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2
votes
1answer
290 views

Problem to a solution

Consider an NP hard problem $\frak P$ which takes an input of length n $\frak P$ can be solved partially by a factor $ p_i = p(n,i)\in$ [0,1)... by a polynomial time algorithm $\mathcal A(i)$ ...
11
votes
4answers
424 views

Mathematics of privacy?

I wonder to which extent the current public debate on privacy issues (not only by state sniffing, but e.g. by microtargetting ads too an issue) offers interesting questions in mathematics? Can we ...
14
votes
2answers
897 views

Switching from pure mathematics (e.g. geometry) to more applied areas (e.g imaging) after Ph.D., as postdoc and chance of getting such a postdoc?

Before I start my question, I should probably mention that this question might not be the right question to ask here, but I tried academiabeta, and stackoverflow, but without getting any to-the-point ...
0
votes
1answer
142 views

Interaction-based approximation for HP-complete λ-theory?

We are looking for a proof or counter-examples for the following hypothesis. Two combinators $M$ and $N$ are solvable and equivalent in the HP-complete sensible $\lambda$-theory iff either $$ \exists ...
3
votes
1answer
267 views

What new primitive recursive functions are needed to reconcile Turing time complexity with Godel time complexity?

Let me begin with an example. Consider the computable function $f(x) = 2x$. A Turing machine can implement this function in $O(|n|)$ steps: simply walk to the end of the input string, write a $0$, ...
12
votes
3answers
550 views

How do we express measurable spaces using type theory?

A measurable space $(X,\mathcal X)$ consists of a set $X$ equipped with a $\sigma$-algebra of subsets $\mathcal X$. I would like to write computer programs involving measurable spaces, but to the best ...
0
votes
0answers
163 views

Extended definition of unambiguous language and the existence of unambiguous grammar

Let's extend the unambiguity of language and grammar as follows: a language $L$ is unambiguous if there is a grammar that generates every word in $L$ in a unique way, the grammar may be of type 0 or ...
2
votes
0answers
51 views

Is the $d$-dimensional Arrangement of Trees still $NP$-hard?

The $d$-dimensional Arrangement Problem for general graphs is known to be $NP$-hard since the special case $d=1$ (OLA) already is (Garey et al, [1976]). For Trees however, the one dimensional case can ...
1
vote
1answer
807 views

The Jones-Sato-Wada-Wiens polynomial for prime numbers and differential calculus?

After works of Davis, Matijasevic, Putman and Robinson between 1960 and 1970, we know that every recursively enumerable set of numbers can be represented by a polynomial. In particular, it's the case ...
3
votes
1answer
126 views

Question about the elementary divisors of a special matrix

I have the following question: Is there a closed formula for the elementary divisors of the Matrix $M=\lbrace (m_{ij})\rbrace_{i=1,...,n,\ j=1,...,k}$, where $m_{ij}$ is the greatest common ...
8
votes
2answers
198 views

Ideal Membership without Certificate?

I have a homogeneous ideal $I=\langle f_1,\ldots,f_r\rangle$ of the polynomial ring $\mathbb C[X_1,\ldots,X_n]=:R$ where each of the $f_i$ is actually over $\mathbb Z$. My computations are usually ...
2
votes
2answers
359 views

What structure has been found for functions with this relationship.

Given $f$ and $g$ $\forall x y. f(x) = f(y) \Longrightarrow f(g(x)) = f(g(y))$ Or equivalently $ker\ f \subseteq ker\ (f \circ g)$. Note: if $f$ is injective then this holds for any $g$. ...
0
votes
0answers
145 views

An interesting version of the problem “balls into bins”

Consider n people, each has k identical balls. Each people choose k different bins from m bins, constrained by the condition that there are no two people choose exactly the same k bins. For instance, ...
11
votes
0answers
291 views

Splay trees and Thompson's group $F$

( I apologize for only indicating some easy to find references, but new users are not allowed to link more than five). This is very speculative, but: Question: Is there a reformulation of the Dynamic ...
1
vote
0answers
91 views

Schönhage's SMM with only one instruction

It is possible to implement $\lambda$-calculus in Schönhage's storage modification machine using an infinite set of nodes and one single program consisted exclusively of (about hundred) instructions ...
3
votes
2answers
183 views

Smallest base to reach partial recursive functions as a closure of unbound search

It is customary to define the class of partial recursive functions by taking the set of primitive recursive functions $PR$ and taking closure over unbound search operation. Do we need the "whole" set ...
17
votes
2answers
832 views

