Foundations

[Sebastien Zany]

Hi everyone,

I'm new to this idea and I'm still trying to understand exactly what's going on from a formal perspective (implementation aside), so forgive me if my questions are somewhat basic.

I gather than the fundamental idea is the infon, and that users are intended to make models, but I lack a complete picture of how everything fits together.

For starters, maybe someone can help me understand how are infons and models different from binary representations and types?

Thanks,

Sebastien

[Bruce Long]

Hi Sebastien,

Great question! I want to spend a little time answering it (I just got back home) so If you don't hear back tonight that's why.

If not later tonight I'll definitely get to it early tomorrow though.

Thanks!

Bruce

--
Give me immortality or give me death!

[Bruce Long]

Hi Sebastien,

Thanks again for your question. If you want just a quick English definition of an infon they are "pieces of information." It was just too long to say that, and Keith Devlin (the math guy) coined this term infon so I borrowed it.  We can give a much more mathematical explanation of what infons are but for now let's just go with "piece of information."

But what KIND of information? One definition I have heard tries to distinguish data from information. I've also heard people speak as if information required a brain consciously processing "data" But what I am talking about has more to do with the way we speak about computers. It also aligns with Shannon's theory of information. (http://en.wikipedia.org/wiki/Claude_Shannon)

To be a little more rigorous, think about the equation that tells us how many states are in a byte:

1 byte = 256 states

States and bytes are just different units for measuring information. So, if infons are pieces of information AND information is measured in bytes or states AND physical systems have states, what does that say about the connection between infons and physical objects or systems?

One of the keys to grasping Proteus is to see that everything is an infon: atoms, the universe, people, businesses, computer memory... If you like ancient philosophy, think "Logos." Infons are physical, really existing systems with states. Some of those states my be located inside a computer in the form of memory but in principle those states are no different than other states outside the computer. And Proteus doesn't care whether the states are inside a computer or outside. SO The things Proteus refers to are real physical systems, not symbols.

You might wonder, if there are no symbols is it even related to mathematics? Yes, because infons behave in a mathematical way. And the language was derived from the math that I understand. Parts of the language are almost certainly improvable but I don't understand all the of math necessary so I hacked. The biggest hack involves types. There is a mapping that I feel strongly should exist and it's related to the way Proteus handles types, repetitions and a few other things. If that mapping is found I suspect the language would become smaller, simpler, and faster. Alas, I haven't had time to think about it for over a year. The good news is the the "hacks" I have in place aren't too ugly and they do and will work.

Does that help a little?

Bruce

On Thu, Apr 7, 2011 at 6:07 PM, Sebastien Zany <seba...@chaoticresearch.com> wrote:

[Davide Del Vento]

I think this is a important discussion, and, Bruce, I like your explanation.

The only point that IMO requires a little more elaboration is the
following (Sebastien and others, feel free to ask us to elaborate on
anything else):

> SO The things Proteus refers to are real physical systems, not symbols.

Bruce, do you think that proteus "acts" on real physical systems?
Consider an infon using data coming from a sensor measuring, say, room
temperature in Proteus. How is it different from some random bytes
used the same sensor fed into a "conventional", say, java process that
is displaying that temperature remotely on my phone - for example.

I think I know the answer, the example with airline ticket on the
website was my ahaha! moment, but for the sake of the reader I'm
asking here anyway.

Thanks,
Davide

[Bruce Long]

Hi All,

Davide, your comment was "spot on."

On Fri, Apr 8, 2011 at 10:06 AM, Davide Del Vento <davide.d...@gmail.com> wrote (edited a bit):

The only point that IMO requires a little more elaboration is the

following:

> SO The things Proteus refers to are real physical systems, not symbols.

To explain what I mean by saying Proteus refers to real systems not symbols it's necessary to understand the Proteus concept of identity in a little more detail.

