# Questions tagged [function]

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### Programming languages with canonical functions

Are there any (functional?) programming languages where all functions have a canonical form? That is, any two functions that return the same values for all set of input is represented in the same way, ...
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### Universal Function approximation

It is known via the universal approximation theorem that a neural network with even a single hidden layer and an arbitrary activation function can approximate any continuous function. What other ...
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### Why was Schönfinkel's work on eliminating "bound variables" in logic so crucial?

AFAIK, The first evidence of using higher order functions goes back to Schönfinkel's 1924 paper: "On the Building Blocks of Mathematical Logic" - where he allowed one to pass functions as ...
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### Existence of injective length-preserving rational function to a smaller alphabet

(This is a simpler rephrasing of an earlier question I have since deleted.) Definitions For this question, a finite-state transducer is like a standard NFA, except at each transition, the transducer ...
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### Universal approximation theorem of second order

The universal approximation theorem (https://en.wikipedia.org/wiki/Universal_approximation_theorem) informally states that up to several conditions, any function can be approximated by a shallow ...
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### A class of functions on a lattice that are easy to optimize

Let $({\cal P}(X),\subseteq)$ be the subset lattice for a finite set $X$. Consider a function $f:{\cal P}(X)\to \mathbb{R}$ with the following property: Given any element $I_0\in {\cal P}(X)$, there ...
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### Combining (block)-sensitivity and Lipschitz conditions?

If we're given a boolean function $f : \{0,1\}^n \rightarrow \{0,1\}$, we can define its sensitivity as follows. The sensitivity $s(f, w)$ with respect to input $w$ is the number of ways of flipping a ...
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### Is there any good literature on the computational complexity of function problems?

There are some cstheory questions that touches function-problems. Like this: Complexity class corresponding to sorting So here is the question: Is there good literature about the computational ...
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### Constructing terms of function types out of the empty type

If a function $f$ is understood as its graph, i.e. a set of pairs $\langle x,y\rangle$ where $x$ is input and $y$ is output, then the empty set $\emptyset$ is a valid function, and for any set $A$, we ...
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### Cook reduction for search problems, by universal property?

A search problem is a relation $R\subseteq \Sigma^*\times\Sigma^*$. A function $f\colon \Sigma^*\to\Sigma^*$ solves $R$ if $(x,f(x))\in R$ for all $x\in\Sigma^*$. Define a search problem to be ...
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### Examples of nontrivial non-discriminatory functions

I am reading Cybenko's "Approximation by Superpositions of a Sigmoidal Function". The paper defines a discriminatory function as: $\sigma$ is discriminatory if for a measure $\mu$, \begin{align} \int ...
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### What is a known sequence for which being constant is not provable?

My question concerns the property of being constant for computable functions ${\mathbb N}\to \{0,1\}$, within any common framework $T$ strong enough to include Heyting arithmetic (and of course not ...
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### What is a "Covering Function"?

In Idris2, I will sometimes get an error telling me that a function "is not covering", which is apparently distinct from it not being total (and I do understand what a total function is). I ...
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### Newbie question: Meta-functions?

Consider a function F that takes a function and produces a function based on structure of the input function. As an example consider F that takes all functions having at least two conditionals and ...
### Multivariable concave function $(n - 1) f(x) >= \sum_{i=1}^{n} f(x_{-i})$
Define the multi-dimension concave function $f(x): \mathbb{R}^n_+ \rightarrow \mathbb{R}_+$ where $x \in \mathbb{R}^n_+$, here I use $\mathbb{R}_+$ to represent the range $[0, \infty)$ and we let \$f(\...