Expected edit distance

The edit or Levenshtein distance between two strings is the minimum number of single symbol insertions, deletions and substitutions to transform one string into another. For example ...
1
vote
1answer
241 views

Distance between vertices in a vertex transitive graphs. [closed]

Can anybody help me in finding out the distances between vertices in a vertex transitive graphs. Is there any specific formula to calculate distance between vertices in this graph. Thanks for your ...
7
votes
6answers
1k views

Giving $Top(X,Y)$ an appropriate topology

I am not sure if its OK to ask this question here. Let $Top$ be the category of topological spaces. Let $X,Y$ be objects in $Top$. Let $F:\mathbb{I}\rightarrow Top(X,Y)$ be a function (I will ...
1
vote
1answer
189 views

Hypothesis: interaction-based model for λKβη

We are looking for a proof or counter-examples to the following Hypothesis. In interaction calculus $\langle \varnothing\ |\ \Gamma(M, x) \cup \Gamma(N, x)\rangle \downarrow \langle \varnothing\ |\ ...
2
votes
0answers
146 views

Is it possible to implement η-reduction in interaction nets?

There are several ways to encode λ-terms in interaction nets; for instance, using the original optimal algorithm by Lamping, or compiling λ-calculus into interaction combinators. However, all the ...
1
vote
0answers
59 views

Optimal Reduction in Interaction Calculus

We work in interaction calculus. Let $\Sigma = \{\lambda, \psi, \delta, \epsilon\}$, $\text{Ar}(\lambda) = \text{Ar}(\psi) = \text{Ar}(\delta) = 2$, and $\text{Ar}(\epsilon) = 0$. For any $\alpha ...
1
vote
0answers
101 views

Turing-complete primitive interaction systems

Let us call primitive an interaction system with the signature $\Sigma = \{(\rho, 0), (\xi, n)\}, \quad n \geq 2;$ and the only rule being of the form $\rho \bowtie \xi[\rho, \xi(a_1, \dots , a_n), ...
1
vote
1answer
143 views

Grzegorczyk-hierarchy, growth-rate and functions with finite image

Grzegorczyk-hierarchy divides primitive recursive functions in distinct classes with respect to their growth-rate. It seems that the higher we go the hierarchy, the more tools we have to define ...
0
votes
1answer
193 views

Deriving the fundamental equation (with regards to computer vision)

I'm having a hard time understanding how a few equations are being derived. So the fundamental equation is an equation that relates corresponding points in stereo images. Anyway, that's the basic ...
6
votes
2answers
260 views

Rigorous numerics for maxima and minima (one variable)

Let $f:\mathbb{R}_0^+\to \mathbb{R}$ be defined by some combination of the four basic operations and square roots. (The argument of square-roots is assumed is to be non-negative, and the value of ...
3
votes
2answers
494 views

Turing-complete primitive blind automata

Let $N$ be the set of natural numbers, $S$ be the set of finite binary sequences, and $Q = [N \rightarrow N] \times [N \rightarrow N],$ where $[N \rightarrow N]$ is the set of all computable ...
2
votes
1answer
158 views

Reducing the error of Algorithms by assigning variables formulas instead of values

Let me first give the intuition for my question: Suppose that you want to use a ruler to mark $n$ points in a line on a page, with 1 cm distance between neighbor points. There are two ways: 1- Mark ...
1
vote
1answer
484 views

Collatz conjecture— finite state machine transducer construction, origination?

wikipedia has an entry on the Collatz conjecture with a section on As an abstract machine that computes in base two. this apparently describes a construction of a FSM transducer computing sequential ...
1
vote
1answer
331 views

Any recommendations on good papers I should read? [closed]

I am a high school student with a particular interest in mathematics and computer science. I have always been intrigued by recreational mathematics (who hasn't been?) and more advance concepts, ...
6
votes
1answer
332 views

Is equality of terms for “real” numbers with roots, logarithm, exponential, sin, cos, and other trigonometric operations decidable with a Turing-machine?

If yes, how? Also, I know you can't do it for arbitrary statements about real numbers, but that's not what I'm asking, and by "real" numbers, I mean the numbers constructible from 1, -, /, and the ...
3
votes
2answers
700 views

How to draw Archimedean-Galileo spiral?

It is known that some plane curves can be drawn with a tool. For instance, I heard at a web site that Archimedes created his spiral in the third century B.C. by fooling around with a compass and ...
4
votes
0answers
102 views

Are there sampNP-intermediate problems?

