Sat, 15 Dec 2018 14:32:02 GMT

And again, I gave a talk for the
best mathematics club of the multiverse, for "younger" students,
meaning that **this text is not scientific**. This is the script for
the talk.

We want to prove the following theorem:

**Theorem:** There is an algorithm of which, with standard set theory,
we cannot decide whether it terminates.

Firstly, we need to look at what an algorithm is. Usually, people introduce this with Turing machines. However, we will use a model which is closer to modern computers, so-called register machines. These are provably equivalent to Turing machines, except that some algorithms might take a bit longer on Turing machines – but at this point, we do not care about efficiency, we just care about whether things can be calculated at all, given enough time and space. If you care about efficiency, that would be in the realm of complexity theory, while we will work in the realm of recursion theory.

A register machine has a finite set of registers `R[0]`

, `R[1]`

, …,
which can contain arbitrarily large natural numbers. A program
consists of a sequence of instructions:

- For every i, there is the instruction
`R[i]++`

, which increases the number that is contained in`R[i]`

. - For every i, there is the instruction
`R[i]--`

, which decreases the number contained in`R[i]`

if it is larger than 0, and does nothing otherwise. - There is the instruction
`End`

, which ends the program. - For every i and n, there is the instruction
`if R[i]==0 then goto n`

. The instructions are numbered, and this instruction jumps to the instruction with number`n`

, if`R[i]`

contains 0, otherwise it does nothing.

The following program checks whether `R[0] >= R[1]`

, and if so, it
sets `R[2]`

to 1:

```
0 if R[1]==0 then goto 6
1 if R[0]==0 then goto 5
2 R[1]--
3 R[0]--
4 if R[2]==0 then goto 0
5 End
6 R[2]++
7 End
```

Line 4 is only here to do an unconditional jump, therefore, we can introduce a shorthand that just does an unconditional jump:

```
0 if R[1]==0 then goto 6
1 if R[0]==0 then goto 5
2 R[1]--
3 R[0]--
4 goto 0
5 End
6 R[2]++
7 End
```

To make it a bit more readable, we can omit the line numbers, which we
usually do not need, and just set *labels* to the lines we want to
jump to:

```
Start: if R[1]==0 then goto Yes
if R[0]==0 then goto No
R[1]--
R[0]--
goto Start
No: End
Yes: R[2]++
End
```

This just increases readability, it doesn't make anything possible
that wasn't possible before. By using this algorithm, we can generally
check whether `R[i] > R[j]`

, and therefore, we can introduce a
shorthand notation `if R[i] > R[j] then goto A`

without being able to
do anything we couldn't do before.

We can do addition by

```
Start: if R[1]==0 then goto Done
R[0]++
R[1]--
goto Start
Done: End
```

and truncated subtraction () by

```
Start: if R[1]==0 then goto Done
R[0]--
R[1]--
goto Start
Done: End
```

Therefore, we can add the instructions `R[i]+=R[j]`

which adds `R[j]`

to `R[i]`

, and `R[i]-=R[j]`

, without being able to do anything more
than before. Having these instructions, we can define multiplication
by repeated addition, and division and modulo by repeated
subtraction. This is left to the reader.

Now, we cannot know for sure whether everything that is computable at
all is computable by register machines. However, we do not know any
computable function that cannot be computed by a register
machine. Hence, it is generally believed that there is none. This is
called the **Church-Turing thesis**.

These programs operate on numbers only. Real computers work with
images and text. However, this is, computationally, no difference, and
there are several injections between these kinds of data. Especially,
programs themselves can be represented as numbers, which is called
**Gödelization**. For this, we use the Cantor Pairing, which
gives a bijective function
by
. This function
enumerates the backward diagonals on the grid of natural numbers, as
this graphic shows. The pairing itself can obviously be
calculated with the above functions by a register machine. Inverting
the function is also easy, and left to the reader.

