Questions tagged [computability]

Computability theory a.k.a. recursion theory.

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Is predicting (in the limit) computable sequences as hard as a dominating function?

Define a "predicting oracle" to be an oracle that does as described in this question. default (weak) version: Is it the case that, for every predicting oracle $O$, there exists an oracle machine $M$...
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Transition Diagram of a Universal Turing Machine

I have searched the web for the transition diagram of a universal Turing machine without luck. Is anyone aware of such a diagram? I need this as a reference, so preferably a book or a published ...
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Decidability of equality of CFL's

Following problem is decidable: Given a context-free grammar $G$, is $L(G) = \varnothing$? Following problem is undecidable: Given a context-free grammar $G$, is $L(G) = A^{\ast}$? Is there a ...
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3answers
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Is there a complexity theory analogue of Rice's theorem in computability theory?

Rice's theorem states that every nontrivial property of the set recognized by some Turing machine is undecidable. I am looking for complexity-theoretic Rice-type theorem that tells us which ...
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3answers
569 views

Is the class of primitive recursion functionals equivalent to the class of functions which Foetus proves to terminate?

Foetus, if you have not heard of it, can be read up on here. It uses a system of 'call matrices' and 'call graphs' to find all 'recursion behaviors' of recursive calls in a function. To show that a ...
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How to make the Lambda Calculus strong normalizing without a type system?

Is there any system similar to the lambda calculus that is strong normalizing, without the need to add a type system on top of it?
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1answer
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Simulation of deterministic turing machines

What are the best known upper and lower bounds for simulating t steps of certain models of deterministic turing machines (1 tape, 1 tape with read only input tape, 2 tape, multi tape, with/without ...
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Intersection between context-free and context-sensitive language decidability [closed]

I'm trying to find a formal proof of the following fact: Given a context-free language, say $L_1$, and a context-sensitive language, say $L_2$, it is NOT decidable if their intersection is empty ($...
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Lovasz Theta as a short certificate

Lovasz Theta Function provides short proof for the question, "is the Shannon Capacity of a graph($\Theta(G)$) greater than $r\in\Bbb R$?" if the answer is NO when $r$ is above a certain value (this ...
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Foundational textbook(s) for Complexity and Computability on Real Numbers

It would be extremely helpful if someone can suggest foundational textbooks on Recursive Analysis (Computability over Reals) which explains connections between Computability and the Topological ...
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1answer
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Arithmetic Analogues of P versus BPP

In the arithmetic hierarchy, is there an analog of $P$ versus $BPP$? Particularly is there a notion of randomness there? If there is no such analogy, why is randomness in the resource bounded case ...
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2answers
441 views

Kolmogorov complexity with weak description languages

We can think of Kolmogorov complexity of a string $x$ as the length of the shortest program $P$ and input $y$ such that $x = P(y)$. Usually these programs are drawn from some Turing-complete set (like ...
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Universal Turing Machines in “Computational Complexity” by Papadimitriou

The first part of this question has been solved (see comments). In the book "computational complexity" by Papadimitriou, a Universal Turing Machine is given. But this machine is not concrete, in the ...
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1answer
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Is an infinite incomputable sequence random wrt a computable measure?

Take an arbitrary infinite binary sequence $\omega$. The interesting case is when $\omega$ is not computable. Is there a computable (semi-)measure $\mu$ such that sequence $\omega$ is $\mu$-random in ...
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6answers
807 views

Geometric Interpretation of Computation

Being from Physics, I have been trained to look into a lot of problems from a geometrical point of view. For example the differential geometry of manifolds in dynamical systems etc. When I read the ...
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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 ...
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Which model of computation to simulate to prove universality?

I am starting out in theoretical computer science. I have a model of computation based on observations of auto-associative memory in the brain. I believe (with little evidence) that I can do ...
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Can chess simulate a Universal Turing Machine?

I am looking to get a definite answer to title question. Is there a set of rules that translates any program into a configuration of finite pieces on an infinite board, such that if black and white ...
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About Inverse 3-SAT

Context: Kavvadias and Sideri have shown that the Inverse 3-SAT problem is coNP Complete: Given $\phi$ a set of models on $n$ variables, is there a 3-CNF formula such that $\phi$ is its exact set of ...
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3answers
327 views

Is there any programming language in which any equivalent program has a unique, decidable normal representation?

Is there any programming language in which any equivalent program has a unique normal representation, and that normal representation is decidable? Is other words, suppose A and B are programs ...
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0answers
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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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1answer
225 views

Is there computable function to compute each computable real number?

Recall that a computable real number is one which can be calculated to any precision, like $\pi$ or $e$. It does not matter that these numbers are irrational, computability is about being able to ...
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Reducing threshold questions to finiteness questions

It is usually simpler to reason about calculus where the limitation is finiteness of computation rather than a threshold like "computable in polynomial amount of time". In formal languages theory for ...
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1answer
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Is it decidable whether the langauge accepted by a reversal-bounded counter machine is deterministic?

I'm wondering if anyone can point me to either an algorithm or an undecidability proof for the following question: Given a non-deterministic reversal-bounded multicounter machine $M$, is there some ...
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4answers
834 views

Can testing show the absence of bugs?

