A Few Questions about sPvP

[Crozame.4098]

@JusticeRetroHunter.7684 said:

@"Crozame.4098" said:Ok, you specified the definition of a complexity metric, which is based on a sorting algorithm, lets assume you are 100% correct. However, the example states you used all consist of 10 "bars". But in from HoT to PoF, we have 7 or 8 more specialisations. Therefore, the number of bars is different? Therefore, PoF by definition is more complex?

it can be deduced that more diversity implies less balance.

I guess I should have explained this more in the proof, but basically, the above observation is explained in the framework, and is responsible for what I believe to be, is the "aha!" moment with this abstraction.

So, It's not that diversity implies less balance... it's a little different. What's actually going on here is that all systems and all states of these systems are heading toward an eventuality. The inevitable "end" of the game which is the maximally complex state. And this framework implies that this eventuality is actually

driven by the act of computation

.

So the most highly diverse (Heterogenous) state of the system is in the beginning ...

before computations

...and the most homogenous state of the system is after the computations. Essentially the framework is like a description for what is happening as a system evolves from a heterogenous system toward a homogenous one!

This to me is the biggest mind blow about this (sorry but it still blows my mind every time I think about it) ...that it's a brilliant descriptor for how any system in any state is always going from a maximally diverse and heterogenous state to a maximally balanced/completely homogenous state through computation.

To put the above realization into the context of your example...

You have Core gw2 as State A and PoF Gw2 as State B. Even though we've added complexity into State B, the act of computation will always move the system towards the eventuality, toward a single metagame. This in my view at least, perfectly describes why POF which is by definition more complex than core, doesn't feel more diverse or balanced then core. It's because it's constantly being driven toward the maximal complex state via players making choices, and that the most diverse state of the game, was when POF was released,...and this makes sense right? People are trying out builds, experimenting and having fun...people are trying all types of combos and exploring all the options...but it is this

VERY ACT

of exploring options and making computations for optimality that is pushing the game closer and closer toward the maximal complex state, and the eventuality of a singular homogenous meta game.

Now there's a bit more to explain here, and this is about computation time.

If we say that it takes a finite period of time to make a computation, then we can also say that time scales with the complexity of the system. Therefor, an increase in complexity, increases the number of computations which increases the time it takes for the system to go from heterogenous to homogenous. If that's the case, then simply increasing the complexity will result in a greater time spent in a heterogenous state, or rather it will take a longer period of time before we reach the eventuality; the maximally complex state.

For the majority of this discussion we've been referring to computations being done by a (singular) computer. But then what happens if we had a parallel process of computers?

Let's say we have a complex system that takes 1 million computations to reach the end maximal state, and that it takes 1 second per computation for a total of 1 million seconds (approximately 11 days) to compute the complexity of a system. But this time, instead of one computer computing the complexity of the system, we had 100 or 1000 computers, with the ability to communicate their findings with one another. This means that the sum of all computers is essentially a more efficient single computer that can perform more computations in the same period of time, thus the complexity of a system among a parallel process of 1000 computers, would reduce the computation time of such a system from 11 days, to 16 minutes!

The analogy is that we are discussing computers making computation but what we are actually talking about are human brains and it's ability to make decisions and choices. If players communicate among themselves about what is optimal and what is not, then they would essentially be pushing the complexity of the system toward maximally complex state exponentially faster then if they couldn't communicate, and this takes us further away from diversity in the system toward a homogenous meta game.

Still feel very confusing and I do not see your point. Dont have time to type anything yet, will do it when I have time.

[razaelll.8324]

@JusticeRetroHunter.7684 said:

@"razaelll.8324" said:Hey, i am not trying to start a second war, but can you please answer 1 question?

Hey, I appreciate that, and honestly, I'm sorry and I sincerely apologize for being a jerk. I don't want a second war either so I'll be civilized this time.

I also apologize for being jerk too. I believe if we leave that aside we can actually have a very interesting and constructive conversation. I appreciate your apology and gladly accept it.

So these are great questions, and before answering this I want to refer you to the post above first, because it's a primer for getting a hold on the entire framework described so far. In addition The proof by Susskind is part of a lecture on Complexity and Black Hole Physics. First part of the lecture is just merely explaining complexity and how it's used and kinda how it manifests in the subject, and the 2nd part is applying it to the physics of black holes and what knowledge is gained from that information. I will link this lecture at the bottom for you to watch in full since really the whole thing is interesting in it's own right.

Complexity metric is the MINIMAL number of simple operations it takes to get from 1 state to another. This metric is used for determining of how "hard" it would be to go from 1 state to another and it is used in many different fields of study as electronics, economics , quantum physics and so on.

But how that proves that "Thus states of a system with the same complexity metric, which are either a homogenous or heterogenous, means that the states are still both equivalent." Can you please elaborate on that?

Right ya. So the minimum number of operations and the maximal complex state are kind of just ways of using this metric and in our example those both take on the same value for the following reason : In our example, we are assuming that we are using the most simple operations already (comparing the size of one bar, to another bar) and we are also assuming that we know how many operations it's going to take to get from one to the other (total of 9 elements means there are a total of 9 computations needed to reach the maximal end state), so we already defined what the minimum is. The minimum in the case of our example is just the same as the number of operations it takes to get to any other state of this system, and so the maximal complex state also happens to be this number. Below I'll explain the key differences between the two things.

Minimal Number of Operations

In the lecture, Susskind is using the Metric to actually find out how many operations are needed in a set of states, where it's not known how many of the elements are in that state. So you have state of some object A where you just start flipping atoms one by one (The Rolls Royce), and after you flipped 100 atoms you now have a state of B (Rust Pile), then that is the relative complexity between the two states, for which at first you didn't know how many things you actually needed to flip to get from one state to the other.

