0
votes
1answer
48 views

Using diagonal argument to prove that H(x)=μyT(x,x,y) has no total computable extension

Hello everyone just like the title says I want to prove that $H(x) = \mu y T(x,x,y)$ has no total computable extension such that if we had a function $BIG(x)$ that is both total and agrees with $H(x)$ ...
1
vote
2answers
48 views

Reduction between two languages and a common one

My question is as following : Let $A$ and $B$ be some languages, there exist a language $C$ such that $A\le C$ and $B\le C$, where "$\le$" means "reducible to", so $A\le C$ means there is a mapping ...
0
votes
1answer
44 views

Prove that [x/y] is a primitive recursive function

Prove that [x/y] is a primitive recursive function using this theorem: If $g(x_1,...,x_n)$ is primitive recursive, then $f(x_1,...,x_n)=\sum^{x_n}_{i=0}g(x_1,...,x_{n-1},i)$ is also a primitive ...
0
votes
1answer
78 views

computability and uncomputability

1) Suppose $f$ is an increasing function from $\mathbb N \to \mathbb N$ $(i.e., if x\ge y, then \space f(x) \ge f(y)).$ Is there necessarily a program which computes $f$? 2) Suppose $f$ is a ...
1
vote
0answers
58 views

Primitive recursive and Turing machines

Can someone give me a hint or the start of a possible proof for the following theorem: A function $f: \mathbb{N}^r \rightarrow \mathbb{N}$ is primitive recursive if and only if there is a ...
1
vote
0answers
8 views

Is the language that consists of machine configurations whose language is a subset of even palindromes semi-decidable?

Let $PAL = \{ww^R\ | w\in\{0,1\}^*\}$. Then let $A = \{\langle M\rangle \ | \textit{M is a Turing Machine and } L(M)\subseteq PAL\}$ Is A semi-decidable (Turing recognizable or recursively ...
0
votes
1answer
156 views

Simultaneous recursion

I have no idea how to even start proving the following theorem: If $f_0, f_1: \mathbb{N}^r \rightarrow \mathbb{N}$ and $g_0, g_1: \mathbb{N}^{r+3} \rightarrow \mathbb{N}$ are primitive recursive, ...
2
votes
2answers
89 views

Prove domain of partial computable function exists

Prove that there is an n such that $W_n$ = {$2n, . . . , 2n + n^2$} Now I don't know where to start with this question, how can I go about answering it? Would I construct a computable function that ...
4
votes
1answer
139 views

The Permitting Method

Define the term late permitting in the following way: $C$ late permits an element $x$ to enter $A_{s+1}$ if for a fixed computable function $f$ with $f(n)>n$, there exists $y\leq x$ such that $y\in ...
1
vote
1answer
83 views

simple sets, cofinite sets, filters

Let $\mathcal S$ be the class of simple sets and $\mathcal C$ the class of cofinite sets. Prove that $\mathcal S\bigcup \mathcal C$ is a filter in $\mathcal E$. Definitions: An infinite set is ...
2
votes
1answer
289 views

Is this undecidable language recognizable?

Is this language: $L = \{\langle M\rangle : \text{$M$ is a Turing machine and $L(M)$ is decidable}\}$ which I know that is undecidable, turing-recognizable? Is its complement recognizable? ...
2
votes
1answer
54 views

Proof that an inverse of a possibly noncomputable function is possibly not decidable

I'm stuck with the following homework: Given an fixed function $f:\mathbb{N}\to\mathbb{N}$. $f$ is an arbitrary (possibly not computable, possibly partial) function. Show that the set $\{f(42)\}$ is ...
1
vote
1answer
108 views

Show that K and complement to K are “1-reducible” to EQ={⟨x,y⟩|φx≃φy}

Where $K = \{x | φ_x(x) \downarrow\}$, $φ_x$ is a $\mu$-recursive function computing $M_x$, $M_x$ is Turing machine with Godel's number $x$. Set $A$ is "1-reducible" to set $B$ ($A \leq_1 B$) when ...
1
vote
1answer
124 views

The set of Turing machines that recognize $\{00, 01\}$ is undecidable

$L =\big\{\langle T\rangle \mid T\text{ is a Turing machine that recognizes }\{00, 01\}\big\}$. Prove $L$ is undecidable. I am really having difficulties even understanding the reduction to use ...
2
votes
2answers
364 views

How to solve this recurrence relation: $T(n) = 4\cdot T(\sqrt{n}) + n$

I was trying to solve this recurrence $T(n) = 4T(\sqrt{n}) + n$. Here $n$ is a power of $2$. I had try to solve like this: So the question now is how deep the recursion tree is. Well, that is ...
2
votes
2answers
244 views

How to define divisibility recursively?

Let $d(x,y)=1$ if $x$ is divisible by $y$, and $=0$ otherwise. How can I define $d(x,y)$ in terms of just the basic primitive recursive functions (zero, successor, identity, projection) and the ...
0
votes
1answer
46 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
1answer
92 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. ...
1
vote
2answers
150 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 ...
4
votes
1answer
230 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 ...
5
votes
1answer
201 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 ...
1
vote
1answer
233 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 ...
2
votes
2answers
153 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..
3
votes
1answer
214 views

Infinite finitely branching recursive tree with no path whose graph is $\Delta^0_2$

I am trying to construct an example of a infinite, finitely branching, recursive tree $T$ such that none of its paths has a graph which is $\Delta^0_2$. I denote the set of paths of $T$ by $[T]$. I ...
2
votes
2answers
103 views

Showing a set $\Sigma^0_n$ subset of $\mathbb{N}$ is $\Sigma^0_n$-complete

This is both a general and specific question in basic computability theory. Broadly speaking, I am not very comfortable with showing whether or not a subset of $\mathbb{N}$ is $\Sigma^0_n$ (or ...
1
vote
1answer
106 views

Relationship between $\Sigma_{1}$ and $\Pi_{1}$ functions (Logic)

I am working on the following homework problem for a logic class on Godel's incompleteness theorems and the following question is asked. Is the converse of Theorem $13.1$ true? Explain. Theorem ...
0
votes
1answer
68 views

how can we categorize m-complete languages of RE (recursive enumerable, re-complete)?

is there any hierarchy for many-one complete languages of re (re-complete languages)? how can we propose a categorization for these languages? depending on what measures?
3
votes
2answers
100 views

Constructing a TM from a grammatically computable function

I have a grammatically computable function $f$, which means that a grammar $G = (V,\Sigma,P,S)$ exists, so that $SwS \rightarrow v \iff v = f(w)$. Now I have to show that, given a grammatically ...
5
votes
3answers
215 views

Recursive function that outputs its own code

This problem is probably a rather trivial one, since I have the impression, that it is a textbook-style one, but nonetheless somehow it won't give in. Here it is: I have to show that there exists a ...
4
votes
2answers
731 views

Is this function a primitive recursive function?

Let $t \in \mathbb{N}$ and consider the function $f: \mathbb{N} \rightarrow \mathbb{N}$, defined by $f_t (m)= 2 \uparrow^{m} t$, where "$\uparrow$" is Knuth's up-arrow notation (which can be ...
5
votes
5answers
685 views

Can a polynomial size CFG over large alphabet describe a language, where each terminal appears even number of times?

Can a CFG over large alphabet describe a language, where each terminal appears even number of times? If yes, would the Chomsky Normal Form be polynomial in |Σ| ? EDIT: What about a language where ...