$(n + 1)$ points are required to uniquely determine a polynomial of degree $n$; for instance, two points in a plane determine exactly one line. How many points are required to uniquely determine a ...
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6answers
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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 ...
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1answer
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Complete problems and universal simulator machines

I'm trying to get straight in my mind the relation between complete problems and universal simulator machines. Some notions of computability have universal machines (Turing-computability) and some ...
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1answer
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n-approximable functions

I came across the following definition in a paper: We can extend the notion of an $n$-c.e. [n-computably enumerable] set to a notion that measures the number of fluctuations of a function as folows: ...
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1answer
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Explanation of 1-generic to prove undecidability of halting problem

This question is about an answer in question Are there any proofs the undecidability of the halting problem that does not depend on self-referencing or diagonalization ? Bjørn Kjos-Hanssen answer ...
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Reference Request: Oracle applications outside cryptography

Oracles have been used to prove results in cryptography where all parties have access to a random oracle instantiated with some cryptographic primitive. I am looking for references to papers that have ...
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3answers
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Template Metaprogramming and Turing Completeness

I am trying to design a Turing Machine using C++ Template Metaprogramming. What steps must be taken to ensure that the code that I'm gonna write will actually build a Turing machine ? I have read that ...
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2answers
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Primitive Recursive Isomorphisms

What is the relationship between invertible primitive recursive functions (that is, a primitive recursive function that is an isomorphism) and all primitive recursive functions? Can every primitive ...
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2answers
255 views

Runtime of a TM enumerator

Is there a way to find out the time bound between 2 consecutive strings enumerated by a TM (the TM that decides this language is promised to run in linear time)? For simplicity let's say the string ...
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2answers
600 views

how do you turn an algorithm for a decision problem into an algorithm for an optimization problem?

It is well-known, I believe, that theoretically, in quite a few cases, an algorithm that solves a decision problem can be turned into an algorithm that solves the corresponding optimization problem. ...
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320 views

Lower bound on number of oracle calls for solving $n$ instances of the halting problem

I encountered the following question, which is an easy exercise (spoiler below). We are given $n$ instances of the halting problem (i.e. TMs $M_1,...,M_n$), and we need to decide exactly which of ...
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1answer
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What is a reasonable representation/encoding of objects? [closed]

Question What is a reasonable representation of objects (for computability)? What is the criteria that we should apply to see if a representation is reasonable? This answer by Andrej suggests ...
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5answers
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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 <...
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1answer
277 views

Are randomly generated infinite patterns computable?

Fix a prefix-free universal Turing machine $U$. Consider the following random process*. The state of the process is a bit-string $s$, initialized with the empty string (say). Suppose the value of the ...
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5answers
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Problem teaching computability

I have difficulty teaching the concept of computable functions. I tried to develop the idea of why researchers like Hilbert/Ackermann/Godel/Turing/Church/... invented the notion of 'computability'. ...
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2answers
281 views

Simply-stated restriction on imperative programming language that captures the elementary functions?

The language of while programs can express the computably enumerable functions. (This is true even if the only arithmetical operations on variables are, say, ...
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2answers
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The relation of Gödel's Incompleteness Theorems to the Church-Turing Thesis

This may be a naive question, but here goes. (Edit -- it is not getting upvotes, but nobody has offered a response either; perhaps the question is more difficult, obscure, or unclear than I thought?) ...
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5answers
531 views

Natural computation based on fundamental forces

Well-known examples of computation inspired by natural phenomenon are quantum computers and DNA computers. What is known about the potential and/or limitations of computing with Maxwell's laws or ...
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Efficient asympotically universal predictors

A computable predictor is an algorithm $A$ computing a function $f_A : \{0,1\}^* \rightarrow \{0,1\}$. We regarding the function as providing a predicted continuation of a finite binary sequence. We ...
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1answer
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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 ...
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Abstract definition of universal computation

There are many universal computation systems. Turing machines, tag systems, rewrite systems, cellular automata to name just a few. The universality of a system is proved via reduction from a known ...
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Is the concept of the Turing Machine derived from automata?

I was just recently having a discussion about Turing Machines when I was asked, "Is the Turing Machine derived from automata, or is it the other way around"? I didn't know the answer of course, but I'...
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597 views

Is it possible to decide $\beta$-equivalence within System F (or another normalizing typed λ-calculus)?

I know that's impossible to decide $\beta$-equivalence for untyped lambda calculus. Quoting Barendregt, H. P. The Lambda Calculus: Its Syntax and Semantics. North Holland, Amsterdam (1984).: If A ...
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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 ...
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2answers
369 views

Difference between Stencil -structures and Cellular Automata Category-theoretically?

Definitions Stencil = "For a given point, a stencil is a pre-determined set of nearest neighbors (possibly including itself)." (source) Wikipedia's definition (source) = It looks ...
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1answer
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How to describe the set of “all computable functions” using Coq?

Would the set of all computable functions be just the set of all maps of the form f : forall n : nat, P n -> nat where ...

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