Questions tagged [computability]

Computability theory a.k.a. recursion theory.

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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 ...
gigabytes's user avatar
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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 ...
user70015's user avatar
9 votes
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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\...
Gro-Tsen's user avatar
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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 ...
Colonizor48's user avatar
3 votes
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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 ...
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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 ...
Corbin's user avatar
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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 $f$-...
Pooya Farshim's user avatar
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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 ...
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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, ...
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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 ...
Alex Meiburg's user avatar
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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 ...
Shane Gervais's user avatar
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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 $y$.&...
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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 ...
ghostOfTomJoad's user avatar
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Can you see that the Linz Halting Problem proof contains a fatal flaw?

Applying a Simulating Halt Decider to the Linz Halting Problem Proof When a simulating halt decider correctly simulates N steps of its input it derives the exact same N steps that a pure UTM would ...
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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 ...
Vivek Joshy's user avatar
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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 ...
Noel Arteche's user avatar
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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 ...
Abhimanyu Pallavi Sudhir's user avatar
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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 ...
Jozef Mikušinec's user avatar
2 votes
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Are there more learnable but undecidable cases except the halting problem

Per request, I cross post the question here which is original from math.stackexchange In the ICML 1996 paper, On the Learnability of the Uncomputables, by Richard Lathrop, he proved that halting ...
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What are the application of Scott-Topology in theoretical computer science?

During a work I came across the Scott-Topology and I see that Scott-continuous functions show up in the study of models for lambda calculi. What I cannot understand is how this enrich the lambda-...
micrus's user avatar
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Understanding the construction of an uncomputable function

The following is from Arora and Barak's "Computational Complexity." I think one does not have to read the second paragraph of the proof to answer this question. Theorem 1.10 There exists a ...
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How can I switch into computability theory when I am part way through my PhD in deep learning?

I am partway through my PhD studying deep learning. I chose it just because it's useful and would yield a lot of industry opportunities. However, I am really missing my previous coursework in ...
JobHunter69's user avatar
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Expressive power of lambda-calculus with restricted application

Consider a syntactic restriction of the (untyped) $\lambda$-calculus in which an application cannot have another application as an immediate subterm. More precisely, restricted terms ($R,S,...$) and ...
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Why are some classes (ALL, ELEMENTARY, R, etc) badly behaved as oracles?

Some classes, such as ALL, ELEMENTARY, and R, are very badly behaved when used as oracles. For instance, all three of these classes trivially collapse P and EXP, even though (by the Time Hierarchy ...
Demi's user avatar
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Detailed proof of Theorem 2.1 in Papadimitrou book (Multitape TM to SingleTape TM)

I want to know if anybody knows a detailed proof of Theorem 2.1 of Papadimitrou's book Computational Complexity. The theorem states "Given any $k$-string Turing machine $M$ operating within time $...
arbolverde's user avatar
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Halting problem for finitary PCF

Is the halting problem decidable for finitary PCF? By "halting problem" I mean the problem of deciding whether a closed PCF term evaluates to bottom under the denotational semantics of PCF. ...
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Is there a fundamental link between Nash's equilibrium and Turing's halting problem?

Since Nash equilibrium exists, is there a computational analogue of this equilibrium point? I am trying to approach Nash equilibrium from computational point of view to see if the equilibrium point ...
Ajax's user avatar
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What is a known sequence for which being constant is not provable?

My question concerns the property of being constant for computable functions ${\mathbb N}\to \{0,1\}$, within any common framework $T$ strong enough to include Heyting arithmetic (and of course not ...
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What is the complexity class of higher-order primitive recursion?

Short and sweet: The complexity class of primitive recursive functions (WP) is PR (WP, Zoo). What's the complexity class of higher-order primitive recursion (WP)? The motivating context is simply that ...
Corbin's user avatar
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A variant of two-counter machine

I would like to show that the halting problem for some variant of two counter machine (Minsky machine) is undecidable: instead of "if c=0 goto i else goto j", there are "if c>d goto ...
Naama Shamash Hal's user avatar
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1 answer
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Is coRE closed under concatenation?

I know that RE is closed under union, intersection, and concatenation (but not complement). It is likewise easy to show that coRE is closed under union and intersection (but not complement). What ...
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Is BigInteger-based Brainfuck Turing Complete?

All of the proofs of Turing-Completeness I've found for Brainfuck rely on its cells being fixed-width integers that wrap around upon over/underflow. The "parent language" P'' on which ...
user513093's user avatar
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Halting behavior of a randomly selected Turing machine?