Now we can represent every program we have in the following way: We first map numbers to the single instructions:

`R[i]++`

is mapped to`R[i]--`

is mapped to`if R[i]==0 then goto j`

is mapped to`End`

is mapped to

Therefore, every instruction has its own code. A program can be encoded as a sequence of these codes; the program with the instructions can be encoded by .

Therefore, it is well-defined to talk about "programs getting other
programs as parameters". And having seen this, it is easy to write an
**universal register machine**, which evaluates such a program, given
a sequence of register values:

The state of a program is entirely determined by its registers and the number of the current instruction. It has a finite sequence of registers , and an instruction line , and therefore, we can encode the state by .

Now, let contain the program code, and contain the current state. Let be the first element of the program state, which tells us, at which position of we are. Let be the current instruction given by .

- If , let and
. is the value of the
-st register in the simulated program. We replace it
inside by and replace
by . We
simulated
`R[k-1]++`

. - Similarily, we can simulate
`R[k-1]--`

. - For , we have to check the register value, and set the instruction number to the apropriate value.
- For , we end, and have the final state of the program.

While it is really intricate, we can see that it is possible to
"simulate" a register program inside a register program. A natural
question which arises is: Is there an algorithm such that, given a
program (or its Gödelization), and an input state ,
the algorithm determines whether terminates. This problem
is called the **Halting problem**. We now show that it cannot be
solved. Formally, we show that there is no program that,
given the Gödelization of a program in and an
input state in , leaves being 0 if and only if
with the given input state terminates. We do this by
contradiction: Assume such an M exists. Then we could, from this M,
generate the following program:

```
"execute M"
if R[2]==0 then goto Loop
End
Loop: goto Loop
```

This program terminates if and only if the given program with the given state does not terminate. We modify this program once again by one line:

```
"set R[1] to <R[0]>"
"execute M"
if R[3]==0 then goto Loop
End
Loop: goto Loop
```

We call this program . only takes one argument. It
terminates if and only if the given program, given its *own*
Gödelization, does not terminate. Such programs are called
**self-accepting**.

Now, as is itself a program, we can Gödelize it, so let be the Gödelization. By setting , we can calculate .

Now assume terminates. This means that in after
executing M there will be `R[2]==0`

. Therefore
would not terminate. Contradiction.

But assuming would not terminate would mean, by the same
argument, that `R[2]`

is not 0 after M. Therefore, would
terminate. Also a contradiction.

Such a program cannot exist. Therefore, cannot exist.

This proves that we cannot generally decide whether an algorithm terminates. However, it is not yet what we want: We want an algorithm, of which we cannot decide whether it terminates, at all. To get it, we need to do a bit of logic. We will mainly focus on Zermelo-Fraenkel set theory here, as it is the foundation of mathematics.

We first define what a mathematical *formula* is, which is essentially
a string that encodes a mathematical proposition.

- We have an infinite set of
*variable symbols*. - The set of strings is the set of
*atomic formulae*: Formulae which just give them -relation between two free variables.

Now, the set of formulae is the smallest set, such that

- Every atomic formula is a formula: .
- If and , then ("for all a X holds") and ("there exists an a such that X holds") and ("not X") are in .
- If , then ("X and Y"), ("X or Y") and ("X implies Y") are in .

We now give the axioms of set theory:

**NUL**: There is an empty set:

We can introduce a common shorthand notation for by , and rewrite this axiom as

If we want to talk about the empty set now, we need to introduce some variable , and add to the formula. Therefore, our system doesn't get stronger if we introduce a symbol for the empty set, instead of always adding this formula, and it increases readability, which is why we do that.

We furthermore define the shorthand notation by , and by .

**EXT**: The axiom of *extensionality* says that sets that contain the
same elements are also contained in the same sets:

**FUN**: The axiom of *foundation* says that every set contains a set
that is disjoint to it. From this axiom follows that there are no
infinite -chains.

or, with additional obvious shorthand notation

**PAR**: The axiom of *pairing* says that there is a set that contains
at least two given elements, meaning, for all , there
exists a superset of :

**UN**: The axiom of *union* says that the superset of the union of all sets in a set exists:

**POW**: The axiom of the powerset: A superset of the powerset of
every set exists:

.

