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Questions tagged [computability]

Computability theory a.k.a. recursion theory.

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Integration of analytic function

It is the continuation of the question: Complexity of analytic functions and integrals. Given an input integer $t$ and a sequence of analytic functions $f_n(...(f_0(x))$, with parameters $t$ itself, $...
5 votes
2 answers
247 views

What are the computations model with a constant slowdown ? (and why do we care about Turing machines)

One problem with Turing machines is that the universal Turing machine $U$ can simulate any machine $M$ but with a $\log$ slowdown, meaning $U(\#M,x)$ runs in time $O(T(n)log(n) + O(n))$, where $n=|\#M|...
-13 votes
3 answers
1k views

What happens when the Linz halting problem proof is based on simulation of the input?

Applying a Simulating Halt Decider to the Linz Halting Problem Proof Of course it is obvious that no halt decider H can possibly return a correct halt status for any input defined to do the opposite ...
0 votes
1 answer
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Deciding whether there are directed paths between two vertices of all possible lengths

I recently read a paper The presence of a zero in an integer linear recurrent sequence is NP-hard to decide by Blondel and Portier, in which they prove the statement The problem of determining for a ...
8 votes
1 answer
168 views

Primitive recursive permutations

How to show that the inverse of a primitive recursive permutation of $\mathbb{N}$ is not necessarily a primitive recursive function?
5 votes
1 answer
190 views

Does every computable function have infinitely many "non-padded" representations?

It's well-known that every computable function has infinitely many representations (when they're expressed via recursive functions, or programs, etc.). I'm trying to understand whether there are ...
2 votes
1 answer
280 views

What kind of string is produced by successive application of argmax M

Fix a version of Solomonoff's universal distribution $\mathbf M$ and consider the following procedure for generating an infinite binary sequence $\omega$. Start with some $\omega_0$. Each subsequent ...
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0 answers
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By Gödel numbering, is the set of computable(partially) transcendental numbers an immune set, productive set? [duplicate]

Every Turing Machine computing(output) real number is encoded as a natural number, namely, admissible numbering. Then what is the set of computable (partially) transcendental numbers? Is it an immune ...
-1 votes
1 answer
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Computability/Complexity of optimization problems in general

Dear StackExchange community, I have a question, or better phrased I am confused and would like to be enlightened by you! So assume we have a (optimization) problem like that: Instance: Let $f:\...
1 vote
0 answers
108 views

Why does the Time Hierarchy Theorem fail relative to promise problems?

Define Program Evaluation (PE) to be the promise problem of determining whether a program (written in a Turing-complete language) returns True or False. The promise is that the program will return ...
13 votes
4 answers
1k views

Decidability of transcendental numbers

I have a question, whose answer is probably well known, but I can't seem to find anything meaningful after a bit of searching, so I would appreciate some help. My question is whether it is known that ...
23 votes
6 answers
5k views

Is it possible to test if a computable number is rational or integer?

Is it possible to algorithmically test if a computable number is rational or integer? In other words, would it be possible for a library that implements computable numbers to provide the functions <...
21 votes
3 answers
2k views

Smallest possible universal combinator

I am looking for the smallest possible universal combinator, measured by the number of abstractions and applications required to specify such a combinator in the lambda calculus. Examples of universal ...
25 votes
3 answers
15k views

How is proving a context free language to be ambiguous undecidable?

I've read somewhere that a Turing machine cannot compute this and it's therefore undecidable but why? Why is it computationally impossible for a machine to generate the parse tree's and make a ...
2 votes
0 answers
79 views

Many-one equivalence of sets that differ finitely

[This is a duplicate of my question from Mathematics Stack Exchange: https://math.stackexchange.com/questions/4792354/many-one-equivalence-of-sets-that-differ-finitely I am posting it here since it ...
3 votes
2 answers
529 views

Technical limitations of Turing machines due to the input and output encoding of values

Convention: Since I will be asking about some technicalities around Turing machines, it behooves to give a precise definition: say, here, “Turing machine” will stand for a $2$-symbol $1$-tape machine ...
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130 views

Computational power of probabilistic automata

I am a bit confused about the proper role of probabilistic automata (PA) in the theory of computation. Informally, I can imagine they can accept more than finite automata (FA) as they, for instance, ...
1 vote
1 answer
63 views

