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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4
votes
2answers
132 views

What if a conjecture were provably unprovable?

Suppose we found a proof that "The Twin Prime Conjecture cannot be proven", without any conclusion as to the conjecture itself being true or false. Is it then possible for the conjecture to be true? ...
9
votes
2answers
189 views

The mother of all undecidable problems

It is usual to show that a problem P is undecidable by showing that the halting problem reduces to P. Is it the case that the halting problem is the mother of all undecidable problems in the sense ...
0
votes
1answer
21 views

Algorithm that takes input desc. of two PDAs and outputs intersection of langs. recognized by two PDAs

Does there exist an algorithm which takes as input the descriptions of two pushdown automata, $P1$ and $P2$, and prints the description of another pushdown automaton which recognizes the intersection ...
0
votes
1answer
19 views

Deciding TM which fails to halt whenever the length of its input string is a prime number

I have the following Statement: "A TM called $A$ which fails to halt (i.e runs forever) whenever the length of its input string is a prime number, and eventually halts for all other input strings" ...
-1
votes
1answer
25 views

disproving union of infinitely many regular languages

I want to disprove the following statement: "if $L$ is the union of infinitely many regular languages, then $L$ is guaranteed to be a regular language." I don't know where to start. Any hint will be ...
4
votes
0answers
94 views

Is it decidable whether the iterates of a polynomial map are bounded?

Let $f:\mathbb{Q}^n\to \mathbb{Q}^n$ be a polynomial map with rational coefficients. Let $p\in \mathbb{Q}^n$. Is there a known algorithm that given this data determines whether or not the iterates ...
4
votes
0answers
49 views

predicate logic with assumption NP $\neq$ CO-NP?

Anyone could describe why: Set of All Tautology in propositional logic with assumption NP $\neq$ CO-NP is CO-NP Complete. Thanks. I ask it here before: Is the language of tautologies NP-complete? ...
0
votes
1answer
32 views

Computable Function and Predicate Question

I See on Our Lecture note on Theory of Computation Course that: .... The basic characteristic of a computable function is that there must be a finite procedure (an algorithm) telling how to compute ...
6
votes
1answer
79 views

Who first discovered that some R.E. sets are not recursive?

Who first discovered that some recursively enumerable sets are not recursive, or equivalently that some semidecidable sets are undecidable? And in what context? Was the earliest formulation of this ...
-2
votes
1answer
75 views

D={ $ deg_T (A) | A \subseteq N$} Problem [closed]

Dear friends I wanted to ask the question that already asked 2 times but it's on-hold and after few days deleted, but I didn't get any answer. I try to solve it but confused. I don't know anything and ...
1
vote
1answer
66 views

set theory, Incompleteness and axiomatic systems

Is the number of theorems that can be proved (decidable) within a certain set of axioms (for instance ZFC) is finite or infinite ? in other words, are we going to fully exhaust that set of axioms ...
1
vote
1answer
60 views

Is the given Language decidable or recognizable?

Let M be a machine that takes a natural number as input and outputs a natural number. Let L = $\{M:\;M(n)\;outputs\;a\;prime\;greater\;than\;n\;for\;every\;n\}$ Is L decidable? Is L recognizable? ...
2
votes
0answers
37 views

Decidability of a language

Let $C$ be a conjecture about natural numbers. Let $$S = \{n\in N: n > m \text{ where $m$ is the first number found for which $C$ is false} \} $$ Is $S$ decidable? If $C$ is true for all ...
2
votes
1answer
86 views

Many to one Reducible & Polynomial time

we know that If $A \le_p B$, then $A$ can be reduced to $B$ in polynomial time. we know that If $A \le_m B$, then $A$ is many to one reduction to $B$ . can we deduce that: if $A \le_m B$ then $A ...
0
votes
1answer
78 views

Why is $x\mapsto x$-th prime number a partial recursive function?

I think that partial recursive functions correspond to all computable functions. Thus, if we can write a computer program to represent a function, the function is partial recursive. In computability ...
0
votes
1answer
70 views

Undecidability of First Order Logic [closed]

friends! I read in Ebraham's Outline of Logic that first order logic is undecidable because it lacks an algorithmic procedure which reliably detects invalidity in every case. It is undecidable ...
0
votes
0answers
140 views

range of one increasing computation function?

