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Questions tagged [partial-functions]

Questions on partial functions, history, usage, properties, significance for computability theory, connections to (inverse)-semi-groups and other algebraic theories.

A partial function generalizes the concept of a function by not requiring that every element of the nominal domain is mapped to an element of the codomain. They became prominent in computability theory, but also have connections to (inverse)-semi-groups and other algebraic theories.

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What kind of 2-pullback is this?

I'm looking for the name of a specific kind of "weak" pullback. I'm curious if anyone might recognize it. Consider a category $\mathbf{ParFun}$ whose objects are sets, and whose morphisms $f:X \to Y$ are partial functions. The composite $g \circ f$…
Joe
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Eager vs. lazy interpretation of recursive functions

One of the ways of defining the set of recursive functions is to define first a language $L$ by induction in the following way: $\mathsf{Z}^1 \in L$; $\mathsf{S}^1 \in L$; $\mathsf{P}^n_k \in L$ for all $n, k$ with $n \ge 1$, $1 \le k \le n$; if…
Luca Bressan
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Compact-open Topology for Partial Maps?

Compact open topology is one of the most common ways of turning the set of continuous maps between two topological spaces into a topological space. Suppose $X, Y$ are topological spaces, and let $K\subseteq X$ be a compact subset and $U\subseteq Y$…
Bumblebee
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Can you define functions which are not primitive recursive, yet total, in Type Theory?

Ackermann's function is total but not primitive recursive. Can one define Ackermann's function in Type Theory, ie: Can you define functions which are not primitive recursive, yet total, in Type Theory? [this post was closed] due to being "not…
RFV
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Definition of (lazy) conditional in partial recursive functions

I'm currently working on formalizing the theory of partial recursive functions, and something seems peculiar about the standard definition. The definition of the primitive recursion clause of $\mu$-recursive functions reads as follows (according to…
Mario Carneiro
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When can a partial associative operation be extended?

Let $X$ be a set with a partial operation $\cdot$ which is associative in the sense that if $x, y, z \in X$ and $x \cdot y$ and $y \cdot z$ are both defined, then $(x \cdot y) \cdot z$ and $x \cdot (y \cdot z)$ are both defined and $(x \cdot y)…
I Eat Groups
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The use and meaning of the "tilde-equal-symbol" for partial (recursive) functions in Girards monograph 'proof theory and logical complexity, volume 1'

English is not my native language, so please forgive if I do not express myself properly. I am working with the book named above on proof theory, and I have a little problem with the authors use of the $\simeq$ symbol. He does not introduce it…
Ettore
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Example function which is partial, injective, and surjective

Can somebody give an example of a function $f : \mathbb{N} → \mathbb{N}$ which is partial, injective, and surjective. I was thinking about $f(x)=x-1$, but I am not sure if it is surjective.
Waseem Anwar
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Number of Partial Surjective Functions from X to Y

The number of total surjective functions from $X$ to $Y$ is known to be $T = y!\left\{{x \atop y}\right\}$, with $|X|=x, |Y|=y$. However, I am interested in the number $P$ of partial surjective functions, that is, elements of $X$ do not necessarily…
doc
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Why can't a base be negative in an exponential function?

The function $f(x) = a^x$ is generally taught to only allow $a > 0$. This is usually justified by giving a few examples of complex points in cases where $a < 0$. For example: $f(x) = (-2)^x$, if $x = 0.5, f(x)=sqrt(-2)$, which of course is…
fedsavi
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Topological perspective of Rice-Shapiro theorem

Given a Gödel numbering $\phi_n$ of the partial recursive unary functions, the Rice-Shapiro theorem states Let $\mathcal A$ be a family of partial-recursive unary functions such that its index set $A = \{ x \in \mathbb N\ |\ \phi_x \in \mathcal A…
Keplerto
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Find P if Integrating factor of $x(1-x^2)dy+(2x^2y-y-ax^3)dx=0$ is $e^{\int Pdx}$.

I have seen one method which correctly evaluates $P=\frac{2x^2-1}{x(1-x^2)}$. But I have seen a method that says if $\frac{\frac{\partial M}{\partial y}-\frac{\partial N}{\partial x}}{N}$ is a function of just x ,i.e, it is equal to f(x) then…
Nivedita Chowdhury
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Analytic Continuation of Fractions

Excuse my lack of expertise, I study natural sciences (physics) and not mathematics so I will be off with my terminology and mathematical vocabulary. Please feel free to poke and build at this idea but I do request at least playing around with it…
Captain HD
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The logic behind adding this sentence to the second condition in this definition of isomorphism

Below is the definition of isomorphism quoted from the textbook Introduction to Set Theory by Karel Hrbacek and Thomas Jech. First, we introduce relevant definitions: An $n$-ary relation $R$ in $A$ is a subset of $A^n$. Then we write…
Akira
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What equivalence relation is being used to define the category of partial maps?

Here's what Awodey says in his Category Theory. For any category $\mathbf{C}$ with pullbacks, define the category $\mathbf{Par}(\mathbf{C})$ of partial maps in $\mathbf{C}$ as follows: the objects are the same as those of $\mathbf{C}$, but an arrow…
D. Brogan
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