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# Questions tagged [computability]

Computability theory a.k.a. recursion theory.

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### Another weird $O(N \log{N})$ Turing machine

This is another question related to the (still open) nice question "Alphabet of single-tape Turing machine" by Emanuele Viola. I describe the question very informally (perhaps it has a trivial ...
2k views

### Alphabet of single-tape Turing machine

Can every function $f : \{0,1\}^* \to \{0,1\}$ that is computable in time $t$ on a single-tape Turing machine using an alphabet of size $k = O(1)$ be computed in time $O(t)$ on a single-tape Turing ...
879 views

### Why doesn't computer science follow biology more closely in computer design?

Nature has proved with the brain that it can create complex computers with very little energy consumed and released, extremely low energy compared to a common computer. However, I noticed the design ...
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, ...
291 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 ...
661 views

### 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 ...
1k views

### 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 $i_1, \ldots, i_K$ ...
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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 ...
464 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 ...
271 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. ...
155 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, ...
1k views

### 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 ...
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 ...
2k views

### 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-...
328 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 ...
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### 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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### 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
399 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 ...
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 ...
347 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 ...
327 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 ...
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### 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, ...
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 ...
2k 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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### 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. ......
374 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. ...
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?
600 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 ...
1k views

### 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: ...
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 ...
240 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 ...
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 ...
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### 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 ...
### 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 ...