Questions tagged [computability]

Computability theory a.k.a. recursion theory.

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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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3answers
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Computing Functions with Dynamical Systems

I was trying to make a set of differential equations "compute" some given function just like a Turing Machine does. Essentially, a given Turing Machine with an initial configuration (which includes ...
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261 views

Is a binary sequence computable iff the Kolmogorov complexity of its initial segments is bounded?

Disclaimer: I am mostly unfamiliar with theoretical computer science, making it hard for me to navigate literature in the field. I ask the following out of curiosity. Background/Motivation: Coming ...
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262 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 ...
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1answer
127 views

A function is lambda-2-definable iff it is HG computable and provably type correct in lambda-PRED2

I'm having a problem regarding Theorem 5.4.40.3 of Barendregt's Lambda calculi with types (1992), a chapter in Handbook in logic in computer science. (I'm referring to the PostScript version available ...
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322 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
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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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Uniform mortality problem for Turing Machines

Consider the following generalisation of the mortality problem for Turing Machines. Given a Turing Machine $M$. Is there a bound $k_M$ such that starting from any configuration $c$ machine $M$ ...
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1answer
357 views

Is there any research on Turing machines with transition relation homomorphic to given algebraic structure?

A Turing machine is defined as a structure $ TM(L,Q,T) $, where $L,Q$ are sets of symbols and internal states of TM respectively, and T is a transition relation: $T: L \times Q \to L \times Q $ for ...
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1answer
376 views

Term that can distinguish beta-equivalent normal forms in the untyped lambda calculus

I'm trying to work through two (non-assessed) class-work questions and am stuck on a question that seems similar to one I could do. The first question was to prove that there does not exist a $\...
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Was Babbage's Analytical Engine really turing-complete?

According to literature, Babbage's Analytical Engine is turing-complete because it supports conditional branching: it can perform different operations depending on the sign of the result last ...
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1answer
598 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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Automata model with undecidable (or non-context-sensitive) languages and no $\varepsilon$-transitions.

Adding extensions to automata has always been a fruitful domain. But usually, one wants to add weak capabilities, as undecidability comes quickly into the picture. Take FSM with added stacks. It is ...
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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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488 views

Prove Post Correspondence Problem Non-Recursive Without Reduction

Given a set of pairs of words $P = \{(\alpha_1, \beta_1), \dots, (\alpha_n, \beta_n)\} \subseteq \Sigma^*\times\Sigma^*$, the Post Correspondence Problem (PCP) is to decide wether or not there are ...
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How fundamental is undecidability? [closed]

What does it mean that some problem is undecidable? For instance the halting problem. Does it mean that humans can never invent a new technique that always decides whether a turing machine will halt?...
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Are there other computer architectures apart fom the von neumann /turing architectures?

Are there other computer architectures apart from the Von Neumann /Turing architecture?
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How to introduce recursion to Simply Typed Lambda Calculus while at the same time keeping strong normalisation?

Suppose you have a version of the STLC with one base type, similar to: data Tree = Branch Tree Tree | Leaf Now, suppose you want to add recursion to that ...
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Isolation in Turing-complete reversible cellular automata

I don't know much about the terminology and the results on cellular automata, but I would like to ask a question about an conjecture I thought. Consider Turing-complete reversible cellular automata. ...
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1answer
276 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\...
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351 views

Hilbert's Tenth Problem and nonrecursive Diophantine sets

In her paper Defining Integers, Alexandra Shlapentokh presents the following as an immediate corollary of the undecidability of Hilbert's Tenth Problem --- that is, the language $\{p : p$ is a ...
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What is a reasonable representation/encoding of objects? [closed]

Question What is a reasonable representation of objects (for computability)? What is the criteria that we should apply to see if a representation is reasonable? This answer by Andrej suggests ...
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1answer
248 views

Random walk returning probability

Consider a two-dimensional random walk, but this time the probabilities are not $1/4$, but some values $p_1, p_2, p_3, p_4$ with $\sum_{i=1}^4 p_i=1$. For example, from $(0,0)$, it goes to $(1,0)$ ...
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271 views

Equivalence problem for one-counter automata

I know that the equivalence problem for deterministic one counter automata is decidable, however does anyone know whether it is decidable for all one counter automata or just the deterministic ones? ...
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1answer
167 views

Minimal information needed for determine some function

From calculus, we know that if someone has a continuous function $f$, it is enough to know $f$'s values on the rationals in order to know $f$ on the entire line. In some sense, a "countable amount of ...
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1answer
256 views

How to describe the set of “all computable functions” using Coq?

