Questions tagged [reference-request]

Reference-request is used when the author needs to know about work related to the question.

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31 views

program search with optimization methods for (resource bounded) Kolmogorov complexity

Are there fields of research that look at finding short programs for generating strings (therefore trying to find the (resource bounded) Kolmogorov complexity of the string), but using optimization ...
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44 views

sophistication or logical depth to detect intelligent extra-terrestrial species

From my understanding, Algorithmic information theory (AIT) gives some ways to define the amount of « structure » in a string: for example sophistication or logical depth (see for instance [1]), can ...
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1answer
91 views

Hardness when restricted to an infinite number of far apart instance sizes

Is there a result that rules out (under common complexity theoretic assumptions) that one can solve an NP-hard problem in polynomial time for an infinite number of possibly very far apart instance ...
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1answer
77 views

Reference for context-free grammar for Martin-Löf type theory

Are the terms and the types of Martin-Löf type theory described by context-free grammars? Have such grammars been written down somewhere?
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127 views

Where is Yao's original proof that distinguishers imply next-bit-predictors?

In the theory of pseudorandomness, there is a well-known lemma that says roughly the following. Let $X$ be a probability distribution over $\{0, 1\}^n$. Suppose there is an efficient algorithm that ...
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1answer
233 views

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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What is known about the stabilizer rank of this simple state?

Consider the uniform superposition of all length-$n$ bit-strings of Hammming weight $w$, $$ |\phi_w\rangle =\sum_{x\in \{0,1\}^n,|x|=w} |x\rangle$$ What is known or conjectured about the stabilizer ...
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1answer
68 views

Given a partition and an element, find the subset that includes this element

I am interested in the following simple problem: Let $X$ be a set and $X_1\cup X_2\cup\cdots\cup X_k$ be a finite partition of $X$. Given $x\in X$, find the subset $X_i$ for which $x\in X_i$. I am ...
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65 views

Terminology for languages of pairs of words

I want to consider $L \subset A^* \times B^*$ as a "language". Is there standard terminology for this? I wrote "double language" first (but that doesn't sound right to me), then &...
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1answer
86 views

Approximating Independent Dominating set on bipartite graphs

I'm interested in the following problem: given a bipartite graph, find the smallest independent set of vertices which dominate all other vertices. My question is: are there any positive results in the ...
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60 views

Multi-round communication complexity of greater than

For the "greater-than" problem in Yao's 2-party communication complexity model, Alice receives $X$ and Bob receives $Y$, and they need to decide whether $X>Y$. I recently listened to an (...
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1answer
370 views

Strongly normalizing type theory beyond induction-recursion

Are there known type theories in the literature, which have strong normalization proofs and their proof-theoretical strength goes beyond strength of type theories with induction-recursion?
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Relative consistency of various Martin-Löf style type theories

I am wondering about relative consistency of various Martin-Löf type theories, when compared to one another, I will use MLTT for the intensional Martin-Löf type theory with $\Pi$, $\Sigma$, $\mathbb{N}...
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89 views

Name of this graph partitioning problem? (related to coloring)

Given a graph $G=(V,E)$ and an integer $k$, find a partition $P_1, P_2, \dots, P_k$ of $V$ into $k$ parts that minimize the total number of edges between two vertices in the same part, i.e. $\sum_i |(...
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88 views

Trading treewidth for depth in Boolean circuits

We know that languages defined by (poly-sized) Boolean formulae equals $\mathbf{NC}^1$: that Boolean formulae can be simulated in $\mathbf{NC}^1$ was shown by Brent/Spira [B,S], and the converse is ...
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242 views

Density of semantics in syntax

Let $L$ be a programming language, and $\cong$ a notion of equality of $L$-programs (in general $\cong$ will be undecidable). Let $syntax(n)$ be the number of $L$-programs of size $n$ (for some ...
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143 views

A variant of transfer learning

Suppose we want to train $K$ linear classifiers based on iid samples. Each classifier is of the form $x\mapsto\mathrm{sign}(w\cdot x+\theta)$, with the constraint that the hyperplane $w$ is the same ...
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24 views

Given an input sequence of real numbers, how to find the closest sequence in a large set of sequences