**INF**: The axiom of *infinity* says that a superset of the set of
natural numbers exists. Natural numbers are encoded as ordinals:
, and . Writing it out
as formula is left as an exercise.

The other two sets of formulae we need are given by **axiom schemes**:
They are infinitely many axioms, but they can be expressed by a
simple, finite rule:

**SEP**: The axiom scheme of *separation* says that, for every formula
and every set , the set exists:

Let a formula be given with free variables among , and not occur freely. Then the formula

.

is an axiom of set theory.

**RPL**: The axiom scheme of *replacement* is a bit more complicated.

A formula is called a **functor** on a set
(which is *not* the same as a functor in category theory), if for all
there is a unique such that
holds. Therefore, in some sense, defines something
similar to a function on , and we write
for this unique . Then the set
, the "image" of ,
exists. Formalizing this scheme is left as an exercise.

**AC**: It should be noted that usually the axiom of *choice* is
added. However, we do not need to care whether it is added or not,
so we omit it here.

We already talked about embedding natural numbers into this set theory. We can also define general arithmetic inside this set theory. Most of mathematics can be formalized inside Zermelo-Fraenkel set theory.

Now, we can formalize propositions. Now we want to formalize proofs. Normally, I would introduce the calculus of natural deduction here, because it corresponds to the dependently typed lambda calculus, so every proof is a term. However, for the specific purpose we need, namely, formalizing proof theory in arithmetic, the equivalent Hilbert calculus is the better choice. It corresponds to the SKI calculus for proof terms.

Firstly, we further reduce our formulae: We can express
as , and as . Furthermore, can be expressed by
. Hence, we only need ,
and to express all formulae. We now
define additional *logical axiom schemes*, where
range over all formulae. (Notice:
is right-associative.)

- P1.
- P2.
- P3.
- Q5. for all variables
- Q6.
- Q7. if is not free in

A *proof* of a formula is a finite sequence of formulae
, such that and for
all , either is an axiom of set
theory, or a logical axiom, or there exist such that
. Essentially this means that
everything in the formula is either an axiom or follows from former
formulae applying modus ponens.

**Completeness Theorem:** If a formula is true in set
theory, then there exists a proof of it.

To prove this, we would need model theory, which would lead too far, so we leave out the proof.

Now, as we did for programs before, we can gödelize formulae and proofs. Let us denote by the gödelization of .

**Diagonalization Lemma:** For every formula with one
free variable , there exists a formula , such
that holds.

*Proof:* First, we notice that, given the formula , we
can express the *substitution* of another variable for
, therefore, we can give a function that satisfies
. Now we can
define . Now,
define . Then we have
. This concludes the proof.

Notice that the definition of is computational: It can be done effectively by a computer. As we can find such a formula for every , we denote it by .

Now, we can also gödelize proofs and their correctness criterion. Therefore, we can give a formula meaning " is the gödelization of a correct proof of the gödelized formula ". Therefore, says that the gödelized formula is provable.

By the diagonalization lemma, there is a formula such that . Now, assume that does not hold. Then also cannot hold, therefore, it would be provable, which is a contradiction. Hence, must hold. But then, it cannot be provable. This is a (sketch of a) proof of

**Gödel's first incompleteness theorem:** In Zermelo-Fraenkel set
theory, there are propositions that can neither be proved nor
disproved.

More generally, this theorem holds for all axiom systems that are capable of basic arithmetic, because this is all we used. Specifically for Zermelo-Fraenkel set theory, there are other examples of such propositions, namely the continuum hypothesis, and the existence of large cardinals.