Would the set of all computable functions be just the set of all maps of the form f : forall n : nat, P n -> nat where ...
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1answer
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. ...
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1answer
398 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 ...
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Are there languages require many variables to achieve $\Sigma_n^0$ completeness?

The proof of Post's Theorem that I am familiar with assumes you have access to as many variables as you wish in your language. Matiyasevich's Theorem by contrast gives a $\Sigma_n^0$-complete formula ...
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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 ...
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Notions of Computability at Higher Type III

I've recently found a very nice survey paper called "Notions of Computability at Higher Type" by John R. Longley. The paper says it is part of a 3-part series, with the 3rd concerning non-extensional ...
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Exact catchup point between SGH and FGH of ordinals?

An ordinal hierarchy is a way to assign a function $f_{\alpha} : \mathbb{N} \rightarrow \mathbb{N}$ to each (recursive) ordinal $\alpha$. The corresponding functions are expected to be monotone and ...
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351 views

Does $\Sigma(n+1)-\Sigma(n)$ eventually dominate every computable function?

Let $\Sigma$ be Radó's Busy Beaver function, and define $\Delta(n) \ = \ \Sigma(n+1) - \Sigma(n)$ for all $n \in \mathbb{N}$. Question: Does the function $\Delta$ eventually dominate every ...
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Why have computer scientists chosen recursor instead of iterator in primitive recursion?

I wonder why computer scientist have chosen recursor instead of iterator (or tail recursor if you like) in primitive recursion, given that function defined in terms of iteration behaves more ...
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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. ......
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Examples of reversible computations

Irreversible computations can be intuitive. For example, it is easy to understand roles of AND, OR, NOT gates and design a system without any intermediate, compilable layer. The gates can be directly ...
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1answer
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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
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Is every busy beaver strictly monotonic asymptotically?

To be specific, let me first define the busy beaver function BB(n)= maximum number of 1's that can be printed on the tape (i.e., the maximum score) by a standard n-state, 2-symbol (0 and 1) Turing ...
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649 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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189 views

Is there any relationship of hardness between the two problems?

Assuming F(x,y,D) is a function, and we can evaluate it in polynomial time with input x, y and D. Consider the problem P1: With D as input, computes $(x^*,y^*)=argmax_{(x,y)}F(x,y|D)$ where x and y ...
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208 views

Is it decidable that a computable analytic function over $\mathbb{R,C} ,$ equals $0$

Is it decidable whether a computable analytic function $f(x_1,x_2,\dots,x_n)$ over $\mathbb{R}$, $\mathbb{C}$ in a semi-algebraic or semi-analytic domain is identically zero? Is there any algorithm? ...
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Mathematical explanation of recursion and lambda (referenced in The Little Schemer)

In the preface of Friedman and Felleisen's book The Little Schemer it states: We could, for example, describe the entire technical content of this book in less than a page of mathematics, but a ...
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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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Primitive Recursive Definition : Binary numbers

Usually primitive recursive functions are define from Zero, Identity and Successor, projectors, composition and recursion. But you obtain algorithms that works with unary numbers. For example, the ...
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1answer
194 views

Arithmetic Analogues of P versus BPP

In the arithmetic hierarchy, is there an analog of $P$ versus $BPP$? Particularly is there a notion of randomness there? If there is no such analogy, why is randomness in the resource bounded case ...
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Transition Diagram of a Universal Turing Machine

I have searched the web for the transition diagram of a universal Turing machine without luck. Is anyone aware of such a diagram? I need this as a reference, so preferably a book or a published ...
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460 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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1answer
523 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
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Representation as sum of unit fractions: primitive recursive?

Consider the following ternary relation $R\subseteq\mathbb{N}^3$: $(p,q,m)\in R$ iff $p,q>0$ and there is a set $S\subseteq \{\frac{1}{n}:n\in\mathbb{N}, n\geq 1\}$ such that $|S|=m$ and $\frac{...
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Explanation of 1-generic to prove undecidability of halting problem

This question is about an answer in question Are there any proofs the undecidability of the halting problem that does not depend on self-referencing or diagonalization ? Bjørn Kjos-Hanssen answer ...