Questions tagged [computability]

Computability theory a.k.a. recursion theory.

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1answer
91 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 ...
11
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1answer
240 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 ...
6
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2answers
459 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
122 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. ...
2
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1answer
67 views

Characterisation of computability of partial functions from infinite words into finite words by functions with prefix-free domain

The following is taken from K. Weihrauch, Computable Analysis, page 21. The notation $f : \subseteq A \to B$ means a partial function. By $\Sigma^{\omega}$ and $\Sigma^{\ast}$ we denote the set of ...
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1answer
293 views

Is the Kolmogorov complexity of the truth tables of the halting problem known asymptotically?

Let $HALT_n$ denote the string of length $2^n$ corresponding to the truth table of the halting problem for inputs of length $n$. If the sequence of Kolmogorov complexities $K(HALT_n)$ were $O(1)$, ...
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5answers
323 views

A definition of computable numbers that requires to “wait an infinite amount of time” to get the correct result; how to make this precise

Consider the following definition: A number $x \in \mathbb R$ is computable, if there exists a (one-tape) Turing machine which (running infinitely long) writes the binary expansion of $x$ onto its ...
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5answers
3k views

Importance of irrational numbers in computer science

Computers can only deal with a few rational numbers. Why is it important as a computer scientist to deal with irrational numbers?
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125 views

What is the largest complexity class that a non-Turing complete proof assitant, like Coq, can achieve? [duplicate]

Given that all functions in Coq will terminate, it cannot cover the entire $RE$. So what is the class it can achieve? Is it $R$?
7
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1answer
217 views

Succinct complete problems in DTIME(EXP(EXP(…)))

I understand complete problems for $EXPTIME$ or $NEXPTIME$ formulated as succinct instances of e.g. $NP$-complete problems such as $3-SAT$. On input $x$, one efficiently computes a circuit $R(x)$ such ...
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165 views

Formal definition of Turing Completeness [closed]

Wikipedia states: In computability theory, a system of data-manipulation rules […] is said to be Turing complete or computationally universal if it can be used to simulate any single-taped Turing ...
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1answer
187 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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139 views

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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3answers
403 views

Defining primitive recursive functions over general data types

The primitive recursive functions are defined over the natural numbers. However, it seems as if the concept should generalise to other data types, allowing one to talk about primitive recursive ...
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1answer
119 views

Are there any known problems that require potential nontermination to solve? [closed]

Apart from problems that specifically have to do with Turing machines, like "Simulate a Turing Machine with the given description", are there any problems that require Turing-complete potentially ...
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1answer
136 views

Could you explain to me the reduction? [closed]

I am looking at the following solved exercise: I haven't really understood at the reduction the part that we construct for each number $a_i$ a package of measurement $(\frac{4}{A}a_i, 5,3)$. Why do ...
3
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1answer
309 views

Can all mathematical operations be encoded with a Turing Complete language? [closed]

In High School Computing I was taught the Structured Program Theorem - that you could implement any mathematical operation using: Sequence Selection Iteration After completing a Computer Science ...
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1answer
133 views

What is the relationship between tail recursion with other recursions? [closed]

I'm rather confused by the recursion theory. From the link, the recursion theory was formed by Dedekind, Gödel and some other famous mathematicians. There are the following types of recursion. But ...
2
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0answers
107 views

Are there analogues of Specker sequences for other complexity classes?

Consider the standard definition of computable real numbers: a real number $r$ is computable just in case $r$ is the limit of a sequence $(a_i)_{i \in \mathbb{N}}$ such that (1) the function $i \...
3
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0answers
147 views

The complexity of the disjoint union of a sequence of complete problems

Suppose that for every $k \in \mathbb{N}$ the decision problem $A_k$ is hard for $\mathsf{N}k\text{-}\mathsf{ExpTime}$. What is the complexity of their disjoint union $A = \{ (k,x) \mid k\in \mathbb{...
3
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1answer
102 views

What is the simplest universal unidimensional interaction net system?

The Interaction Combinators are possibly the simplest multidimensional system of interaction nets that is Turing-complete. What about interaction nets with only 2 ports - 1 principal, 1 auxiliary? ...
2
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1answer
425 views

Langton's ant questions

I'm a mathematician currently working on the Langton's ant conjecture, just for fun. I have some result but I don't know if they are meaningless. So that is why I'm asking. 1) Is there a mathematical ...
0
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1answer
181 views

In regards to the tautologies of a polynomially-bounded propositional proof system

In the book 'Logical Foundations of Proof Complexity', co-authored by Stephen Cook, the following definition is given: A proof-system $F$ is said to be polynomially-bounded if there is a polynomial p(...
7
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1answer
138 views

A word anticorrespondence problem

A problem instance is a finite list of 4-tuples $(\alpha_1, u_1, v_1, \beta_1), ..., (\alpha_N, u_N, v_N, \beta_N)$, where $\alpha_i, \beta_i \in X$ come from a finite set, and each $u_i,v_i \in A^*$ ...
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3answers
696 views

Is there a result in computability theory that does not relativize?

I was reading Andrej Bauer's paper First Steps in Synthetic Computability Theory. In the conclusion he notes that Our axiomatization has its limit: it cannot prove any results in computability ...
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2answers
227 views

Are there any open problems concerning decidability? [duplicate]

I am learning computability theory. I am just interested to know some famous problems (Formally languages) whose decidability is in question.
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2answers
320 views

Are there any propositional proof systems which are not Cook-Reckhow proof systems?