We know that that the range of any recursive partial function is recursively enumerable. Also we know the fact: Set A is recursive if and only if it is range of some increasing section partial ...
0
votes
1answer
42 views

Problems On Many-one Reducible [closed]

In computability theory and computational complexity theory, a many-one reduction is a reduction which converts instances of one decision problem into instances of a second decision problem. ...
-1
votes
1answer
41 views

Set of Logical Result Problem [closed]

If we have a set of predicate formulas $A$, and there is an algorithm such that for every predicate formula $X$, (with input $X$), output YES iff $X \in A$. My question is about set of logical result ...
1
vote
0answers
37 views

Is a set $\{ e \in \mathbb{N} | \#\{x \in \mathbb{N} | \phi_e(x) \downarrow \} = \#\mathbb{N}\}$ computable?

Denote every partial computable function $f$ with its Godel number $e \in \mathbb{N}$ by $\phi_e$. Then let the halting set of $\phi_e$ be $W_e=\{x \in \mathbb{N} | \phi_e(x) \downarrow \}$ where ...
1
vote
2answers
58 views

How many recursively definable groups are there on $\mathbb{N}$?

How many non-isomorphic, (non-free), non-trivial, recursively definable groups are there on $\mathbb{N}$? I know we can at least get 1. Let $F:\mathbb{N} \to \mathbb{Z}$ be the "natural bijection". By ...
2
votes
1answer
36 views

Primitive-recursive functions and polynomial equations

I am looking for examples of primitive-recursive functions $f:\mathbb{N}\rightarrow\mathbb{N}$ that can not be written as a pair of polynomials, i.e. $$f(n) = m \Leftrightarrow P(n,m) = Q(n,m)$$ ...
4
votes
2answers
75 views

Uncomputability of subset relation

I suppose this obvious question should already be answered in plenty of places, but for some reasons I cannot find a proof of this anywhere. Prove or disprove that their exist a set $X$ that is ...
3
votes
1answer
70 views

Bijection between computable reals and rationals?

This wikipedia article http://en.m.wikipedia.org/wiki/Computable_number#Properties suggests that there is such a bijection. How does it look like? And how to map computable transcedentals like pi to ...
1
vote
1answer
70 views

Is there a more general proof for the halting problem?

Note:If this question is better suited for a different site, please tell me in the comments. Summary:Is there a proof for the impossibility of the halting problem that doesn't involve calling it on ...
1
vote
0answers
51 views

Is this proof for the undecidability of $\beta$-normalisation in $\lambda$-calculus valid?

The proofs I have so far seen for the undecidability of $\beta$-normalisation all make use of Gödel numbering in order to first prove the more general Scott-Curry theorem. As an exercise, I have tried ...
-1
votes
1answer
108 views

Many-one Reducibility Understanding Problem [closed]

We know for every set $B$, that be r.e have: $$B\leq_mK$$ (The set $B$ is many-one reducible, or m-reducible, to the set $K$) we know $K$ is r.e and define: $$K=\{ e:e\in W_e\}$$ my challenge is: ...
1
vote
1answer
74 views

Big Questions in First Order Logic

if $\Sigma$ is a r.e set (half decidable) of sentence in first order logic, the set of logical result of $\Sigma$ is Recursively Axiomatizable. why this is false? or maybe it's true? ...
2
votes
1answer
92 views

Is it decidable: is there an input for which turing machine will move its head left?

$L=\{\langle M \rangle | M $is a Turing machine and $\exists$ input $x$ such that in $M(x)$ running $M$ moves its head left at least once $\}$ Is $L$ decidable?
3
votes
2answers
87 views

Distinguishing sets according to more fine-grained notions than cardinality.

I'm interested in distinguishing sets according to more fine-grained notions than cardinality. Now I don't know a thing about computability theory, but it seems to me that considering sets up to ...
-1
votes
1answer
56 views

Computable Set & Function

we know that i read this sentence are true? can anyone say an example for following sentence? there are a non computable set A such that
0
votes
1answer
52 views

Logic & Computability Problem

i read this sentence in one exam that be false. anyone could say why? if predicate H(x) become false when a program with code r(x) halt on input l(x), then H be a computable predicate.
0
votes
1answer
91 views

Turing & Computability & Computation

We know if we have: we can show (T=t= Turin Redu.) but i have no idea why this relation be correct? any idea?
0
votes
3answers
31 views

Recursive Set in Partial Computable Function Problem

Suppose $A, B, C$ are disjoint set such as shown on this figure. $f_1(x), f_2(x), f_3(x)$ is partially computable function. why $A,B,C$ is recursive set?
2
votes
0answers
22 views

Computable models of ($\omega$, <) without computable isomorphism

I read somewhere that "it is easy" to construct a computable presentation for the model ($\omega$, <) so that any computable isomorphism between this construction and the usual presentation of ...
1
vote
1answer
31 views

What techniques are there to search for first order sentence equivalence?

