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Questions tagged [computability]

Computability theory a.k.a. recursion theory.

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86 views

Solving the Halting problem for most inputs [closed]

Is it possible to solve the following version of the Halting problem : given any Turing machine and some input tape, the program should answer if this pair halts or not except possibly for one Turing ...
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0answers
112 views

Can two-tape read-only Turing machines recognize any recursive language?

Suppose that a $k$-tape read-only Turing machine receives its input on each $k$ tapes. It cannot write on the tapes, but it can move on them in both ways, even move off from the input. So for example, ...
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2answers
246 views

Algorithm for identifying unprovable statements

I understand that this may depend on the specific set of axioms, but is there a general way (algorithm) for automatically detecting unprovable statements within a set of axioms? For example: If there ...
2
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1answer
153 views

Is there a difference between incompleteness and unknowable? [closed]

Godel proved that there are statements that are true but not provable. Unproven conjectures such "Twin Primes" might fall into such a class, as I understand. If so, does it mean that we will never ...
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3answers
189 views

Why/when do we ever need transfinite loop variants?

I do not understand why one would ever need a transfinite loop variant. Why do natural-number-valued variants not suffice? My simple (but perhaps too naive) argument is: if a loop $L$ terminates ...
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1answer
257 views

Is there a useful notion of being “approximately computable”

It seems that we can define a notion of being “approximately computable” where a set, $S$, is approximately computable if there is a family of computable functions $f_n(x)$ such that $$\lim_{n\to\...
8
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1answer
254 views

Non-comparable natural numbers

The "name the biggest number game" asks two players to write down a number secretly, and the winner is the person who wrote down the larger number. The game commonly allows players to write down ...
4
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1answer
77 views

Complexity of type-checking in relation to complexity of normalization

In order to verify that a terminating program terminates, one thing that can be done is to actually run the program. That may take a lot of time. If the program is typed in a total type-system, we can ...
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3answers
79 views

Reference request: Arithmetic circuit complexity

I am completely new to this field. I want to read the prelims and unfortunately, I don't see any book written for a beginner in this area. Can anyone give me some basic starting points/references(text/...
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2answers
423 views

Are these problems in NP class?

${\bf New\ version}$ [Version 1.2] Let $f: \mathbb{N} \to \{0,1\}$ be a computable function, ${\bf Fin}(\mathbb{Z})$ be the set of all finite subsets of $\mathbb{Z}$, and $W: {\bf Fin}(\mathbb{Z}) \...
6
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2answers
508 views

a polynomial representation of boolean functions

I came up with this linear transformation to map boolean functions to polynomials and it seems to have some nice properties. I was wondering if there is any reference describing this (and/or similar) ...
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2answers
179 views

Is true randomness and the physical Church-Turing thesis incompatible?

As follow up to Does the physical Church-Turing thesis imply that all physical constants are computable?, I ask if true randomness (as predicted by QM) and the physical Church-Turing thesis are ...
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2answers
300 views

Does the physical Church-Turing thesis imply that all physical constants are computable?

The physical Church Turing thesis is a conjecture that any physically computable algorithm can be computed by a Turing machine. Let us create a machine that, for example, outputs the digits of the ...
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1answer
136 views

Is the function undefined everywhere computable? [closed]

We could build the function like these: $\phi = \lambda x,y.5$ $\psi = \lambda x.(\mu y.(\phi(x,y)=0)$ $\psi$ is a $\mu-recursive$ function so it is computable and its undefined everywhere, but I ...
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0answers
40 views

Define recursive binary successors and usual successor mutually

It can be read in "Computation Models and Function Algebras", Peter Clote that the class of Polynomial Time functions (PTime) can be defined through the following Function Algebra: $[z,s_0,s_1,smash,\...
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1answer
116 views

Are there proofs which show that AIs are bound to be “worse” than the human brain? [closed]

I was talking with two PhDs (both teach IT related subjects) about artificial intelligence the other day. They were in agreement that an AI can never reach the level of the human brain, but failed to ...
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2answers
535 views

Enumerating decidable languages

[The assumption in this question is wrong. It is possible to enumerate exactly the decidable languages with semideciders.] Lets say we have a TM $M_E$ enumerator that writes out codes of TM's on a ...
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4answers
325 views

Is there a notion of computability on sets other than the natural numbers?