Many-one degrees of some particular sets

Let $W_0, W_1, W_2,\dotsc$ be an effective numbering of r.e. sets. Consider sets $\text{Emp}=\{x\mid W_x=\emptyset\}$, $\text{Tot}=\{x\mid W_x=\mathbb{N}\}$ and $S_n=\{x\mid W_x=W_n\}$ (for some fixed ...
1 vote
0 answers
174 views

Decidability of the complexity of decision problems

This might be a question that is related to some of the existent questions on the topic in the title, but I still find some answers either not full, or the topic still slightly different (maybe due to ...
1 vote
1 answer
129 views

Primitive recursion with varying parameters

Suppose $g\colon \mathbb{N}^k \to \mathbb{N}$, $v_1,\ldots,v_r\colon \mathbb{N}^k \to \mathbb{N}^k$ and $h\colon \mathbb{N}^{k+r+1} \to \mathbb{N}$ are all primitive recursive, and define $f\colon \...
1 vote
1 answer
132 views

Complexity of analytic functions and integrals

There exist polynomial - time computable functions, log - space computable functions, and NC - functions. Given this: To which class do analytic elementary functions, including trigonometric ones, ...
1 vote
0 answers
69 views

Primitive recursion relative to a logical system

In various places I have read that the normally considered non-primitive recursive Ackermann function is primitive recursive in higher-order logic. It's claimed to be due to "Reynolds, 1985",...
40 votes
7 answers
9k views

What's the simplest noncontroversial 2-state universal Turing machine?

I'm wanting to encode a simple Turing machine in the rules of a card game. I'd like to make it a universal Turing machine in order to prove Turing completeness. So far I've created a game state ...
2 votes
0 answers
62 views

Complexity measures for semi-decidable problems

Is there any sensible complexity measure that makes sense to compare the "hardness" of undecidable semi-decidable problems? Time and space are of course not suitable, because they cannot be ...
7 votes
1 answer
301 views

Is there a well-defined notion of an “R/poly” complexity class?

This would be the complexity class of all problems that are decidable in finite time with a polynomial length advice string that can be arbitrarily hard to compute. But potentially undecidable without ...
9 votes
1 answer
418 views

Is P=NP relative to the halting oracle?

Consider the following variant $\mathscr{H}$ of the halting oracle: given the code $e$ for an ordinary Turing machine and an input $n$ to it, we let $\mathscr{H}(\langle e,n\rangle) = \langle 0,0\...
0 votes
0 answers
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What is the meaning of the additive epsilon term in the definition of a time constructible function?

There is a well-known theorem that states that a function $f$ is time constructible if and only if $f$ can be computed in time $O(f)$. But this theorem comes with some conditions: $f$ must be a ...
3 votes
2 answers
153 views

What is formal definition of non-deterministic algorithm in context of primitive/general recursion?

I want to understand general method for formally defining non-deterministic algorithm. But all formal definitions I see are related to FSM/Turing-machines. What is the reference for non-deterministic ...
1 vote
2 answers
129 views

Do realizable systems always have some non-well-founded sets?

Suppose we are standing outside a realizable system which admits CZF or a similar constructive set theory. Then consider the following: LEM is not realized (e.g. this MSE answer) The traditional ...
16 votes
1 answer
369 views

Is there a language of first-order logic such that every r.e. set is Turing-equivalent to some finitely axiomatizable theory in that language?

I hope that mathematical logic / recursion theory type questions are welcome here. I am sorry this question is so long and technical, but I believe that if you read it you will find that it is well-...
10 votes
3 answers
512 views

Complexity results for Lower-Elementary Recursive Functions?

Intrigued by Chris Pressey's interesting question on elementary-recursive functions, I was exploring more and unable to find an answer to this question on the web. The elementary recursive functions ...
3 votes
0 answers
83 views

Generalizing Quines: Outputting an Arbitrary Function of Source Code

​​A Quine is a (non-empty) program $P$ that takes no inputs and returns its own source code $\langle P\rangle$ as the only output. For a function $f$ (with appropriate domains and range) define an $f$-...
4 votes
0 answers
320 views

How to prove that a language that allows infinite loops is still not Turing-complete?

