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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Proof that mappings $K$ and $L$ are primitive recursive.

Let $J$ be the function: \begin{equation*} J(m,n)= \begin{cases} n^2+m \text { if } m\le n \\ m^2 + m + (m-n) \text { if } m > n \\ \end{cases} \end{equation*} Let $K, L$ such that $K(k)$ is ...
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3answers
75 views

Can we simplify analysis by getting rid of the uncountable reals? [duplicate]

Since the entire observable part of the universe can only be in a finite number of physically distinguishable states, it seems rather strange that an efficient formal description of the universe would ...
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0answers
127 views

Recursive and Primitive recursive functions

According to the book that I'm reading, we can define the $\mu-$recursive functions inductively, as follows: The constant, projection, and successor functions are all $\mu-$recursive. If $g_1, ...
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1answer
44 views

Proving the principle of definition by generalized recursion using the inductive closure of an induction system

I'm working through Hinman's Fundamentals of Mathematical Logic in order to review some things, and got stuck in an exercise from section 1.2. Specifically, he asks us to prove (what he calls) the ...
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0answers
130 views

Decidability of determining the definition of a function

Let's say a property is an SMT formula. Let's say a function has a property iff, with addition of the function symbol to a monadic predicate calculus formula over the signature of Presburger ...
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2answers
64 views

What's the significance of the Church-Turing Thesis?

My understanding is that the thesis is essentially a definition of the term "computable" to mean something that is computable on a Turing Machine. Is this really all there is to it? If so, what makes ...
2
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1answer
74 views

Pointclass of $\text{dom}(F)$ where $F:\omega^\omega\rightarrow\omega^\omega$ is partial recursive.

The definition I am working with: A partial function $F:\omega^\omega\rightarrow\omega^\omega$ is said to be partial recursive iff the partial function $G:\omega^\omega\times\omega\rightarrow\omega$ ...
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1answer
50 views

Finding the sum of special multiplications

Let $n$ be an integer and $a_1, \dots, a_n$ positive reals. $\forall 1 \leq i < j \leq n$ let $a_{i, j}$ be a positive number. Let $k \leq n$ be a positive integer. I would like to find an ...
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1answer
82 views

Is there an algorithm that probably solves the Halting problem?

Such an algorithm takes as input any program and returns a probability that it halts. In the limit of many programs, it must answer on average in the correct proportion. But im interested in other ...
4
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2answers
111 views

How should the relativized Kleene pointclass $\Sigma^1_1(A)$ be defined?

How should the relativized Kleene pointclass $\Sigma^1_1(A)$ be defined, when $A$ is a set of reals ($A \subset \omega^\omega$)? I assume that there is a standard definition, but I can't seem to find ...
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1answer
82 views

Which mistake(s) in my argument re: representability, definability and the halting problem?

I'd like to ask for your help in showing me the (quite likely: several) flaws in my argument below, relating weak and strong representability in a formal system and the halting problem. At least ...
4
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1answer
87 views

Do you need true randomness to beat the two-envelope game?

A well-known (non-)paradox in probability involves a two-envelope game played between two players, $A$ and $B$: $A$ selects two distinct (real) numbers, $x$ and $y$, writing each one down on a card ...
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2answers
25 views

Computability: is there an alternative method to decide this language?

For my computability revision I am trying to decide the language, $$L = \{ \text{all binary strings containing the pattern 001 (not necessarily in consecutive places)} \}.$$ I believe that I can do ...
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0answers
28 views

Show the following languages are not recursive

Show that the language $$L = \{ M : M \text{ is a Turing Machine that halts on input $M$ } \} $$ is not recursive. Show that the language $$ L = \{M : M \text{ is a Turing Machine such that $L(M)$ ...
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1answer
63 views

determining recognizable or decidable (TM that accepts a TM)

I'm having an issue determining whether certain languages are decidable, recognizable or neither. The specific languages I'm referring to are of the following form L = {<M> | for every w, M accepts ...
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1answer
86 views

is differ between distributive lattice vs semi-lattice on Turing Degrees

We know a Posed Closed under suprema but not necessarily under infima is an upper semi-lattice. We now r.e set forms a distributive lattice. But my question is why following statement is hold? I ...
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1answer
35 views

decidability of a given language

The language EGAL is $\{(A,B): A \text{ and } B \text{ are DFAs with } L(A) = L(B)\}$ How do I prove that such language is decidable by testing every word of $A$ and $B$ until a defined length ? i ...
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0answers
108 views

Does removing the (1) in $\Phi{(1)}$ affect the proof that $ K_0 \leq_m K$ or not?