Now, something we always implicitly assumed is that set theory is
**consistent**: If is provable, then
cannot be provable. This is, however, unknown, which follows from:

**Gödel's second incompleteness theorem:** Set theory cannot prove its
own consistency.

*Proof:* We use our from the proof of the
first incompleteness theorem. Furthermore, we can define
such that
. Now, we can
define what it means to be consistent, namely: . Now, we know that
, and therefore, since
false propositions imply anything,
for all formulae ,
and obviously this implies
. Therefore,
. But this contradicts what
we proved in the first completeness theorem. Hence,
cannot be provable.

Let . Obviously, if and only if set theory is inconsistent (since it is wrong). Now consider the following algorithm:

```
Retry: if ν(R[0], a) then goto Found
R[0]++
goto Retry
Found: end
```

Does this algorithm terminate?

If it terminates, it has found an inconsistency in set theory. Assuming that set theory is consistent, it would not terminate. But if we could prove that it does not terminate, we would be able to prove that set theory is consistent, and this contradicts the second incompleteness theorem.

Hence, we have an algorithm of which we cannot decide whether it terminates.

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Fri, 07 Dec 2018 14:42:45 GMT

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Nerd Culture:

- MIT researchers create a robot houseplant that moves on its own (via) – has science gone too far?
- A Portal Port Programmed For Platforms Of The Past
- Six Degrees of Wikipedia (via)
- The blue eyed islanders puzzle
- Mensa Online-Test

STEM/MINT:

- Mitochondrial DNA Can Be Inherited From Fathers (via)
- Record-breaking atomic clocks precise enough to measure spacetime distortions

Software/Programming:

- Google, Mozilla working on letting web apps edit files despite warning it could be 'abused in terrible ways' (via)
- The Kernel Underground
- Microsoft to Silently Remove the CONCATENATE Function from Excel, Offers Paid Alternative: CONCAT
- Quiz zum Tag der Computersicherheit
- Compiler basics: lisp to assembly (via)

WTF:

Art (Comics etc.):

Sat, 24 Nov 2018 22:10:02 GMT

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Politics/Law:

Popular Culture:

Nerd Culture:

STEM/MINT:

Software/Programming:

WTF:

- „Cyber Kitchen“: Deutsche Telekom bringt Kochbuch raus
- Katholische Kirche bringt Pokémon-Klon
- A bra that falls off when you clap your hands (new patent)

Art (Comics etc.):

Fri, 09 Nov 2018 03:04:41 GMT

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Politics/Law:

- Straftäter dürfen Bitcoins nicht behalten –
*"Auch der Einwand, dass die Behörden ohne Kenntnis des privaten Schlüssels nicht auf die Wallets der Angeklagten zugreifen könnten, ändert hieran nichts. Das sei eine reine Vollstreckungsfrage, so das Gericht."* - Kommentar zur IoT-Sicherheit: Europas Verordnung ist zahnlos – was ich ja immer nicht verstehe ist, wieso es nicht einfach standardmäßig so ist, dass die Hersteller in vollem Umfang haftbar gemacht werden können?

STEM/MINT:

- Lumen in Watt richtig umrechnen
- BIGNUM BAKEOFF contest recap
*"The aim of this contest was to write a C program of 512 characters or less (excluding whitespace) that returned as large a number as possible from main(), assuming C to have integral types that can hold arbitrarily large integers. [...] The final and winning entry, {loader.c}, diagonalizes over the Huet-Coquand `calculus of constructions'."* - Ein BIOS-Update von Lenovo Ziegelsteint die Rechner – es geht um irgendwas mit Thunderbolt. Man sollte halt das BIOS vielleicht nicht zu oft verändern, und vielleicht auch nicht zu komplex gestalten.
- Typesetting modal logic
- iPhones are Allergic to Helium
- Psychedelic psilocybin therapy for depression granted Breakthrough Therapy status by FDA
- World-first aerogel is made from plastic bottles
- How to implement strings – maybe interesting for beginners (or people who never used any low-level programming language).
- Cookie Editor for Chrome-Bases Browsers
- GSM-Codes: Befehle für das Mobilfunknetz
- C-QUIZ
- pointers are more abstract than you might expect in C
- Orgmode for GTD

WTF:

Art (Comics etc.):

Quotes:

- It's terrifying that both of these things are true at the same time in this world: 1. Computers drive cars around. 2. the state of the art test to check that you are not a computer is whether you can successfully identify stop signs in pictures.

