# Tagged Questions

For question borderline with, or having application to, computer science. Consider also posting http://cs.stackexchange.com/ or http://cstheory.stackexchange.com/ instead of here, if appropriate.

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### Self-similarity in the theory of computability

Let $M = w_0w_1... \in \{0,1\}^*$. For any computable function $f$ define $M_f = w_{f(0)}w_{f(1)}...$ Let for any computable strictly increasing function $f$ there is continuous computable mapping ...
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### Can We Decide Whether Small Computer Programs Halt?

The undecidability of the halting problem states that there is no general procedure for deciding whether an arbitrary sufficiently complex computer program will halt or not. Are there some large $n$ ...
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### About “natural proof” of Razborov and Rudich

The famous "Natural Proof" paper ,http://www.cs.umd.edu/~gasarch/BLOGPAPERS/natural.pdf , ‎of Razborov and Rudich gives a barrier for any proof that try to separate P and NP. It mainly shows that if ...
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### Place N points in a 3d cube in a way that maximizes the minimum of their pairwise distances

Place $N$ points in a 3d cube in a way that maximizes the minimum of their pairwise distances. The problem can easily be solved for $N\lt5$, but how to proceed for larger $N$?
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### Why is there no product type in simply typed lambda-calculus?

Consider simply typed $\lambda$-calculus that has only the unit type as primitive. We would like to encode the product and the sum types. An encoding of the product type in the untyped $\lambda$-...
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### How many 2L-bit numbers are the product of two L-bit numbers?

If I multiply two integers $x, y$ in $[0,2^L)$, I get an integer in $[0,2^{2L})$. Clearly, this map from $[0,2^L) \times [0,2^L) \to [0,2^{2L})$ is not bijective. I am interested in the size of ...
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### Two questions from combinatorics on words

Question 1. Assume that an infinite word $u\in\{0,1\}^{\mathbb Z}$ is not balanced. Is it true that there exists a finite 0-1 word $w$ such that $0w01w1$ or $1w10w0$ is a factor of $u$? Is it true ...
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### What prefix and factors determine a ultimately periodic word uniquely

Let $\xi$ be an ultimately periodic sequence, i.e. there exists finite sequences $p, q \in X^*$ such that $\xi = pq^{\omega}$. Does there exists a $n > 0$ such that the prefix of length $n$ and all ...
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### Securing privacy of “who communicates with whom” under Orwell-like conditions

Assume that there is a big and powerful country with an information-greedy secret service which has backdoors to all internet nodes throughout the world which permit him to observe all exchanged data ...
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### Subsets of $\omega$-regular lanuages accepted by automata with special acceptance condition

Let $\mathcal A = (X, Q, \delta, q_0, F)$ be a deterministic finite automata with the following acceptance condition on infinite words: The automata accepts $\xi \in X^{\omega}$ with respect to $F$ ...
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### Categorical Semantics for Second-Order Logics

I am currently doing some work using a categorical semantics of first-order logic. The specific semantics I am using is due to Andrew Pitts, as described in: Categorical Logic, Andrew M. Pitts, ...
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### Physical Disturbances to Computations [closed]

In this paper, page 7 (160 of the Journal), Fig 3, there is a particularly amusing (not to the authors!) caption: "... On April 1 of year 2 in the $S_0$ experiment, the computer was hit by a cosmic ...
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### Proof of the lower bounds of time of algorithm working [closed]

I have asked this question on math.stackexchange already: http://math.stackexchange.com/questions/515920/lower-bounds-on-the-running-time There are some problems, when there is non-trivial lower ...
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### A simple language and systematic computations

The following somewhat popular simple computer language was enjoyed on sci.math, sci.math.research, pl.sci.matematyka, and perhaps before and after at several places (I wish I knew it's exact history)....
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### What new primitive recursive functions are needed to reconcile Turing time complexity with Gödel 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 $0$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...
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 $g$. ...