An abstract proof system is a polynomial time function $f$ whose range is equal to the set of tautologies. If $\tau$ is a tautology, then an $f$-proof of $\tau$ is any value $\pi$ such that $f(\pi) =...
2
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0answers
113 views

A variant of the tiling problem

A classic tiling problem with Wang tiles has the form: Given $n$ tiles $T=\{t_1,...,t_n\}$ and some constraints $H,V\subseteq T\times T$, is there a way to tile a $w\times h$ rectangular grid with $...
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189 views

Applications of “Seemingly Impossible Functional Programs”

What are some practical applications (existing or potential) for Martin Escardo's "Seemingly Impossible Functional Programs"? For starters, here are a few from: Alex Simpson’s Lazy functional ...
2
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1answer
82 views

Proof software for Primitive Recursive Arithmetic

Primitive Recursive Arithmetic is a critical foundational system in mathematics at large, and all the more so in areas studying constructive reasoning and/or computability such as Theoretical Computer ...
2
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1answer
153 views

Can interaction combinators implement any interaction net efficiently?

It is widely known that interaction combinators can implement any interaction net. My question is, can they do so efficiently? I.e., is it possible to prove that there is no interaction net system ...
3
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1answer
191 views

Polynomial-time reductions between undecidable languages

The Turing degree $\mathbf{0}'$ is defined as all languages Turing-equivalent to the halting problem. In fact any recursively enumerable language is polynomial-time reducible to the halting problem. ...
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42 views

Proof for multiplicative dominance of universal probability distribution

I'm looking for the proof of the Leonid Levin theorem that states that the universal prior distribution function multiplicatively dominates all other functions of its type. The original article is ...
0
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1answer
847 views

Computing the DAG of a program given source code or AST

I've seen many papers on scheduling components or tasks once a DAG for the program is known, either by user-input or by domain restriction (i.e. all cross shaped 5-pt stencil codes have a known DAG). ...
11
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1answer
352 views

An example where smallest normal lambda term is not fastest

Let the $size$ of $\lambda$-terms be defined as follows: $size(x) = 1$, $size(λx.t) = size(t) + 1$, $size(t s) = size(t) + size(s) + 1$. Let the complexity of a $\lambda$-term $t$ be defined as the ...
2
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0answers
55 views

Simple version of Wang's tessellation problem

I'm reading about Wang's tessellation problem and the text mentions a simpler version: If we consider a finite set of tiles $W_{n}=\{w_{1},...,w_{n}\}$ where $n$ is bounded then the claim is that now ...
10
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0answers
197 views

The halting problem in computational models weaker than Turing machines

What are the main results and/or literature on the (self) halting problem for other machines than Turing machines? Alternatively, what would be the right keywords or tags to search for it. I am ...
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2answers
203 views

Information-theoretic Diffie-Hellman

The following non-standard description of Diffie-Hellman is entirely my own, by which I mean that I came up with it having not read about it anywhere else beforehand. In Diffie-Hellman Alice and Bob ...
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2answers
935 views

Smallest possible universal combinator

I am looking for the smallest possible universal combinator, measured by the number of abstractions and applications required to specify such a combinator in the lambda calculus. Examples of universal ...
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1answer
145 views

Undecidable Single Programs [closed]

So the halting problem basically states that there cannot exist any finite length algorithm for automatically verifying if other finite length algorithms terminate. But suppose I start listing out ...
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4answers
853 views

Is there algorithmic mathematical analysis?

There are algorithmic graph theory/number theory/combinatorics/information theory/game theory. Is there algorithmic mathematical analysis? According to wiki, mathematical analysis includes the ...
7
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1answer
181 views

Formalized priority argument

A priority argument, an important proof technique in recursion theory, was introduced by Friedberg and Muchnik, to solve Post's Problem, i.e., the existence of two r.e. sets that do not Turing reduce ...
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1answer
186 views

Is there any research on approximation of reals with computable numbers

I was wondering if there is any research in the field of Diophantine Approximation using the computable numbers. It seems to be a good fit, a dense countable set with a variety of different potential ...
2
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1answer
125 views

Turing degree of Solomonoff semi-measure

We define the Solomonoff semi-measure $m$ on finite strings $x$ by $$m(x) = \sum_{p: U(p) = x} 2^{-l(p)},$$ where $U$ is a universal prefix Turing machine, $U(p) = x$ means $U$ outputs $x$ on input $...
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2answers
2k views

Proof for Kolmogorov complexity is uncomputable using reductions

I am looking for a proof that Kolmogorov complexity is uncomputable using a reduction from another uncomputable problem. The common proof is a formalization of Berry's paradox rather than a reduction, ...
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0answers
143 views

What can you do with a moving knife besides cutting a cake?

In the fair cake-cutting, two different computational models are used: A discrete model, in which the algorithm issues queries to the players and proceeds according to their replies; A continuous ...
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1answer
93 views

Results about computability power or limitations of shared read/write registers

I want to know more results about the computability power or limitations of shared $\texttt{read/write}$ registers/objects in distributed/concurrent computing theory. Two typical examples are: [1]. ...
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1answer
2k views

Why study type theory?

After reading the literature on type theory (especially the constructive kind - CTT) I'm left wondering "why" should one study type theory, specifically within the confines of "computing" in general? ...
10
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1answer
490 views

Can differential equations be classed into their own complexity classes?

Problems have been, as a whole, classified, thanks to Computational Complexity. But, in differential equations, is it possible to classify differential equations depending on their computational ...
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2answers
2k views

What functions can System F not compute?

In this wikipedia article on Turing Completeness it states that: The untyped lambda calculus is Turing complete, but many typed lambda calculi, including System F, are not. The value of typed ...