The concept of identity in Proteus is the most important concept to understand if you want to deeply understand the language. Any two infons are either identical or not identical.
Here's the story I tell: Suppose I ask you what your shoe size is and you give me a number then I ask how many miles you must drive/ride to work and you give me another number.
And suppose that those two numbers happened to be the same number. If we are using numbers under the traditional definition we would say that they are "equal."

    ShoeSize = MilesToWork

But, they are not the same piece of informaition; they aren't the same infon. So as infons

    ShoeSize != MilesToWork

This changes the way the algebra works a little. It means that we can't tell by looking at the symbol for two numbers A and B which of the following is an accurate description: A=B or A!=B. This is why I say Proteus doesn't use symbols but infons. The properties of the symbol are not useful when we can't use it to tell whether two infons are identical.

(Notice in that paragraph that I didn't say which statement is true but which is an accurate description. Pragmatically, it's a minor point but logically it's important: infons don't have truth values built in. We could define them in Proteus, but it's not that useful because the idea of accurate descriptions is easier and more intuitive once you get used to it.)

Let's use the term 'identity' for the Proteus concept of identity and 'equality' for the traditional concept. How does the switch from equality to identity change the algebra?

To answer, consider that I give you 5 bytes of memory and ask you to communicate to me some information I need; it doesn't matter what for this example, only that you have just 5 bytes and no more.

Now if you tell me what 4 of the bytes are then tell me that the 5th one is equal to the first one I now know all five. And we still have five bytes. It's neither here nor there that two of them are equal, is it? If we represented your message in bits instead of bytes MANY of the bits would be equal to each other.

Now, suppose that just after I tell you that you have 5 'slots' into which you can place one byte of information, I also say "BTW, slot 5 will be identical to slot 1". You test it out by placing several values in slot 1, and sure enough, slot 5 updates each time with the same value as slot one. Now there may be a problem. You just lost a byte. You really have only four now.

So the point of this is that identity in the Proteus sense represents a loss of information. Every new identity that you learn about in those state systems which you can control gives you fewer states to work with.

This has a MAJOR consequence and it's the number one reason that Proteus can do things that say, Haskell, cannot.

I'll need to send this off but I'm not finished so I'll get back to it soon.

Bruce, do you think that proteus "acts" on real physical systems?

I'll get to that question too. It's related.

Bruce

[Bruce Long]

Hi folks,

I've created a new wiki page about infons. The new page is intended to walk you though thinking about the structure of infons. It's a work in progress. If you have insights or find mistakes please edit the page.

https://github.com/BruceDLong/ProteusCore/wiki/Infon-Theory-Intro

[Davide Del Vento]

Hi,

How cool.

However, I don't get the following:

> In other words, can we say that there is no action
> that moves the system from state 1 to state 3,
> for example? No, because we could either get
> to that state through a combination of actions
> (in which case just put it in), or we can never get
> to it (in which case why list it in the first place?)

What if state 3 can be only reached from state 4 or 2 and not from
state 1? Think of a switch like the one on the right of this picture:
http://media.rsdelivers.cataloguesolutions.com/LargeProductImages/R193573-01.jpg

I see that this is essential to go to the group theory (which I am
familiar with, although not fluent in), so I believe some more
elaboration is needed.

Bye,
;Dav

[David Van Duzer]

Hi guys.

On Sun, Apr 10, 2011 at 7:05 AM, Davide Del Vento
<davide.d...@gmail.com> wrote:
> However, I don't get the following:
>
>> In other words, can we say that there is no action
>> that moves the system from state 1 to state 3,
>> for example? No, because we could either get
>> to that state through a combination of actions
>> (in which case just put it in), or we can never get
>> to it (in which case why list it in the first place?)
>
> What if state 3 can be only reached from state 4 or 2 and not from
> state 1? Think of a switch like the one on the right of this picture:
> http://media.rsdelivers.cataloguesolutions.com/LargeProductImages/R193573-01.jpg

This is partially a statement about mathematical closure.
Definitionally, if you can't get to state 3 from state 1, then you
don't have a group. In your example, it is possible to reach state 3
from state 1, but you have to traverse through state 2 first. This is
perfectly compatible with group theory (and infon theory).

