Facts about Bash/Dash/sh

  • Arch Linux uses Bash(Bourne Again Shell) for POSIX shell (/bin/sh)
  • When Bash is called with sh, it tries to emulate the POSIX shell
  • Dash is POSIX compatible, making it Bash incompatible.

Facts about of V-Table

In C++, the virtual table pointer is located in the head of the class instance.

The virtual table is a table of pointer to virtual(overridden or not) functions.

Therefore, to invoked these virtual methods, a mechanism called thunk(a jocular form of think) is needed. It means doing something before the actual function is called. And functional language relies largely on this mechanism.

Fact about Zeromq

  • It is a distributed communication library.
  • It sits on top of the TCP/IP layer, means the possibility is huge that it sends message by syscalls like send.

Facts about lttng

  • Means Linux Tracing Tool, next generation
  • In Artful repo, for older Ubuntu versions, go to ppa:lttng
  • Can be used to trace syscall, or even user application(need to write code into the program)
  • babeltrace could be used to translate the file.

Facts about Ubuntu

  • The life span of Zesty(17.04) has ended in Jan. 2018. Artful will meets its end in Jul. 2018.
  • The life span of non-LTS versions is 9 months.
  • 18.04 is called Bionic, will upgrade later

Facts about Opam

The package manager of OCaml. Not a kind of mineral.

Facts about type algebra

Refer to this page.

The type could be expressed in an algebra-like manner:

0: None type, there is 0 way to construct an empty type object

1: Singleton

2: Two ways to construct the type, boolean type is an example

$+$: Sum type, the object of type $a+b$ could be either type a or type b, but not both, there are exactly $a+b$ ways to construct the object. Injection is used to construct an instance, if it is called an instance.

$\times$: Product type, the object of type $a\times b$ is a combination of an object of type a and one of type b. OCaml adopts this notation, an constructor applied to type $a\times b$ is written as Foo of a*b.

And as a surprising yet unsurprising coincidence, the sum and product obey the rules:

Communicative: $a+b=b+a$

Associative: $a+(b+c)=(a+b)+c$

Distributive: $a\times(b+c)=a\times b+a\times c$

For recursive types like the tree, which is

data Tree a = Leaf a | Branch (Tree a) (Tree a)

The formula is:

$T = a + T^2$

Therefore, we have:

$$T^2-T+a=0 $$

$$T = \frac{1-\sqrt{1-4a}}{2}$$ (Why choose the minus sign?)

Use Taylor series, we could obtain:

$$T=1 + a + a^2 + 2 a^3 + 5 a^4 + 14 a^5 + 42 a^6 \cdots$$

And the coefficients of a are Catalan numbers.

Zippers means we could turn the pocket inside out and focus on one element in a data structure, in a way that updating the focused element , as well as moving the focus around, takes $O(1)$ time.

And we can focus on the holes of a type.

$a\times b$ has one hole of a

$a+a$ has two holes of a, that is data _, a and data a, _

It is likened to partial differentiation, for example:

$$\frac{\partial (u \times v)}{\partial a} = \frac{\partial u}{\partial a} \times v + \frac{\partial v}{\partial a} \times u$$

If $\frac{\partial t}{\partial a}$ means the hole numbers of a in t, and let the g of constant be zero, and g of other variables be one, then magic!

It can be applied to the list type:

data List a = [] | Cons a (List a)

$$L=1+aL$$, take both sides’ derivative of a, we got:

$$\frac{\partial L}{\partial a} = L + \frac{\partial L}{\partial a} \times a$$

$$\frac{\partial L}{\partial a}=\frac{L}{1-a}$$ and$\frac{\partial L}{\partial a}=L^2$. Bravo! That is the zipper type, one pointing to left and the other right.


Added 3/29:

But the same thing could not be perfectly applied to the binary tree type.

Recall, we have the binary tree type:

data Tree a = Leaf | Branch a (Tree a) (Tree a)

Then the formula is:

$$T=1+aT^2$$

The best I could get is about:

$\frac{\partial T}{\partial a}=T^2L$, and the $L$ is a list of type $2aT$.

According to LYAHFGG, the zipper of a binary tree should be something like this:

data Crumbs a = LeftCrumb a (Tree a) | RightCrumb a (Tree a)
type BreadCrumbs a = [Crumbs a]
type Zipper a = (Tree a, Breadcrumbs a)

It is easy to see that Crumbs a has a type of $2aT$ and BreadCrumbs a has a type of $\frac{1}{1-2aT}$.

But wait! That gives $T$ rather than a $T^2$ in the numerator.

Maybe we’ve just started wrong, like, we took the wrong differentiation.

That would be left for future investigation.

Misc

PEBKAC (Problem Exists Between Keyboard And Chair) is the word that describes user error.