Is there a notion of computability on sets other than the natural numbers? For the sake of argument, let's say on sets $S$ that biject with $\mathbb{N}$. It's tempting to say "yes, they are those ...
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0answers
31 views

references for optimal computation under memory constraint?

Can someone help me find some references for finding good execution schedule given memory constraint? Assuming computation graph is simple in some sense (ie, small tree-width) There is this reference ...
8
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4answers
443 views

Impossibility in Computability and Complexity: always ultimately due to diagonal arguments?

In Computability, if we want to prove that a problem is not recursive or not recursively enumerable, we can use e.g. reductions from other non-recursive or non-r.e. problems, Rice's theorem, Rice-...
5
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1answer
294 views

Show that minimal CFG is undecidable via mapping reduction

Actually the problem below is an exercise in Sipser's textbook (Problem 5.36). However, from my perspective, it is not so trivial so that I post this question on this site instead of CS.SE. The ...
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0answers
226 views

Computability of a “weird” set

The starting point of this question is the observation that the smallest positive integers $a,b,c$ satisfying $$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 4$$ are absurdly high. This leads to ...
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1answer
203 views

Equilibrium in a Halting Game

Consider the following 2-player game: Nature randomly picks a program Each player plays a number in [0, infinity] inclusive in response to nature's move Take the minimum of the players’ numbers, and ...
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1answer
76 views

What is the relation between P-immune languages and NP-complete languages? [closed]

Can a NP-complete language be P-immune? Why can't existence of P-immune languages separate NP from P?
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1answer
100 views

Set of languages that can represent every c. e. languange

Could we find any set of languages $S$, such that it can represent every c. e. languange as it's union, intersection, complement, production(times of element ), and $S\subset X$, where $X\subseteq c.e....
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1answer
61 views

Can any c.e. language with infinite words be decomposed into infinite CFLs with infinite words?

Suppose $L$ is a computably enumerable language, can it be decompose into infinite CFLs with infinite words ? $$L=\bigcup_{L_i\in CFL }^{\infty}L_i$$ Second question: if it is possible that every $L$...
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1answer
71 views

Is there an inherently ambiguous language which can not be recognized by Deterministic LBA?

Is there inherently ambiguous language which can not be recognized by Deterministic LBA? For example, $L=\{wv: w,v=(x|y)^*, w=w^R,v=v^R\}$, is there any deterministc LBA that recognizes $L$ ?
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1answer
95 views

Is the relation decidable?

Given an ideal $I$ over $\mathbb{C}$ and P, a polynomial, is it decidable whether $P\in I$?
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1answer
269 views

Computability Theory prerequisites

What are the prerequisite disciplines for Computability theory? How much is Theory of Computation, Automata Theory, etc and how hard would it be studying it without those prerequisites?
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1answer
1k views

Is Hartmanis-Stearns conjecture settled by this article?

The paper "On the computational complexity of algebraic numbers: the Hartmanis--Stearns problem revisited" by Boris Adamczewski, Julien Cassaigne, Marion Le Gonidec https://arxiv.org/abs/1601....
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1answer
398 views

Original proof that “almost all decision problems are uncomputable”?

Who gave the original proof that "almost all decision problems are uncomputable"? Any hint at the original paper appreciated, thanks!
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1answer
357 views

Compressing information about the halting problem for oracle Turing machines

The halting problem is well-known to be uncomputable. However, it is possible to exponentially "compress" information about the halting problem, so that decompressing it is computable. More precisely,...
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167 views

What are considered to be the most canonical and important consequences of the recursion theorem?

