# Questions tagged [computability]

Computability theory a.k.a. recursion theory.

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### 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 ...
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### The range of Busy Beaver Function is immune set？

I am not familiar with Busy Beaver function ---BB(n). Some body assert that the range of BB(n) is not c.e. set, somebody even say that the range of BB(n) is not c.e. set and it is an immune set. But ...
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### Hardwiring advice (bit string) into Turing machine [closed]

In paper, page 5, 1st paragraph, it is stated that: Notice that an n-state Busy Beaver, if we had it, would serve as an O(n log n)-bit advice string, “unlocking” the answers to the halting problem ...
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### 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 ...
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### 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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### Computability of Time Complexity of Recursive Sets

Is every recursive set's worst-case time complexity a total recursive function?
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### 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, ...
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1 vote
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### 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 ...
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### 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 ...
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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 ...
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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 ...
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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 $f$-...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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-...
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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 ...
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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 ...
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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 ...
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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, ...
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### 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 ...
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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 ...
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