Questions tagged [computability]

Computability theory a.k.a. recursion theory.

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9
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4answers
2k views

Why are linear bounded automata not as popular as other automata?

In my experience, context-sensitive languages and linear bounded automata are frequently skipped or breezed over in computability theory courses, and are even left out of some notable text books, ...
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2answers
289 views

Reference for checking primitive recursiveness

There is a theorem that states a function $f$ can be computed with a Turing-machine in time $O(g)$ with primitive recursive $g$ (of the length of input) iff $f$ is primitive recursive. Where can I ...
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1answer
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Undecidable problems not Turing-complete?

are there systems whose nontrivial properties can't be decided by Turing machines, but for which a Turing machine with an oracle able to find out these properties isn't able to solve the Halting ...
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2answers
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Post Correspondence Problem variant

This is probably pretty simple, but consider the standard Post Correspondence Problem: Given $\alpha_1, \ldots, \alpha_N$ and $\beta_1, \ldots, \beta_N$, find a sequence of indices $i_1, \ldots, i_K$ ...
27
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4answers
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Church's Theorem and Gödel's Incompleteness Theorems

I have recently been reading up on some of the ideas and history of the ground-breaking work done by various logicians and mathematicians regarding computability. While the individual concepts are ...
3
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2answers
461 views

Does an infinitely long tape make any difference when proving theorems about Turing Machines?

Standard accounts of Turing Machines in the literature assume an infinitely long tape in at least one direction (and indeed infinitely time long to perform its computations). Clearly in practice no ...
5
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1answer
269 views

The Complexity of Advice in Computational Indistinguishability

One of the cornerstones of the modern cryptography is the definition of computational indistinguishability: It is used in definition of cryptosystems, pseudorandom generators, zero-knowledge, etc. ...
8
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1answer
153 views

Is testing two SO-Horn queries for equivalence decidable?

It follows from Rice's theorem that you cannot determine whether or not two Turing machines decide the same language. My question is: Does this also apply in descriptive complexity settings, ...
-5
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1answer
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Alternative Turing Machine Proofs

I am asking this question again. I am aware of, and have read the other similar "alternative proof TM" questions, but unfortunately, they do help me. I am looking for a TM Halting Problem proof that ...
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2answers
1k views

Is there an alternative proof of the TM Halting Problem other than the “standard” one? [closed]

I'm wondering if anyone is aware of a proof of the Halting Problem that is not just a permutation of the "standard" proof. Since there are so many formulations of this proof, rather than pick a ...
44
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5answers
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Historical reasons for adoption of Turing Machine as primary model of computation.

It's my understanding that Turing's model has come to be the "standard" when describing computation. I'm interested to know why this is the case -- that is, why has the TM model become more widely-...
8
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0answers
323 views

Are there any models of computation currently being studied with the possibility of being more powerful than Turing Machines? [duplicate]

Possible Duplicate: What would it mean to disprove Church-Turing thesis? Are there any models of computation currently being studied with the possibility of being more powerful than Turing ...
7
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2answers
746 views

An example of a totally computable function that is not definable in system T?

Could you give me an example of a totally computable function of type N × N → N that is not definable in System T? Thanks.
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4answers
846 views

Survey Article on the Theory of Recursive Functions?

Could you recommend a survey article or textbook chapter that introduces the theory of recursive functions? Thanks
5
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1answer
392 views

Efficient Algorithm for bilinear pairing on ECC

Bilinear Pairing in the elliptic curves is a wonderful mathematical mapping which is usually defined by the map $e:G_{1} \times G_{1} \rightarrow G_2$ for some groups of $G_{1}$ and $G_{2}$. For ...
7
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3answers
1k views

Are there languages that are not in RE nor CO-RE?

I am familiar with the theorem which states that some languages are not in the RE (Recursively Enumerable) class of languages, but that can mean either that they are all in CO-RE (or rather, the part ...
10
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1answer
346 views

Computational consequences of Friedman's (unprovable) Upper Shift Fixed Point theorem?

Harvey Friedman showed that there is a neat fixed point result that cannot be proved in ZFC (the usual Zermelo-Frankel set theory with the Axiom of Choice). Many modern logics are built on fixed ...
10
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1answer
325 views

Is predicting (in the limit) computable sequences as hard as the halting problem?

Question: Is predicting (as defined below) computable sequences as hard as the halting problem? Elaboration: "Predict" means successfully predict, which means make only finitely many errors on the ...
10
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1answer
221 views

Uniform hierarchy of problems that span complexity and computational hierarchies

Does anyone know of a set of problems that vary uniformly and span one of the "interesting" hierarchies of complexity and computability? By interesting, I mean, for example, the Polynomial Hierarchy, ...
15
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0answers
1k views

Can a Penrose tile cellular automaton be Turing-complete?

This question was based on an incorrect premise ... see Colin's comment below. Forget it. This was inspired by the discussion on this Math Overflow question. First, I need to define our terms. In a ...
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2answers
1k views

What do we know about restricted versions of the halting problem

(UPDATE: a better formed question is posed here as the comments for the accepted answer below show that this question is not well-defined) The classical proof of the impossibility of the halting ...
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3answers
516 views

Is one definition of the word paradox, “something that can be used to prove the halting problem undecidable?”

I've been toying recently with the idea that the set of all "real-world" paradoxes...such as the Liar's Paradox, Russell's Paradox, the Unexpected Hanging Paradox, the Berry paradox, etc. ......
8
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2answers
373 views

Relativization with Respect to Non-Recursive Oracles

In the paper Relativizations of the P = ? NP Question, Baker et al. showed that there are relativized worlds in which either P = NP or P ≠ NP holds. All oracles in their settings were recursive sets. ...
8
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3answers
449 views

Repeated Quine program

A Quine is a computer program which produces a copy of its own source code as its only output. Is there any Quine program that could print itself out n times, with n specified some way in the program?
6
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1answer
591 views

Definition of a monotone machine.

There is a definition of a 'monotone machine' in Li & Vitany's Book, and another one which is for example stated in this paper via c.e. (computably enumerable) sets. I can't see why these ...
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1answer
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Question about Mapping Reductions (Clarify Example)

I cannot for the life of me wrap my head around these reductions. Specifically, the example I'm wrestling with: ...
29
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4answers
3k views

Can a probabilistic Turing machine solve the halting problem?

A computer given an infinite stream of truly random bits is more powerful than a computer without one. The question is: is it powerful enough to solve the halting problem? That is, can a ...
12
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3answers
236 views

Is there a natural restriction of VO logic which captures P or NP?

The paper Lauri Hella and José María Turull-Torres, Computing queries with higher-order logics, TCS 355 197–214, 2006. doi: 10.1016/j.tcs.2006.01.009 proposes logic VO, variable-order logic. This ...
3
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2answers
690 views

Is the following language in RE?

There's a computability theory exercise that had me stuck for a long time. I have the language L={ M | there exists a reduction from HP to L(M) } I managed to ...
39
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7answers
4k 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 ...
6
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2answers
599 views

Computable function $f = \Theta(g)$ with $g$ uncomputable

This question most likely has a simple answer; however, I do not see it. Let $g:\mathbb{N} \rightarrow \mathbb{N}$ be an uncomputable function and $c$ a positive real number. Can there be a ...
15
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3answers
717 views

Can it be determined if language L lies in NP?

Given a language L defined by a Turing Machine that decides it, is it possible to determine algorithmically whether L lies in NP?