The recursion theorem in computability states that, for any computable map $f : \mathbb{N} \to \mathbb{N}$ there exists $n \in \mathbb{N}$ such that $\varphi_{f(n)} = \varphi_n$, where $\varphi$ is a ...
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0answers
84 views

Small universal monotone Turing machines

This paper surveys small universal Turing machines. What are some examples of small universal monotone Turing machines, as described by Schmidhuber? Which of these are efficient (polynomial time) ...
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0answers
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Question on Turings Dissertation *Systems of Logic based on Ordinals*, Axiomatic Properties [closed]

I have a question on Alan Turing's Dissertation Systems of Logic Based on Ordinals, a scanned copy you can find here, or rewritten in LaTeX here, and also a copy of the published version here (but in ...
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0answers
61 views

Relation between MDPs and non-deterministic finite automatons

I'm confused as to the relation (computability-wise) between markov decision processes and NFAs. Are finite state MDPs expressible as regular grammars? If so, are markov decision processes thus ...
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0answers
120 views

An alternative model of a probabilistic Turing machine [closed]

A probabilistic Turing machine operates with an additional tape of random bits, and its output is a random variable with some distribution over the random bits. Is it also useful to talk about the ...
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1answer
85 views

Is there a relativized form of Rice Theorem?

Suppose $P_1$ and $P_2$ are nontrivial semantic properties of Turing Machines, and suppose that $P_1\wedge P_2$ is nontrivial given $P_1$. Can one claim that $P_1\wedge P_2$ is undecidable given an ...
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1answer
507 views

Is simply typed lambda calculus equivalent to primitive recursive functions

It's well known that the computation models untyped lambda calculus and $\mu$-recursive function are equivalent in terms of computability (in fact they are both Turing complete.) It is also well known ...
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0answers
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Which computational models support bigotous programs?

A bigotous program is a program which decides if its input is semantically equivalent to itself. Of course, this is impossible in a Turing complete language due to Rice's theorem. In fact, its pretty ...
2
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1answer
112 views

About the decidability of sets enumerated in non decreasing order

It is well known that a set of numbers enumerable in nondecreasing order is decidable. However, the typical proof, by cases on the finiteness of the enumerated set, is not constructive. In general, it ...
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0answers
241 views

A sequence wherein the Kolmogorov complexity of the terms does not increase

I am looking for an algorithm $A$ - which for any non-null input string $s_1$ produces a sequence $s_1, s_2...$ such that : It can be proved in some axiomtic system $S$ that: $\...
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1answer
291 views

Gap between $BB(n)$ and “second largest” $BB(n)$

If $HT(n)$ is the set of halting times of $n$-state Turing machines on a binary alphabet with empty initial tape, then $BB(n) = \max HT(n)$. What can we say about the second largest number in $HT(n)$...
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1answer
308 views

Where does the “intuitive” understanding of Kolmogorov complexity fails

Often, the Kolmogorov complexity of some string $x$ is defined as the length of the shortest program producing $x$, for example on wikipedia. So to give this more formal meaning, define $$ K'(x) := ...
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1answer
173 views

Is there a formulation of Rice's Theorem that does not involve admissible (or Gödel) numberings?

It is an easy task to devise an encoding for a computing formalism (such as Turing Machines) that has a function with fixed encoding. That is, the encoding is a numbering $\psi : \mathbb{N} \...
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1answer
579 views

Does the uncomputability of Kolmogorov complexity follow from Lawvere's Fixed Point Theorem?

Many theorems and "paradoxes" - Cantor's diagonalization, undecidability of hatling, undeciability of Kolmogorov complexity, Gödel Incompleteness, Chaitin Incompleteness, Russell's paradox, etc. -...
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1answer
85 views

Enumerator for the language w#w^R? [closed]

I'm trying to build a Turing machine diagram for the language w#w^R, where w^R is the reverse of w, and w is a word made up of 0's and 1's. I'm trying to think of an algorithm but I can't think of ...
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1answer
237 views

Entscheidungsproblem vs. Unvollständigkeitssatz (soft question)

The first term is used by Hilbert in his 1928 work, but in Gödel's later work, the same thing is referred to as Unvollständigkeitssatz ("incompleteness theorem"). For today's German CS researchers, it ...
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2answers
365 views

In the context of regular languages, must the alphabet be finite?

In The Theory of Parsing, Translation, and Compiling, Volume I, Section 0.2.1 (p.15 / 1972), Aho and Ullman casually write that "[a]n alphabet need not be finite or even countable, but for all ...
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1answer
118 views

About the origin of the names “immune” and “simple”

I have been wondering for a while about the origin of the names "immune" and "simple". I also posed the same question to Andrea Sorbi, who in turn involved a few more colleagues in the discussion. ...