Thu, 01 Nov 2018 15:43:26 GMT

Nie dürft ihr so tief sinken, von dem Kakao, durch den man euch zieht, auch noch zu trinken.

- Erich Kästner

So I bought a new smear^H^H^H^H^Hsmartphone. My old phone was fine, it was a Moto G5, but somehow the developers thought that an exchangable battery is an excuse for not including a compass – but I often use it to navigate, and that sucks without a compass. Now I bought a Moto G6, which has a compass, but no easily exchangable battery, and a fingerprint sensor that is even worse than the one from the Moto G5.

Not being able to easily exchange the battery is the current trend. I chose the Moto G6 because at least it seems to be doable to exchange the battery at some point. I want a phone which I can have for longer than two years. And I think one reason for not having an exchangable battery is to make people buy new phones after about two years, because that is usually the time, in my experience, when batteries lost a relevant amount of duration. It is an instance of Planned Obsolescence.

Which brings me right to the point: The whole ecosystem around Android is a big capitalist circlefuck. Android has created an immune system against software freedom and personal freedom.

So when starting my new phone, the shiny motorola start animations shalmed right back at me. Then I was asked to enter my SIM code and connect to Wifi. Then I was asked whether I wanted to import settings from another phone, which I wanted, so I started the procedure and hoped that my app settings would be synced.

After that, I was kindly asked to connect my Google account with
Outlook, because reasons. I accidentally did that, now Microsoft can
access my Mails from Google I guess(?). Well, why not. I mean, Outlook
appears to be the default Mail application on this phone. I have no
idea why anyone would *want* that instead of the original GMail app,
but just as a wild guess, one could think of the possibility that
Microsoft might have payed for that.

After that, there was a system upgrade, and lots of "system apps" were updated. "System apps" are apps you cannot uninstall. One would assume that these apps are essential for the system to work. But to be honest, I do not see why "LinkedIn" is an essential app. Again, one might think of the possibility that LinkedIn &c payed for that.

Then I was asked to configure the fingerprint reader, and got some messages from these "system apps" telling me that they are there and why I should use them or something … I just closed them. There were two reasons for choosing a Moto G6: The first reason was that I hoped it would be similar to the Moto G5. I don't see that. The UI is entirely different. The second reason is that the older Moto G phones are supported by Lineage OS, so I hope that in the future, it will also be supported. That is important, because the vendors stop supporting their phones at some point – which is also planned obsolescense, but also creates a huge security hazard. There should be laws against this practice – but that will probably not happen in Europe.

**Update:** I was told that Lineage OS still uses proprietary blobs as
its drivers, which may contain security holes but can only be fixed
by the companies that made them. Replicant doesn't do that,
but won't run on as many devices, therefore.

Now even though I used the official sync functionality, most things just have not been synced. Like, everything useful was not synced. My app settings for Conversations and K9 Mail for example. Also my WhatsApp contacts and logs were not synced. I restored a backup, but only to learn that it lost messages. Because this is not how Android works. In theory, as it is a single-user system, there could be some central place where apps store their configuration. Think of the windows registry or dconf. Instead, apps get their directories in which they place SQLite3 databases.

Which brings me to the next point: The filesystem. Yes, Android has
one, but tries to hide it. I do not understand why hiding the
filesystem is so trendy right now. Hierarchical filesystems are
**good**, **clean**, **simple** and **easy to understand**. They are a
perfect example of an abstraction that is easily usable by humans, as
well as efficiently implementable for computers. On Android, it can be
hard to create a file with one app, and open it with another app, even
though they are on the same computer. This got on my nerves several
times. And people start to think it has to be this way. It
hasn't. Having access to a cleanly structured filesystem is **strictly
better** than whatever Android does.