Bruce's point is that we can't conjure up a system with impossible
states. We can wax poetic about a possible state of affairs, but if
the system can't logically be arranged that way, then we are speaking
nonsense. Consider a Rubik's Cube. If you take two colored stickers
on the corner and swap them, you have a Rubik's Cube that can never be
solved without changing the stickers again. You can rotate it through
as many combinations as you like, but you can't reach a solved state
by working within the defined operations of the system.

However this line of thought raises a separate question for me. When
examining systems in the physical world, we appear to have one-way
operations. If you break a teacup, you can't put it back together.
Entropy is directional, right? How does Proteus address this?

dvd

[Bruce Long]

On Sun, Apr 10, 2011 at 7:05 AM, Davide Del Vento <davide.d...@gmail.com> wrote:

However, I don't get the following:

> In other words, can we say that there is no action
> that moves the system from state 1 to state 3,
> for example? No, because we could either get
> to that state through a combination of actions
> (in which case just put it in), or we can never get
> to it (in which case why list it in the first place?)

What if state 3 can be only reached from state 4 or 2 and not from
state 1? Think of a switch like the one on the right of this picture:
http://media.rsdelivers.cataloguesolutions.com/LargeProductImages/R193573-01.jpg

We can consider sequences of state changes, So, while we may not have a transition directly from state 1 to state 3,  we can compose such a transition: 1->2->3.

An example of this is travel through space. I cannot get directly from here to Denver (consider that my spatial location is part of my state), but I can get there be transitioning through other bits of space.

This applies to the switch you mentioned too. There is not direct transition off->8R,  but you can take the switch from off to 8R by off->RY->Y8->8R.

Cheers,

Bruce

[Bruce Long]

This is partially a statement about mathematical closure.

  In your example, it is possible to reach state 3

from state 1, but you have to traverse through state 2 first.  This is
perfectly compatible with group theory (and infon theory).

Exactly.

 

Bruce's point is that we can't conjure up a system with impossible
states.  We can wax poetic about a possible state of affairs, but if
the system can't logically be arranged that way, then we are speaking
nonsense.  Consider a Rubik's Cube.  If you take two colored stickers
on the corner and swap them, you have a Rubik's Cube that can never be
solved without changing the stickers again.  You can rotate it through
as many combinations as you like, but you can't reach a solved state
by working within the defined operations of the system.

And it depends on what we are modeling. Perhaps there is a system where we can go directly from any state to any other state, Then we can model it that way,

 

However this line of thought raises a separate question for me.  When
examining systems in the physical world, we appear to have one-way
operations.  If you break a teacup, you can't put it back together.
Entropy is directional, right?  How does Proteus address this?

There may be actions that are too hard for us to compose. In theory, you could reverse the teacup breaking -- it would not violate laws of physics. But in practice it's too hard to do because the state transitions required are too unlikely.

I'll add some more about this to the wiki page soon.

[Davide Del Vento]

>> Entropy is directional, right?  How does Proteus address this?
>
> There may be actions that are too hard for us to compose. In theory, you
> could reverse the teacup breaking -- it would not violate laws of physics.
> But in practice it's too hard to do because the state transitions required
> are too unlikely.

In fact this is the case. However, if that's the level of modeling we
want to achieve, I think we will be out of luck. Let's make an example
of a gas expanding in a room, which is much easier to model than the
cup breaking (conceptually there is just a little difference). Imagine
the gas expanding from being all concentrated in point A (a small
volume) to the whole room. Microscopically, gas expansion is
reversible. Macroscopically it's not. What this means microscopically
is that the number of gas molecules is so large (say, of the order of
10^23) that the state "all the gas goes back to point A" is very
unlikely (let's say: really impossible) to happen. So a microscopical
model (with ~ 10^23 molecules) does not need to bother with
irreversibility. However, if we would like to have a macroscopic
model, with much less states (and memory usage) we have to model
irreversibility explicitly, This is Physics (did I tell you that I am
a Physicist?) Now what to do in proteus, I don't know.