The fragment below from Martin Davis' book shows $ K_0 \leq_m K$ and also proves $ K_0 \leq_1 K $. My question is if we remove the $(1)$ of $ \Phi^{(1)}$ in the definition of $Y$ (i.e fifth line in ...
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1answer
50 views

How to computably reduce the number of colors in (infinite) Ramsey's theorem

Suppose we have an "oracle" that gives a homogeneous set for a 2-coloring $\hat c : [\omega]^2 \rightarrow 2$ of pairs of integers. Using this oracle, can we "compute" a homogeneous set for a ...
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1answer
67 views

Which way is best to solve: $T(n)=5T(n/5) + n\;?$

I'm not sure which way is best to solve $$T(n)=5T(n/5) + n$$ (recursion tree/master method/recurrence?) I would like some assistance, which way is easier and how can I be sure I got the right answer ...
2
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1answer
100 views

Have we found a Turing Machine for which halting/non-halting is unprovable?

The undecidability of the Halting Problem implies that there exist Turing Machines such that you can't prove whether they halt or not in whatever logical system you're using (let's say ZFC)$^1$. Have ...
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1answer
25 views

strongly hh-immune sets

I'm trying to do exercise X.2.16 from Soare's Recursively Enumerable Sets and Degrees, but I have no idea how ro solve it. Any hints would be appreciated. An infinite set is strongly hh-immune or ...
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1answer
43 views

Convert the regular expression to a NFA

I have to convert the following regular expressions to a NFA: $$(0 \cup 1)^{\star} 000 (0 \cup 1)^{\star}$$ $$(((00)^{\star} (11)) \cup 01)^{\star}$$ $$\emptyset^{\star}$$ $$a(abb)^{\star} \cup ...
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2answers
58 views

Convert NFA to DFA

I have to convert the following NFA's into the equivalent DFA's. I have done the following: Could you tell me if it is correct??
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1answer
298 views

Primitive Recursive on Some Functions?

We took an entrance exam on Set and Complexity Course, The question says: if $g$ be a primitive recursive, $1)$ $f_1(0)=c_1, f_1(1)=c_2, ...
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2answers
185 views

Set of One-Variable Computable Function and one Local Contest Questions?

I prepare for local complexity contest and review some old question banks. I get stuck in one problem and no idea how we can solve it. please share your idea or help with this question: Suppose ...
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2answers
71 views

Are there any known noncomputability proofs that do not rely on the halting problem?

I have looked around and thought of this for a while, and I have not found or been able to construct any proof that a problem is not decidable, without said proof being fundamentally equivalent to ...
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1answer
59 views

an strange set $ \Xi_A =$ {$ n \in N | \exists k^2 \in A $ s.t $ k^2 \leq n$} is decidable ?, an Interview questions?

We are some student that had an Interview for M.sc Entrance Exam. This interview has two part and one multiple choice question. We see 1 strange question that some definition is so strange for us, we ...
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1answer
45 views

$A_n$={$x \in \mathbb{N}\mid n \in W_x $} and computation questions?

I‌ prepare for Final-Exam on Complexity Course. in one of my prof. old-exam I see this question: Suppose $A_n$={$x \in \mathbb{N}\mid n \in W_x $}. Which of them is false? 1) Set $A_n$ for each ...
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1answer
235 views

Arithmetical hierarchy and complexity course note?

In my note, our professor talk about Arithmetical hierarchy. at the end he wrote all of these is True. My main problem is how these are True? ($N$ means Natural ...
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2answers
108 views

Why there is no recursive enumerable set such as B that: > $B \nleq_m K$?

I get stuck in one fact that I see on old-mid exam. Why there is no recursive enumerable set such as B that: $B \nleq_m K$ Def: K means Halting Set and $ ...
3
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1answer
119 views

Can the empty set be an index set?

I ran into a question, encountered in a computational course. Could anyone tell me why the empty set $ \emptyset $ can be an index set? My source is this book
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1answer
84 views

Creative and Simple Set and $S \leq_m C$? [closed]

I see in an old-exam that wrote if C is a Creative Set and S be a Simple Set we have: $S \leq_m C$ (i.e. m: many to one reducible ). How we can conclude this?
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1answer
165 views

Range or Domain of Primitive Recursive Function? [closed]

We are given that $A$ is R.E set. I think all of the following are equivalent to that: (1) A is the range of one primitive recursive function, (2) A is the domain of one strictly increasing ...
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1answer
42 views

A={$i+1 | i \in N \varphi_i(1393)=2015 $} is Recursive?

