's T predicate, I was unable to find why Kleene's T predicate is primitive recursive. Can anyone show why? (I know what primitive recursive is.)
0
votes
1answer
301 views

Recursively inseparable sets

I'm trying to show that there is a pair of $\Sigma_1^0$ recursively inseparable sets. From the definition, recursive inseparable is if there is no recursive set $C$ such that $A\subset C$ and $B\cap ...
0
votes
1answer
123 views

A real number being computable

In my text, it says that a real number $r \in \mathbb{R}$ is computable iff given $n$ one can compute $q \in \mathbb{Q}$ such that $\left|r-q\right| \leq 2^{-n}$. Can anyone show why it is the case? ...
2
votes
2answers
110 views

$\omega_1^{CK} - \omega$ - infinite or finite set? And boundary

I am curious whether $\omega_1^{CK} - \omega$ would result in a finite set or infinite set. Does anyone know what happens? Edit: OK, let me add one more question: Suppose that we take $\omega \cdot ...
4
votes
2answers
80 views

Proving induction from basic recursion lemma

Induction Principle: Let $A$ be a set such that $0 \in A$ and $n \in A \implies n + 1 \in A$. Then for all $n \in \mathbb{N}$, $n \in A$. Basic Recursion Lemma: For all sets $X, W$ and given ...
1
vote
1answer
54 views

analytical hierarchy and individual variable quantifiers

For analytical hierarchy, $\Sigma^1_0$ is usually defined as the class of formula that does not have any set quantifier - but does this mean that there can be any number of quantifiers for individual ...
0
votes
1answer
64 views

$\mid$ in simply typed lambda calculus

$e = x \mid \lambda x\!:\!\tau.e \mid e \, e \mid c$ So, what is $\mid$ in this example of simply typed lambda calculus? The syntax of the simply typed lambda calculus is essentially that ...
1
vote
2answers
178 views

Writing Fermat's last theorem in arithmetic hierarchy

Somehow connected with How natural numbers can be defined using primitive recursive $\Sigma_0^0$: OK, so here's how Fermat's last theorem is formulated: $$\forall x,y,z,n>2 \quad (x^n+y^n + z^n ...
1
vote
1answer
194 views

How natural numbers can be defined using primitive recursive $\Sigma_0^0$

$$S=\{x\mid (\exists y_1)\cdots (\exists y_r)P(x,y_1,\ldots,y_r)\}, \qquad P \text{ primitive recursive.}$$ I do get how some set of natural numbers (or numbers) can be defined with $\Sigma_1^0$ ...
4
votes
1answer
99 views

Why is computable function in $\Delta_{1}^0$?

I am not sure why computable functions are in $\Delta_{1}^0$. Can anyone explain this?
1
vote
2answers
127 views

The computability of Kleene's $T$-predicate

Why is Kleene's T-predicate computable? how to argue this using turing computability? would that be useful or writing it as some function
4
votes
1answer
110 views

$\Sigma^0_n$ complete sets

Does anyone know of a way of showing that a $\Sigma^0_n$-complete set is not $\Pi^0_n$ without having to appeal to $\Sigma^0_n$-universal sets? For instance a more direct diagonalization argument ...
2
votes
1answer
107 views

Computable function example

Suppose $p(x)\in\mathbb{Z}[x]$. How can we show that the function $b\to$ the least non-negative integer root of $p(x) - b$ is computable (if there is no such root, then the function is undefined)? ...
1
vote
1answer
95 views

Godel numbering and Turing jump

Given a set $X$ and a Gödel numbering $φ_i^X$ of the $X$-computable functions, the Turing jump $X'$ of $X$ is defined as $X'= \{x \mid \varphi_x^X(x) \ \mbox{is defined} \}.$ OK. But then ...
5
votes
2answers
197 views

How does second-order logic relate to lambda calculus?