Have a good night!
Davide

[Bruce Long]

Let's not confuse the mathematical theory with the Proteus language.
For what we are doing we don't need to model such complex systems. The theory should handle them, but it isn't necessary for us to model them in Proteus (for our current purposes).

[Sebastien Zany]

Thanks for the substantial discussion everyone. My academic quarter is finally ending so I have more time to think about this.

Is there a comprehensive account of the theory anywhere? All I've seen are scattered and incomplete.

Sebastien

[Sebastien Zany]

P.S. While googling for background reading on infons, I ran across a mysterious project page, which seems at least superficially along the same lines as this project: <http://www.sirius-beta.com/sHeherazade.html>. Does it relate at all?

On Apr 11, 2011, at 0:30, Bruce Long <qst...@gmail.com> wrote:

[Sebastien Zany]

P.P.S. Irreversible changes could formally be added by generalizing to monoids instead of using groups.

[Bruce Long]

Hi Sebastien,
I've been working on the implementation but I would like to fill in some of the gaps in the documentation of the theory. Probably this weekend.
I am curious about monoids. How would that work? Are monoids more general than groups?

BTW, the sHeherazade project isn't connected; I've never seen it. It's interesting how we appear to be converging on the same thing.

~BL

[Sebastien Zany]

Monoids are groups minus the invertibility axiom, so by dropping the assumption that all state transitions are reversible you would get something that looks like a monoid instead of a group. See <http://en.wikipedia.org/wiki/Monoid>.

[Bruce Long]

Hi all, I have a question about the math I'm using in Proteus. (Sebastien?)

There are several aspects of the language where I could not tie my solution directly to the mathematics, though I did show that the solution would work for most purposes.

One of those is the notation that gives us "loop" like functionality:

{`an infon here` | ...}

 

The idea is that the infon on the left of the '|' applies to every member of the list. (In this example the members are not known, hence the '...').

For example, here is a list of unknown items where each item has 16 states:

{*16+_ | ...}

The following would be a contradiction:

{*16+_ | *8+2, *10+3}

It's a contradiction because the two items in the list do not both have 16 states. To not have a contradiction it would have to look like this:

{*16+_ | *16+2, *16+3}

or this:

{*8+2, *10+3}

In other words, the left side infon cannot contradict any element of the list (i.e., those on the right side).

It's a little like "For each element of this list, "*16+_" applies."

==================

SO, my question is this: I'm trying to connect that construct to group theory.

If you look at this page on "group extensions" http://en.wikipedia.org/wiki/Extension_problem#Extension_problem we have

  a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence
he definition that a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence

I wonder if the left part of the Proteus syntax (the "*16+_" or whatever is to the left of the '|') is analogous to the normal subgroup (N) of a group extension where G is the entire list infon. In other words, is N an infon description that would have to apply to all (or some set) of sub-groups of G? Can you use group extensions to say something like "all of the ??? sub-groups in G match the pattern N" And if so, what should be in the "???" part of that sentence?

If group extensions are NOT like that, then what aspects of group theory are used that way?

-- Bruce

[Sebastien Zany]

I would have to understand the rest of your group theory ideas to comment on this. What does it mean for an infon to apply to another?

[Bruce Long]

Thanks Sebastien,

OK. Infons are quite group-like. As a starting point, I'll paste the definition from wiki then talk about the differences.

A group (G, •) is a set G closed under a binary operation • satisfying the following 3 axioms:

• Associativity: For all a, b and c in G, (a • b) • c = a • (b • c).
• Identity element: There exists an e∈G such that for all a in G, e • a = a • e = a.
• Inverse element: For each a in G, there is an element b in G such that a • b = b • a = e, where e is an identity element.