I see that my prof. wrote: A={$i+1 | i \in N \varphi_i(1393)=2015 $} is Recursive, but B={$n^2 + n | n \in N \varphi_n(n)= \uparrow $ } is not an r.e set. Who can learn me, about this two example?
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1answer
65 views

Some questions about Church's Theorem

On page.238 of Enderton's "A Mathematical Introduction to Logic", Church's Theorem is stated (The set of Gödel numbers of valid sentences (in the language of R) is not recursive.) My question is ...
3
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1answer
70 views

Is there a Turing Machine that can distinguish the Halting problem among others?

Can there be a Turing machine, that given two oracles, if one of them is the Halting problem, then this machine can output the Halting problem itself? Clearly, if the first oracle is always the ...
3
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1answer
45 views

What is the least ordinal $\beta$ for which the function $f_\beta(n)$ in fast-growing hierarchy is incomputable?

Fast-growing hierarchy consists of a transfinite succession of faster growing functions $f_\alpha$: $f_0(n) := n+1$, $f_{\alpha+1}(n) := f^n_\alpha(n)$, $f_{\alpha}(n) := f_{\alpha[n]}(n)$ if ...
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1answer
43 views

Natural Numbers and $A_x=\{y \in A \mid y \leq x\}$ [closed]

Suppose A is a arbitrary subset of Natural Numbers and $A_x=\{y \in A \mid y \leq x\}$ with respect to $ n \in A \Longleftrightarrow n \in A_n $ and $A_n$ is finte, which of them is True? a) A and ...
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2answers
44 views

Recursion, Truncation, and “coding.”

The example is "left to the reader", but I am having trouble approaching this problem. There is a primitive recursive function $tr$ such that if $s$ codes a sequence $(a_{0},...,a_{n-1})$, and $m\le ...
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1answer
32 views

Showing the “converse” of a relation is semirecursive.

I feel that I intuitively understand why this statement is correct, and I "think" I can explain it, but I don't know how to make it rigorous. I will show the problem, explain what I think the ...
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95 views

Path to categorical realizability theory

I'm trying to understand the sorts of things found on this page: http://ncatlab.org/nlab/show/realizability In particular, I want to read Oosten's Realizability: An Introduction to the Categorical ...
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1answer
125 views

Is there a relationship between Turing's Halting theorem and Gödel Incompleteness

Turing's proof that a Halting oracle is impossible and Gödel's proof that and omega-consistent first order theory of arithmetic must be incomplete are similar in that they use self-referential ...
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1answer
73 views

Dedekind Cuts and Computable Numbers

Can someone please tell me where I'm wrong? I'm not able to figure out where is the error: First let's define what a computable number is: a number is computable if there is a Turing Machine that ...
2
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1answer
60 views

Are there any Martin-Löf random reals that are computable?

For example, Chaitin's constant is both Martin-Löf random and uncomputable. Are there any examples of numbers that are Martin-Löf random but computable?
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1answer
51 views

Showing that a set is primitive recursive.

I've been having a lot of difficulty even beginning this problem. I believe that I would have to use the min and max functions, but I'm not entirely sure how to actually write this down rigorously, or ...
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0answers
27 views

Enumerating general recursive functions with a primitive recursive function

I am reading "Set Theory and the Continuum Hypothesis", a monograph by Paul Cohen. In the preliminary chapter, he gives a proof that not all recursively enumerable sets are recursive. He begins by ...
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1answer
33 views

Show that the language is regular - Closure

For languages $A$ and $B$, let the perfect shuffle of $A$ and $B$ be the language $$L=\{w \ \mid \ w=a_1 b_1 \dots a_k b_k, \text{ where } a_1 \cdots a_k \in A \text{ and } b_1 \cdots b_k \in B, ...
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1answer
32 views

Show that the language is regular

Let $$B_n=\{a^k \ \mid \ k \text{ is a multiple of } n\}$$ Show that for each $n \geq 1$, the language $B_n$ is regular. $$$$ Could you give me some hints how we coukd show this?? Do we have ...
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1answer
32 views

Construct the DFA of the language

I have to construct a DFA for the language $$\{w \mid w \text{ has exactly two } a's \text{ and at leat two } b's\}$$ To construct it we have to construct first the DFA's for the languages $$\{ w ...