We are given a set $S$ of $m\gg 1$ sequences (arrays) of $n$ elements, where each sequence $s\in S$ belongs to $\mathbb{R}^n$. In the problem I am trying to solve, in a sequential fashion, we obtain a ...
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1answer
115 views

An invariance theorem for algorithmically random data in statistical learning

Motivation: The following invariance theorem for statistical learning in the setting of algorithmically random data occurred to me yesterday. This theorem uses the fact that the property of ...
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50 views

Logarithmic queries to $\Sigma_i^P$ oracle and the Boolean hiearchies

If I understood correctly (the complexity zoo, wikipedia, and some of the cited articles), the class $\textsf{P}^{\textsf{NP}[\log]}$, also known as $\Theta_2^{\textsf{P}}$, sits at the top of the ...
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77 views

Is there a standard interpretation of this quantity?

Define an encoding scheme by a pair of functions: $$\mathsf{encode} : \mathcal{M}\to\mathcal{X},\quad \mathsf{decode} : \mathcal{X}\to\mathcal{M}$$ Typical examples of encoding schemes are things like ...
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1answer
124 views

Structural normalization algorithm for the simply typed lambda calculus

I would like to know if there is a (piecewise) structural normalization algorithm for the simply typed lambda calculus. By structural I mean a recursive function that only calls itself on subterms of ...
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346 views

Complexity of Unknotting problems

The complexity of the Unknotting problem is known to be in $\mathrm{NP} \cap\mathrm{co\text-NP}$, see references: The Computational Complexity of Knot Problems. Knottedness is in NP, modulo GRH. . ...
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51 views

Hessian of non differentiable convex function

The motivation of the question is the following: Let $P$ be a set of $n$ points in $\mathbb{R}^d$. Consider the following objective(convex and differentiable) function $f:\mathbb{R}^d\rightarrow [0,\...
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47 views

A variant of randomized co-ordinate descent

Let us consider the following optimization problem. $\mathcal{P} =\{P_1,\cdots,P_n\}$, where $P_i\subset\mathbb{R}^d$. Let $m = max_i\lvert P_i\rvert$. The goal is to find a point $c$ such that ...
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2answers
234 views

Intuition behind nested positivity and counterexamples

I'm looking at the nested positivity conditions for inductive types stated in the Coq manual. First off, are there any other references (not necessarily for Coq, but in dependent type theories ...
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1answer
83 views

Survey on Quantum error correction

Are there any standard recent survey articles on quantum error correction (and may be including fault Tolerant computing)? The most standard ones that many people refer to are this and this. Both of ...
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465 views

One-way randomized communication complexity of Greater-Than

Let $\mathrm{GT}_n:\{0,1\}^n \times \{0,1\}^n \to \{0,1\}$ be the greater than function: $\mathrm{GT}_n(x,y)=1$ exactly when the positive integer whose binary representation is $x$ is greater than the ...
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1answer
107 views

Survey on Erdős-Pósa?

Does anyone know of any good surveys on Erdős-Pósa? I am particularly interested in what are the latest results for the bounding function for directed and even cycles in planar and minor free graphs ...
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1answer
87 views

“Parity testing set” for disjoint pairs of sets

I'd like a construction of the following description. Let $V$ be a set of $n$ elements. I'd like a collection $X$ of subsets of $V$ such that for any pair $(P,Q)$ of disjoint subsets of $V$, there ...
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2answers
134 views

Can concurrency models be compared in terms of some metrics?

In Seven Concurrency Models in Seven Weeks by Butcher, it compares Actor Model and Communicating Sequential Processes (CSP): CSP is more flexible than actor model: In actor model, the medium of ...
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1answer
98 views

Generalisations of the Fundamental Theorem of Statistical Learning to different tasks and losses

The fundamental theorem of statistical learning gives an equivalence between uniform convergence of the empirical risk to learning in the PAC framework. I have only seen this stated in the case of ...
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1answer
120 views

Upper bound on Independence Number of Random Regular Graph with degree $\Theta(\sqrt{|V|} \log^2 |V|)$

Let $G=(V,E)$ be a random $\Delta$-regular graph with $\Delta \in \Theta(\sqrt{|V|} \log^2 |V|)$. I'm analysing an algorithm having asymptotic running time crucially depending on the Independence ...
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1answer
108 views