Infons meet all the above requirements. The root difference is that there is a different meaning for the '=' symbol.

------------------------------

IDENTITY

Some background reading is in http://frontiersinai.com/ecai/ecai2000/pdf/p0219.pdf. (Identity, Unity, and Individuality: Towards a Formal Toolkit for Ontological Analysis; Nicola Guarino and Christopher Welty)

The short version:

In set theory we have the definition "Two sets are equal (are the same set) if they have the same elements."  This is true of infons too, but it's a little recursive. In ZFC notation we would say that {1,2,3}={1,2,3}. That's because 1=1, 2=2, and 3=3. They are the "same" since they are the "same" symbol.  This recursive problem is resolved differently for infons (and thus in Proteus.)

My pet example is: suppose I ask you how many miles you drive to work/school and you give me a number (D).  Then, I ask you what your shoe size is and you give me a number (S). And suppose they both happen to be the "same number." So in ZFC your-distance-to-work = your-shoe-size. D=S.

In proteus these two pieces of information, these infons, are not the same piece of information. One is information about your spatial situation and the other is information about your shoe size. So in Proteus, D=S may not be true. In fact, in this case, D != S.

So, in Proteus we only act on identities when they are asserted. They cannot ever be assumed by the shape of the symbol or the state of the underlying bits. 

Now just rebuild number-theory and group theory with the new meaning and you'll have infons! :)

------------------------

SOME WAYS INFONS ARE SIMILAR TO GROUPS:

1. Numeric infons in Proteus, (+1, +5, +800, etc.)  that is, infons measured using "states" as a unit of measure. Are isomorphic to cyclic groups.

2. Strings and Lists have more complex (but finite) group structure. List infons with expressions for unknowns inside them can act like infinite groups.

------------------------

SOME WAYS INFONS ARE DIFFERENT:

NOTE: For any two infons they are either identical or not. If not, their respective sub-infons may have identities among them.

That fact means that when we combine two infons through multiplication they can only combine one way, even if we reverse the order of operation. That means that infons are ALL commutative. Even "cyclic group" infons. We can still speak of "abelian infons" which would be commutative even without identity relations reducing the way infons can combine.

Here's a short way to imagine why all infons are commutative: 

1.    Suppose I have a closed system (a closed infon) of 16 states.
2.    Suppose I divide that system into sub-infons A and B of 4 states each. (Remember that when we measure in bits we add but when we measure in states we mulitply; 8 bits + 8 bits=16 bits. But 4 states + 4 states = 16 states,
3.    Now when I combine A and B like this:  A * B  I should get a system with 16 states -- my closed system.
4.    But if  A * B  != B * A, then I have just gained an extra bit of information and effectively now have 32 states.
5.    Which contradicts the assertion that I have a closed system. 
6.    To fix this paradox you have to use the identity trick described above.
7.    If you can picture the actual infons and their identities of some real 4 state systems and how they combine you'll notice that they are indeed described by the identity trick.

So, in Proteus as it works today the "universal quantifier" operation { DESCRIPTION-INFON | ...} lets us assert that all of a certain collection of sub-infons share the DESCRIPTION.

But I want to connect it to the math if possible, I think that there are likely some enhancements to the language that would be possible making it a smaller but faster and more powerful language.

In a list infon like {*16+2, *16+9, *16+7} has three infon members of 16 states each. They are in states 2, 9, 7 respectively.

These sub-infons are very independent. They are like cosets or normal sub-groups. They have only the identity element (state 0) in common. They act like a hexadecimal odometer with only three (hex) digits. The current reading is hex "297".

So when we combine infons using * and +, imagine that they are already combined and are already sub-infons of a larger infon. The expression merely describes how they are related. Don't imagine two infons moving closer together and combining into one; the math won't work correctly.