Let $TM(k,2)$ be the set of Turing machines with $k$-states and $2$ symbols. Let $h(k)$ be the number of machines in $TM(k,2)$ that halt when run on the blank input. Is $\lim_{k \to \infty} \frac{h(k)...
user101010's user avatar
3 votes
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Is relativization statement-dependent or proof-dependent?

I'm relearning some computability theory, and have encountered the idea of relativization of results to arbitrary subsets of $\omega$ and the subtlety of figuring out what the correct relativized ...
Alex Mine's user avatar
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1 answer
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Where, if any, is there currently any research being done on the subject of ternary computers? [closed]

I had the experience several years ago of working with a team that had developed a ternary computing system. It ran out of funding and was abandoned but I felt it was ahead of its time. Currently, ...
tale852150's user avatar
3 votes
1 answer
296 views

Proof and computational complexity

I couldn't find documents elaborating on this: if the Curry Howard correspondence is to be interpreted as establishing a strong relation between proofs and programs, should there not be a strong ...
user1868607's user avatar
1 vote
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A proved computationally-irreductible function

In arXiv:1111.4121 and arXiv:1304.5247, Hervé Zwirn and Jean-Paul Delahaye propose a formal definition of computational irreducibility. Is there a (possibly artificial) function that can be proved to ...
Bob's user avatar
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What is the difference between a model of computation and a programming language?

https://en.wikipedia.org/wiki/Model_of_computation includes sequential models, functional models and concurrency models. Sequential models include finite state machine, Turing machines, random access ...
Tim's user avatar
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Is this variant of bitwise cyclic tag Turing-complete?

Cross-posted to MathOverflow. CT is an extremely minimalist programming language that can simulate arbitrary tag systems, and is therefore Turing-complete. A program consists simply of a string of 3 ...
user76284's user avatar
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3 votes
1 answer
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What is the meaning of 'uniform' in Blum's paper "On the Size of Machines"

In Blum's "On the Size of Machines" he uses the word 'uniform' in several places. For instance, in Theorem 1, he writes Let $f$ be any recursive function. Then there exists $i,j\in \mathbb{...
user10108's user avatar
15 votes
1 answer
693 views

Computability and continuity

Say $L_1$ and $L_2$ are computable languages. Let $f$ be a function $L_1 \rightarrow L_2$. Let $C$ be the statement, "if $l \subseteq L_2$ and $l$ is a computable language, the preimage $f^{-1}(...
causative's user avatar
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1 answer
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Are all computable functions monoidal from a category theory POV? [closed]

I have been studying some category theory only to understand the "why" behind the various categorical constructs we use ("borrow") in the world of functional programming. One of ...
PhD's user avatar
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16 votes
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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-...
Gary Hoppenworth's user avatar
1 vote
1 answer
268 views

How come Wikipedia says that Random Turing Machines can provide uncomputable output?

Wikipedia article mentioned : Hypercomputation The third paragraph starts off with: Technically, the output of a random Turing machine is uncomputable; however, most hypercomputing literature focuses ...
Novicegrammer's user avatar
1 vote
1 answer
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What kind of computational model is the brain? [duplicate]

I was wondering what kind of computational model is the human brain (as it seems superior to a Turing machine). Another thing that should be a separate question, What would be a perfect computer model ...
Aether's user avatar
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12 votes
2 answers
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Complexity relative to the graph of the Busy-Beaver function

This question is inspired by the comments made on this other question that I asked, and by an attempt to provide an explicit example of a complexity question beyond the Turing degree $\mathbf{0}$. (...
Gro-Tsen's user avatar
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Has there been any development of complexity theory for other Turing degrees than 0?

(I'm not sure if this question is better suited for MathOverflow or here. I'll try here first, and move over to MO later if it appears to be more appropriate.) Complexity theory can be very broadly ...
Gro-Tsen's user avatar
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Given a program specification, S, what can be said about the size and efficiency of programs that exactly satsify S, with respect to the size of S?

Suppose we are given a program specification, $S$, and we want to reason about programs $P$ that satisfy $S$. One might like to think that if the specification is 'simple', the the program should be '...
user10108's user avatar
4 votes
0 answers
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Logic of learning

Does Robust logic (Leslie Valiant), Default logic (Raymond Reiter) and Circumscription logic (John McCarthy) have any relation? I was Mathematician and Computer Science (dual degree undergraduate) ...
Mahdi Heidarpoor's user avatar
-2 votes
1 answer
116 views

Definition of Omega Jump [closed]

In the book Computability theory (Rebecca Weber) I stumbled about Exercise 7.1.24 with the definition of the omega jump. The book says: The omega jump of $A$, $A^{(\omega)}$ is the join of all $A^{(n)}...
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