This questions is approximately cross-posted from theoretical computer science stackexchange Ladner's theorem establishes that if $\mathsf{P} \ne \mathsf{NP}$ then $\mathsf{NPI} := \mathsf{NP} ...
12
votes
6answers
1k views

SAT and Arithmetic Geometry

This is an agglomeration of several questions, linked by a single observation: SAT is equivalent to determining the existence of roots for a system of polynomial equations over $\mathbb{F}_2$ (note ...
1
vote
0answers
131 views

A Multiplicative version of McDiarmid's Inequality like the one of Chernoff-Hoeffding Bounds

McDiarmid's Inequality basically says the following: Let $X_1, X_2, X_3, \ldots, X_n$ denote independent random variables and $f$ is a function of $n$ real arguments. If changing the value of the ...
1
vote
1answer
248 views

Non-uniform complexity of the halting problem

This question is approximately cross-posted from Theoretical Computer Science Stack Exchange: http://cstheory.stackexchange.com/questions/14445/complexity-of-the-halting-problem What can be said ...
2
votes
1answer
165 views

Is there research on the notion of co-accessibility?

I want to start off with a disclaimer that I am only a mathematical amateur. Please forgive me for ignorance or any non-standard nomenclature I use here :) Let's start off with some context. Let X ...
3
votes
1answer
137 views

fast approximate k-nearest neighbors in high dimensions?

Hi, I've been scanning the literature trying to find an adequate approximate k-neighbour for my outlandish data set, but I remain stymied. Perhaps someone can help? The dataset is huge, both in ...
3
votes
1answer
170 views

Various notions of Turing reduction for partial functions

If $f$ and $g$ are partial functions $\mathbb{N} \to \mathbb{N}$, define six preorder relations $f \preceq g$ as follows: $f \mathop{\preceq_{\mathrm{S}}} g$ ("$f$ is strict/Sasso reducible to $g$") ...
2
votes
0answers
149 views

A primal-dual (double) circle packing (coin graph) question

I know that any 3-connected simple planar graph with a designated outside face (outer face) has a primal-dual (double) circle packing (Brightwell-Scheinerman Theorem). Q1- But I am not sure whether ...
1
vote
0answers
276 views

The used symbols for equality and equivalence

Background: I am currently developing a general purpose programming language which allows formal verification (i.e. correctness proofs) of programs. During the development it came out that a lot of ...
3
votes
2answers
244 views

Definition of continuous functions in order theory

If we have a complete partial order (i.e. directed complete) I find frequently the following definition of a continuous function. A function $f:A\to B$ where $A$ and $B$ are cpos is called continuous ...
9
votes
0answers
242 views

Various definitions of recursion from ordinal machines

Background: I'm trying to get an intuitive understanding of α-recursion and related concepts in higher recursion theory. Once nice book is Peter Hinman's Recursion-Theoretic Hierarchies, available ...
0
votes
1answer
179 views

hypergraph cartesian join operation (over same vertex set)

consider two hypergraphs $H_1 = (V, \mathscr{E}_1), H_2 = (V, \mathscr{E}_2)$ over the same vertex set $V$. am interested in what could be called a "cartesian join" operation building a new hypergraph ...
3
votes
2answers
224 views

What categories correspond to the typed lambda calculus with parametric types?

the unadorned typed lambda calculus correspond to the closed cartesian categories, but if we add in dependent or parametric types how are they then characterised?
2
votes
0answers
296 views

Groupoid interpretation of type theory

Hello, I read the paper on groupoid interpretation of type theory by Hofmann and Streicher and I have a question. According to the authors $Tm([[\text{Set}\:[\Gamma]\: ]])$ is the same as ...
0
votes
1answer
240 views

Counterexamples for this algorithm for recognizing lexicographic product of graphs?

Found a possible reduction from recognizing lexicographic product of graphs to 2SAT (since 2SAT is polynomial, the algorithm is polynomial). Can't prove completeness of the algorithm and since it is ...
0
votes
0answers
85 views

Exact Length Problem in a directed graph

I have a directed graph that consist of N^2 vertices (like a square) and each vertex is connected to at most 1 node (not bidirectional) and every connections have length 1. There are no cycles in the ...
4
votes
1answer
426 views

Deciding equivalence of regular languages

Given two regular expressions $R$ and $S$ on an alphabet $\Sigma$ it is possible to decide their equivalence as follows: build two finite automata $M_R$ and $M_S$ such that $L(R) = L(M_R)$ and $L(S) ...
5
votes
1answer
570 views

How much does a quantum oracle to find a needle in a haystack really cost?

Among the basic algorithms of quantum computations Lov Grover's result on quantum search stands out, both in regards to its intrinsic interest, and for its undisputable elegance. Grover's algorithm ...