Computability theory a.k.a. recursion theory.

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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 ...
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135 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 ...
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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? ...
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73 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 ...
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1answer
90 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 ...
7
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126 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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1answer
280 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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140 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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284 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 ...
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92 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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137 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 ...
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1answer
51 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 ...
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1answer
58 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 ...
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1answer
88 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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28 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 ...
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1answer
145 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). ...
7
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1answer
223 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 ...
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39 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 ...
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125 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
177 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 ...
13
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2answers
428 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
92 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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483 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 ...
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1answer
132 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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80 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 ...
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1answer
92 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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456 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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116 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
80 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
264 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? ...
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1answer
202 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
983 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 ...
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610 views

How exactly does lambda calculus capture the intuitive notion of computability?

I've been trying to wrap my head around the what, why and how of $\lambda$-calculus but I'm unable to come to grips with "why does it work"? "Intuitively" I get the computability model of Turing ...
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272 views

Was bombe machine turing complete?

In the recent movie called The Imitation Game, there is a affirmation that Turing was building his theoretical machine. That machine is the Bombe Machine. Is this machine really equivalent to a Turing ...
4
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1answer
106 views

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 ...
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2answers
107 views

Where can I find the proof of the theorem and what is the computational complexity of the computably isomorphic map?

"any two representations of reals which are acceptable are actually computably isomorphic",please see here for reference where may proof of this theorem be found, and what is the the computational ...
4
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1answer
167 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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249 views

How to judge the definition of computational complexity of reals is natural or suitable?

As we know, definition of computational complexity of algorithm is almost without controversy, but the definition of computational complexity of reals or the computation models over reals is not in ...
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566 views

Fixed points in computability and logic

This question has also been posted on Math.SE, http://math.stackexchange.com/questions/1002540/fixed-points-in-computability-nd-logic I hope it is ok to also post it here. If not, or if it is too ...
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3answers
232 views

How high are the higher types that appear in practice?

This is admittedly a rather naively put and vague question, and I'm not sure how much more specific I want or can make it, but I'll try. By "practice" I mean surely in actual programming practice (of ...
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507 views

For a random oracle R, does BPP equal the set of computable languages in P^R?

Well, the title pretty much says it all. The interesting question above was asked by commenter Jay on my blog (see here and here). I'm guessing both that the answer is yes and that there's a ...
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1answer
78 views

Conflicting definitions regarding TM and Recursively Enumerable languages

In Lewis's and Papadimitriou's book "Elements of the Theory of Computation" the transition table is a function $\delta: Q \setminus F \times \Gamma \rightarrow Q \times (\Gamma \cup \{L,R\})$. ...
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1answer
125 views

Computability of infinite-dimensional vector space

So there is a talk about infinite-dimensional vector space being computable. But then I find it hard to understand. Apparently, dimension is infinite, so how would the operations of the space be ...
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1answer
96 views

What is necessary and/or sufficient requirement for a subring of a field to be computable? [closed]

As title asks, what is necessary and/or sufficient requirement for a subring of a field to be a computable ring? Conditions on either field or subring are fine.
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1answer
581 views

To what extent can the mathematics of Reals be applied to Computable Reals?

Is there a general theorem that would state, with proper sanitization, that most known results regarding the use of real numbers can actually be used when considering only computable reals? Or is ...
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1answer
90 views

Consistency and completeness of any arbitrary 3-valued logic? [closed]

Based on the explanations here [1] I know that 3-valued first order logic has many different versions, which differ in the definition of their operations (e.g. implication). All of these (as far as I ...
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1answer
106 views

Decidability of first-order theory of real closed fields with functions

By a famous theorem of Tarski, the first-order theory of real closed fields is decidable, as it admits quantifier elimination. Can this result be extended so that propositions can be quantified over ...
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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
158 views

How can you prove that all halting probabilites are normal real numbers?

Wikipedia claims that any halting probability (Chaitin's constant) is a normal number. Since Chaitin's constant is uncomputble, how is a proof the the normalcy of the number possible? Computable ...
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1answer
183 views

Does hyper-computational power of infinite time Turing machines also require infinite memory?

Can a infinite time Turing machine perform hyper-computation like checking the consistency of the set theory ZF without using infinite memory?