CPS translations will always use pairs, either explictly or by currying. Though I can't find a reference for that, I'm assuming this is a necessary condition (I'd appreciate a reference if someone has ...
0 votes
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Input-Output Machines

From what I know, there is a vast literature on language recognizers in computer science. Language recognizers are machines (e.g., Finite State Automata, Pushdown Automata, Turing Machines, ...) that, ...
10 votes
1 answer
615 views

A simple proof that decidability of typability in System F ($\lambda 2$) implies decidability of type checking?

Suppose we don't know Joe B. Wells's result from 1994 that both typability and type checking are undecidable in System F (AKA $\lambda 2$). In Barendregt's Lambda calculi with types (1992) I found a ...
4 votes
0 answers
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Reference for "A Turing machine cannot be equipped with an oracle for itself"

Starting with a basic Turing machine, successive applications of the Turing jump produce successively more powerful machines, that can be indexed by any ordinal. A tempting error for students, and one ...
0 votes
0 answers
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Computability for universal quantum turing machines

I would like to ask if anyone has any ideas about what a universal quantum turing machine (UQTM) can do as supposed to a classical universal turing machine (UTM) (i.e. quantum computer vs classical ...
13 votes
4 answers
836 views

Can any program be implemented mechanically?

Is it possible to build a single purpose (non Turing complete) mechanical implementation of say, Microsoft Word? Is it possible to implement such things as iterators, first-order functions, the whole ...
9 votes
1 answer
172 views

What is known about reduction by "$P_1$ interprets $P_2$" for generalized programming languages?

Inspired by this answer, let's say that a programming language is given by the data $L=(P,ev)$ where $P$ (the set of "valid programs") is a computable subset of $\Sigma^*$ and $ev$ (the "evaluator") ...
8 votes
0 answers
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Does Bellantoni-Cook safe recursion (or any other implicit characterization of P) admit Kleene's second recursion theorem?

Abstractly, by a programming language that operates on binary strings I mean a set $P$ of programs along with a semantics relation $[p](x) = y$, ``the program $p$ on string $x$ halts with output $y$.&...
27 votes
6 answers
3k views

How are real numbers specified in computation?

This may be a basic question, but I've been reading and trying to understand papers on such subjects as Nash equilibrium computation and linear degeneracy testing and have been unsure of how real ...
12 votes
1 answer
5k views

Why study type theory?

After reading the literature on type theory (especially the constructive kind - CTT) I'm left wondering "why" should one study type theory, specifically within the confines of "computing" in general? ...
1 vote
1 answer
112 views

Extension of primitive recursion, that is as powerful as System-T

I know that System-T restricted to first-order types is exactly as powerful as primitive recursive functions, because I proved it in Agda. I asked myself, if there is a extension of primitive ...
44 votes
7 answers
6k views

Truly random number generator: Turing computable?

I am seeking a definitive answer to whether or not generation of "truly random" numbers is Turing computable. I don't know how to phrase this precisely. This StackExchange question on "efficient ...
20 votes
2 answers
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What is the "nearest" problem to the Collatz conjecture that has been successfully resolved?

I am interested in the "nearest" (and "most complex") problem to the Collatz conjecture that has been successfully solved (which Erdos famously said "mathematics is not yet ...
0 votes
0 answers
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Are there parsers that can parse languages in more expressive grammars than context-free grammars?

The Earley Parser is able to parse all context-free languages. Are there parsers that can parse say languages in context-sensitive grammars? I realize ambiguous grammars are non-deterministic and ...
21 votes
5 answers
5k views

Can a computer simulate itself as part of a simulated world?

Let's say you build a computer that will calculate the state of all atoms in the Universe at certain future point in time. Because the Universe is, by definition, everything that exists (and anything ...
2 votes
0 answers
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Gurevich's theorem on primitive recursive functions being logspace-computable

I recently came across the following result attributed to Gurevich, according to which I understood that the class of problems solvable by primitive recursive functions is precisely the class L of ...
1 vote
0 answers
61 views

Generalization of computability to continuous for loops? [closed]

A computable function, formulated in the sense of mu recursion, can compute a for or do loop over some (possibly infinite) integer range. I was wondering if a suitable generalization exists that ...
6 votes
1 answer
280 views

An analogue of Scott continuity for infinite-time-Turing-computable functions

$\newcommand{\tb}{2_{\!\bot}}\newcommand{\tbO}{\tb^{\,\omega}}$Let $2 = \{0,1\}$ and $\tb = \{0,\bot,1\}$. Scott continuity is important for defining models of lambda calculus, a formalism for ...

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