And since sending a file to another app on your same phone is so hard,
it can be **almost impossible** to get some file from your computer to
your phone (let alone saving it in the right place and making the
corresponding apps open it). I have seen people use DropBox for
this several times, even though the two computers were in the same
room. And for the providers, this makes sense: It makes you use more
traffic, and it makes you upload more files, so they can be scanned
into your ad profile.

This brings me to another point: The connectivity. It is assumed that you have a fast internet connection with no traffic limits. Many apps assume this. YouTube does not properly cache its videos, except when you explicitly download them. Chrome always reloads tabs when they went out of focus too long. There is no caching done. I have also seen some apps profiling your network connection to decide how much bloat to download. All of this assumes that you have an infinite amount of traffic. But of course, this is good for your internet providers: As there is still no affordable real flatrate for mobile internet in Europe, you will have to pay for additional traffic.

However, still, a smartphone is a highly portable computer, and as such, often changes places, and therefor often switches between networks. That is a problem for chat applications which need to manage their persistent connection. Also, sending keep-alive-packets will drain battery. In theory, TCP with the right parameters should be able to handle this. In practice, programmers do not know this. Hence, Google invented cloud messaging: Your app registers a server, which connects to Google servers, and sends messages. Google play services will itself keep an XMPP-connection to the Google servers, and forward those messages to the registered apps.

The problem is that your app needs to be from Google play. F-Droid apps cannot do this. Another problem is that the push server is hardcoded into the app. That is especially bad for free decentral services like XMPP: I am hosting an XMPP server, and Conversations is a very good client, but the version supporting cloud push costs a few euros, because they have to host an own cloud push server. To prevent this, I would also host an own cloud push server. But to do this, I would need to recompile the app and put it into the Google play store. Which is stupid, considering the fact that it would cost me money, and it would publish my Conversations fork for everyone, while I just want the people with an uxul.de-Account to use it. In the meantime, there might be an alternative, HTML5 Web Push. Maybe there will be support for this in some web client like converse.js in the future. At the moment, there isn't.

The App store is a problem of its own. In theory, having an app store is a good thing. It can improve security, because one can quickly react to security holes. It also can do dependency tracking. In theory, it is like a good old package repository, think of Debian/Ubuntu. In practice, there appears not to be any dependency management, and every app just bundles all of its dependencies, because a few hundred megabytes for a mail client is not a reason not to use it, aparently. Also, it costs money. Not much money, but it costs money. Commercial apps have no problem: Hosting a push server in AWS is cheap. And WhatsApp and Telegram are free as in free beer, and work out of the box.

Of course, they collect your data. Yes, they are encrypted, but at
least they know your friends, and they know when you are awake. I
would *guess* that automatically reading your address book entries
should be illegal. The Facebook Messenger at least asks whether he may
access the address book. They make *you* the criminal.

Android kills background applications, except when you explicitly
allow them to run. This is one further reason for cloud push. And this
*really* gets on my nerves sometimes. I often use Google Maps and the
Deutsche-Bahn-Navigator. I want these to stay open, so I can look up
things again afterwards. However, it seems that they are reaped from
time to time. And they will not always go back to the state where I
left them. This is annoying. Of course, you do not have to worry about
closing programs anymore, as you would have on a normal computer. But
I do not really see the great advantage in that. I also do not see why
they do not support swapping.

People often argue that my opinions assume that smartphones are
computers. They do. But I don't see why smartphones are *not*
computers. They are small, highly portable computers, with lots of
sensors and a touchscreen. The touchscreen is the main difference to
laptops and notebooks. And for computers, there is a set of principles
that **work**: The UNIX principles.