-------------------------

You asked "what does it mean for an infon to apply to another." The answer is in the above text. Someday I'll have time to put all of this together into a nice, organized book or PDF. Until then, thanks for your patience, and what are your questions?

BL

 

PS, you might be able to partly answer my question without knowing about infons. Does group extension assert the "N" group over all the normal sub-groups in G? Or something similar? Do G's normal sub-groups have to be like N in some way? Group extensions are a new concept to me. Somehow I missed it before. 

[Sebastien Zany]

A few immediate thoughts:

• "Identity" shouldn't be an issue that requires any new mathematics; with groups we almost never talk about equality, only isomorphism.
• If A,B are infons with a,b states respectively, A*B != B*A does not mean that A*B has 2*a*b states, but that A*B and B*A are different infons. What you're describing is a direct product.

I think it would be helpful to come up with axioms (an infon is a set with ...) and rigorously define what all these terms mean. Until there is some sort of mathematical foundation, it won't be productive to make high level analogies with group theory (or other algebraic theories).

I can't really answer your question about group extensions since I haven't studied them. But again I think you'll have a better chance of finding an answer once infons are axiomatized.

[Bruce Long]

The problem is somewhat of a chicken-and-egg problem. I'm working on creating axioms and have most of them (which is how I came up with Proteus). But I am missing something. The article on group extensions (which I hadn't seen before; are they a new thing?) seems like it might help but I can't figure out what they do. Hence my questions. I do know that infons are groups but without any assumptions about identity; that is, you can't assume 2=2 because they may be different pieces of information. (You can ASSERT that they are identical.) You can also assume that if they are identical then they are also equal/isomorphic. In other words, if A is identical to B and A's value is in state '2' then B's value will be state '2'.

It would really help to know if extensions are a way of asserting "all the normal sub-groups of G are isomorphic to N" Or if not, what concept in group theory (or number theory) plays that role?

Maybe you know someone who would know? David or Davide do one of you know? 

Currently, I'm basically creating a bald assertion that the normal sub-groups share a pattern. But there are inferences that cannot be automated this way. If I can connect the assertion to the rest of the theory then I can make them happen in normalize. Like this:

Let's call N (from the wiki article) a "centrino" of the list G. It's the pattern I want to assert.

Instead of {PATTERN-TO-ASSERT | ...} having special code to apply the pattern to all the elements of the list I would do:

G's centrino = PATTERN-TO-ASSERT. 

Now it's an identity statement which the normalize algorithm can process.

Cheers!

[David Van Duzer]

I'll chat with some of my friends in the UCD math department to see if
they can offer any insight into group extensions.

Quick question, though. We can't always assume 2=2, but can we always
assume *16+2=*16+2?

[Bruce Long]

On Tue, Jun 28, 2011 at 10:01 PM, David Van Duzer <d...@tennica.net> wrote:

I'll chat with some of my friends in the UCD math department to see if
they can offer any insight into group extensions.

Quick question, though.  We can't always assume 2=2, but can we always
assume *16+2=*16+2?

No, I just used 2 by itself as a shortcut. We can't assume *16+2 = *16+2. (This just means we have a 16 state system that is permanently in state 2.)  We don't know what information is stored in that state. One could be "2 miles" and the other "2 cups of sugar"

I found a real-life example a few days ago. I was debugging the program and I had the debugger set to tell me when a memory location changed, An assignment statement was executed that copied a new number into that memory location. But as it happened, the new item copied in had the same numeric value as what was already in the memory location. So the debugger didn't register the change.  The debugger was using the non-Proteus meaning of "same" / "changed"  Under the Proteus definition of identity the debug trigger would have occurred because new information was in the cell. The fact that the new information had the same numeric value as the previous information is a coincidence. 

[Davide Del Vento]

> Maybe you know someone who would know? David or Davide do one of you know?

I don't and my math friends specialize in different fields, sorry :-(