How does second-order arithmetic/logic relate to lambda calculus? By lambda calculus, I mean both typed and untyped. And is there any relationship with recursive and recursively enumerable sets? ...
0
votes
1answer
239 views

NonRecursive Sets

I'm trying to show that the following are nonrecursive: $\{x \in \mathbb{N} \mid \phi_x(y) \uparrow\}$ $\{(x,y) \in \mathbb{N}^2 \mid \phi_x = \phi_y\}$ $\{(x,y) \in \mathbb{N}^2 \mid y \in ...
-3
votes
1answer
257 views

Computability of busy-beaver sequence? [closed]

We can draw a parallel between cellular automata and busy-beaver numbers. For example the initial case occupies some kxk square in the plane,leaving all the other cells emty, after how many ...
3
votes
0answers
303 views

How to derive Church-Kleene ordinal

Crossing-out: (How does one prove the existence of Church-Kleene ordinal? Also, why is it labeled as $\omega_1^{CK}$? And why is it first ordinal not hyperarithmetical, and is the first admissible ...
1
vote
1answer
185 views

Halting problem on 2 registers

How can we show that the halting problem on register machines equipped with only two registers is unsolvable. My intuition stems from assuming it is unsolvable on $n$ registers, but if we take a ...
1
vote
1answer
208 views

Recursively enumerable sets

How can we show that any diophantine set is recursively enumerable? To be r.e., we require that the characteristic function is recursive and thus we require set to be decidable. Only thing hat may ...
2
votes
1answer
161 views

Diophantine sets

I'm trying to show that the following sets are Diophantine: $\{(x,y)\mid x \leq y\}$ $\{(x,y)\mid x < y\}$ $\{(x,y)\mid x\text{ divides }y\}$ $\{(x,y,z)\mid x\equiv y \pmod z\}$ $\{(x,y,z)\mid x ...
1
vote
2answers
62 views

In complexity, Is the relationship between P and R known?

The relationship between P and NP is unknown; However, we can ask an "easier" question, what is the relationship between P and R (=decidable languages)? In other words, is there a (decidable) problem ...
2
votes
2answers
69 views

When is the complement of a diophantine set in the naturals also diophantine?

A diophantine polynomial is a (multivariable) polynomial with integer coefficients. If we write this polynomial as $p(x, y_1, \dots, y_n)$, then it defines the diophantine set $D_p = \{ x \in ...
2
votes
2answers
127 views

Complements of recursively enumerable subsets.

Let $A,B \subseteq \mathbb{N}$. If $A$ and $B$ are recursively enumerable, can we say anything about expressions like $A^c \cup B$, $A^c \cap B$, etc.?
0
votes
2answers
147 views

Church's Thesis

If we let $f$ be a computable function and define $h(x) = 1$, if $x$ is an element of $\operatorname{dom}(f)$ and undefined otherwise. I am trying to prove that h is computable via Church's Thesis. ...
2
votes
2answers
132 views

Is Turing-completeness decidable?

This may be a silly question, but is there an algorithm that decides whether any given model of computation is Turing complete?
5
votes
0answers
80 views

Is the measure induced by the Mandelbrot set computable on rational rectangles?

Is there a computable function that, given a positive rational number $\epsilon$ and a rectangle with rational corners $A$ returns a number $f(A,\epsilon)$ such that $|\mu(A \cap ...
3
votes
1answer
135 views

Are the brackets in formal box notation of recursive functions omittable?

So we know all recursive functions can be expressed as a finite sequence of symbols for the basic functions and processes composition, primitive recursion, and minimization. What I'm wondering is if ...
7
votes
2answers
200 views

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 ...
1
vote
2answers
198 views

Recursive relation using successor function

What is the recursive relation for $$H(m)=2^{(m^2)}$$ using successor function recursive relation for multiplication: $$mult(x,0)=0; mult(x,S(y))=add(x,mult(x,y))$$ recursive relation for addition: ...
1
vote
1answer
93 views

Recursive functions, successor function

How to show that the power function $\displaystyle A=2^{m^2}$ is primitive recursive based on successor function? Thanks much in advance!!!
3
votes
1answer
191 views

Explain why if the language A is recursive, then A is reducible to 0*1*

I'm in a theory of computation class and there is a problem that I think I am way overthinking. Can anyone point me in the right direction with the following: Give a short justification of the fact ...
1
vote
1answer
367 views

Converse of Collatz Conjecture

How to write a pseudocode program that halts only if the Collatz Conjecture is false. Thanks much in advance!!!
1
vote
1answer
176 views

Recursive relation and predicate

If we let P(x,y) be a primitive recursive relation and g(x) be a primitive recursive function. Then how to show that there exists a y < g(x)*P(x,y) is a primitive recursive relation? And how can ...