VC-dimension of infinite set of triangle wave

I am searching for the VC-dimension of the following: What is the VC-dimension of the infinite set of triangle wave functions with amplitude 1 and period parameter p on points on the line? 2πarcsin⁡(...
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85 views

Request for an update on a discussion about coinductive types in HoTT (or anywhere else)

Googling something else I stumbled on a conversation titled "coinductives" initiated by Vladimir Voevodsky on Google groups in 2014. It lasted for three days, invloved a dozen people, and ...
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1answer
113 views

Extending cographs with product operation

Let $\mathcal{C}$ be the class of undirected graphs defined inductively as follows: A single vertex is in $\mathcal{C}$; If $G\in\mathcal{C}$ then its complement $\overline{G}$ is in $\mathcal{C}$; ...
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137 views

Computing and maintaining the minimum of a set $S$ of integers while allowing updates on $S$

This question is about computing and maintaining the minimum of a set $S$ of integers while allowing updates on $S$. The computation model we are considering is the unit-cost RAM machine with linear ...
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170 views

Regular languages accepted by an automaton with at most one transition per letter

I'm interested in the (very restricted) subset of regular languages for which there is an automaton having the following property: for every letter $a$ of the alphabet, the automaton has at most one ...
3
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1answer
104 views

Generalizations of Dyck languages?

The "narrowest" generalization of Dyck languages that I am aware of is Visibly Pushdown languages. Are there any useful classes of languages that are intermediate between Dyck languages and ...
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2answers
150 views

A clear and rigorous explanation of critical pairs and the Knuth-Bendix completion algorithm?

I'm looking for an explanation of critical pairs and the Knuth-Bendix completion algorithm that is at once rigorous and of high pedagogical value, i.e. clear, detailed, containing illustrative ...
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1answer
229 views

The number of clauses in an unsatisfiable CNF

I am interested in generalisations of the following observation: An unsatisfiable $k$-CNF has at least $2^k$ clauses. A special case of the observation is when $k=n$, where $n$ is the number of ...
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19 views

Notation for scheduling problems with multiple release times/deadline intervals

I am interested in the classical scheduling problems where we are given jobs with a release time and a deadline, we must find a schedule for all jobs. Specifically, I am interested in the variant ...
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33 views

Nominal Tree Languages i.e. with Binders and Infinite Symbols?

I'm wondering if there has been any research done into automata that accept languages of trees that can bind arbitrary variables, and are considered equal under alpha equivalence. I've found so far: ...
3
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1answer
118 views

Number of maximal cliques in a ($2C_4$, $C_5$, $P_5$)-free graph

So far, I have found out that chordal graphs have linear number of maximal cliques with respect to the number of vertices. In general case, it is exponential. I am trying to determine whether the ...
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140 views

Reference for computing the rank of a matrix in polynomial time

In a recent paper, I need to use the fact that computing the rank of a matrix over the integers has polynomial complexity. Given the context, I don't particularly care about the exact asymptotics, as ...
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1answer
146 views

Looking for an online community specializing in the Z specification language, where I can ask questions

Where can I find an online community specializing in the Z specification language, where I can ask specific questions about the ISO standard for Z?
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1answer
248 views

Entropy-like quantity

For $p\in[0,1]^{\mathbb{N}}$ and $\alpha\ge1$, define $$ H_\alpha(p) = \sum_{i\in\mathbb{N}}p_i|\log(p_i)|^\alpha. $$ When $\sum_i p_i=1$ and $\alpha=1$, $H_1(p)$ is just the Shannon entropy of the ...
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78 views

Harmonic analysis of sequences of Boolean functions (i.e. of words in $(\{0,1\}^n)^*$)

Is there any research on harmonic analysis of sequences of Boolean functions, which represent the application of a Boolean function on a word in $(\{0,1\}^n)^*$? I'm looking for any reference on this, ...
7
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1answer
261 views

Testing for finite expectation

The mean of a positive random variable $X$ is either finite or infinite; define $J(X)$ to be $0$ in the former case and $1$ in the latter case. Claim: there does not exist a function $J_n$ from the ...
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103 views

Exact algorithms for $k$-means

Lets recall the definition of $k$-means clustering for euclidean spaces. Let $X$ be a set of $n$ points in $R^d$ and $k$ a given natural number. Let $C$ any $k$ clustering of $X$. Define the cost of $...

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