Suppose we have a first order sentences $\phi$, $\psi$, and $\chi$ such that: $\phi$ $\longleftrightarrow$ ($\psi$ $\land$ $\chi$) And $\phi$ and $\chi$ are known or fixed. How can we search for a ...
1
vote
2answers
35 views

Proof that INF (the set of indices of Turing Machines that halt on infinitely many inputs) is not computably enumerable, I.e. $\not \in \Sigma_1^0$

I got curious about this today when looking for sets to proove aren't $\Sigma_1$ as exam prep. Unlike with its complement, FIN, a run of the mill contradiction was not easy to come by (perhaps I'm ...
3
votes
1answer
68 views

Proof that $\{ e \ | \ \forall p$ prime$: \varphi_e (p) \downarrow \}$ is not $\Delta_2$

This is a problem I've come across in my exam studies, and neither me nor my friend in the same course have been able to solve, so it would be good to see how it's done before the exam in a couple of ...
1
vote
0answers
48 views

“Building blocks” for computable functions

In an (otherwise very enlightening) answer to another question of mine the question came up What functions are allowed as building blocks for computable functions? I was astonished that there ...
2
votes
0answers
29 views

Generating interesting random TMs

To get a more intuitive understanding of the halting problem I want to generate some random TMs and see how they behave, what some heuristics can tell about them, etc. The problem is that, if I ...
2
votes
1answer
77 views

Functions with and without distinction of cases

Consider computable functions $f: \mathbb{N} \rightarrow \mathbb{N}$, given as formulas. I assume that it is clear for at least some of them whether they contain a distinction of cases: $$f(n) = n$$ ...
1
vote
0answers
36 views

Meaning of Biinterpretability.

I'm reading this paper: http://www.math.cornell.edu/~shore/papers/pdf/hyp9.pdf and I am struggling with the meaning of Biintereptability, to quote the paper A degree structure $D$ is ...
0
votes
1answer
57 views

Function Combination on Computer Science

I read some material on Computational Function, every one could describe the result of following combination? suppose $g_1(x)=3x$, $g_2(x)=4x$, $f(x,y)=x+y$, how we compute combination of $f$ with ...
0
votes
0answers
72 views

Primitive Recursive Predicate Challenge

I'm an Computer scientist, and I recently ran into a challenge. If we have primitive recursive predicate $P(x), Q(x)$, I think that all of following 4 expressions can be primitive recursive. Any hint ...
0
votes
0answers
48 views

Recursive Set and Complement Problem

if we have $$A=\{x:|W_x\ne\phi\}$$ can we say always my tight listed below is true? $A$ is recursive , $A$ is r.e, complement of $A$ is r.e, complement of $A$ is not recursive?
0
votes
1answer
80 views

Recursive Set Challenge

we knoe also we know for example if A be any arbitrary r.e set. can we always Necessarily the following is TRUE ? (always) any description is good. (bar sign means complement)
0
votes
0answers
46 views

Proof: All recursive functions are arithmetic (logic)

So I'm trying to understand the proof of the following statement: > All recursive functions are arithmetic The proof begins with: "It is sufficient to show that all arithmetic functions satisfy ...
2
votes
1answer
32 views

Hausdorff dimension of computable real numbers

This might be a trivial question but do the computable numbers have a positive Hausdorff dimension?
1
vote
0answers
63 views

Is the Mandelbrot set computable?

This is a weakened version of Is the measure induced by the Mandelbrot set computable on rational rectangles? ; Given a (computable, or rational) rectangle in the complex plane, is it computable ...
1
vote
0answers
57 views

Efficient algorithm for calculating the tetration of two numbers mod n?

I'm trying to study the algebraic properties of the magma created by defining the binary operation $x*y$ to be: $ x*y = (x \uparrow y) \bmod n $ where $ \uparrow $ is the symbol for tetration. ...