Maximal Complex State

The maximal complex state, is essentially the maximum number of states that are possible in a system. This is determined using that metric, by knowing all the elements in the system, and then identifying how many simple operations it will take to cycle through all possible states before returning you to a state you've already "visited." In our example, we had 9 states, and if each operation is just an analysis of 1 element to another element, then that maximal complex state is going to happen after 9 operations, as a 10th operation would mean you are now cycling through a state you've already visited.

---

So in the examples , we are showing that, given we know what the minimum number of operations are, and knowing all the elements in the system, then the maximum we can determine, and by proxy if we compare this result to all other results using the same parameters, we can determine that all states of the same complexity lead to the same "value" of a maximally complex state, which is just 1 build. We can start with any number of elements, and any number of computations, and it will all eventually collapse to just 1 eventually, and therefor all states of the system are equivalent in this regard.

I understand and agree with all said above, just 1 think i still fail to understand. Lets use your chart for the example.Lets assume that Pink bar is warrior and Blue bar is Necro. So in state A from your chart the Warrior(pink) is the optimal choice and in State B the necro is the optimal choice (blue bar) , which means that if you are in State A then you have the best chances for wining if you play war, but in state B you have the best chance of wining if you play Necro do we agree here? So the number of optimal choices (classes) is equal in State A and in State B and it is 1, if that's what you are saying i completely agree.But State A is not equivalent to state B in respect to optimal choice since the optimal choice in this 2 states are different to each other, which can be used to say that state A is different from State B.

"and therefor all states of the system are equivalent in this regard." this can be interpreted in the both ways i mentioned above that's why i asking what specifically you mean by all states of the system are equivalent. Or maybe the right question is they are equivalent in terms of what specifically?

As far as i know equivalent states are states which have the same optimal choice , but in our example the optimal choice is different , that's why i claim that this 2 states or not equivalent to each other. Do you agree with that?

In the video of the lecture he defines thoroughly the parameters of this metric. Here in this picture he presents a list of axioms regarding the states and the equalities of the metric:

1) If State A is equal to State B, then then the number of computations needed to go from A to B must be 0. (And by proxy, the number of computations needed to get to the maximal complex state will also be the same, like in our example.)

2) A and B are symmetrical, in that it doesn't matter which order the states are in you can "flip atoms" in any direction to reach the same value...This is essentially the analog of what was debated in the war thread, about how any operation used can have any inverse operation that can be used to achieve the same result.

3) "Satisfies triangle inequality" <- idk I don't care about this but hey it's cool too I guess.

In the video which you posted the Lecturer Leonard Susskind (great professor btw) says that State A is orthogonal to State B and also State A is orthogonal State C (means State B and State C are both at distance P/2 from State A) so measuring the distance is not a good way to measure how "complex" is to go from State A to State B and from State A to state C thats why other metric is needed and namely that metric is relative complexity. With other words measuring how "complex" is the difference between State A and State B compared to how "complex" is the difference between State A and State C. In matter of facts neither in the video nor the chars balance is mentioned nor equivalency of states so i fail to understand how that video and the charts prove that any of the states are equivalent to each other or how homogenous system is equivalent to heterogenous in terms of balance? So can you please elaborate on that?

Okay, this is a bit harder to talk about since I'm not 100% sure, but The Orthogonal states is referring to Eigen Vectors of particles and stuff (Quantum states), in which when these vectors are orthogonal, the dot products of these vectors add up to 0. This is actually required in physics to do because it's sort of like a conservation law for particles...Where vectors add up (or negate each other) to encompass the entirety of the system in order to equal 0. Essentially, you can't push a ball forward without the ball exerting a force back on you. So when looking at 2 different objects, one would EXPECT there should be some kind of meaningful imbalance that describes why the quantum states of these two distinctly different objects are different, but there isn't...everything is just orthogonal to each other and it's like...well how is this different if all the vectors are adding up to 0?

Anyway, here's the full Lecture. It's well worth the hour and a half watch if you have the time.

&

I hope i will have enough time to watch the full lecture tomorrow. Thank you for the link!

Have a good night!

[Sigmoid.7082]

@razaelll.8324 said:hehe that does not mean that you cannot have a good discussion about science, mate. In matter of fact this type of discussions are more constructive imo instead of the typical nerf this nerf that ...

I wouldn't be so sure.

[razaelll.8324]

@Sigmoid.7082 said:

@razaelll.8324 said:hehe that does not mean that you cannot have a good discussion about science, mate. In matter of fact this type of discussions are more constructive imo instead of the typical nerf this nerf that ...

I wouldn't be so sure.

thats just my opinion mate i do not force anybody to agree with it.

[JusticeRetroHunter.7684]

@"razaelll.8324" said:I understand and agree with all said above, just 1 think i still fail to understand. Lets use your chart for the example.Lets assume that Pin bar is warrior and Blue bar is Necro. So in state A from your char the Warrior(pink) is the optimal choice and in State B the necro is the optimal choice (blue bar) , which means that if you are in State A then you have the best chances for wining if you play war, but in state B you have the best chance of wining if you play Necro do we agree here? So the number of optimal choices (classes) is equal in State A and in State B and it is 1, if that's what you are saying i completely agree.But State A is not equivalent to state B in respect to optimal choice since the optimal choice in this 2 states are different to each other, which can be used to say that state A is different from State B.

"and therefor all states of the system are equivalent in this regard." this can be interpreted in the both ways i mentioned above that's why i asking what specifically you mean by all states of the system are equivalent. Or maybe the right question is they are equivalent in terms of what specifically?

Okay, ya I think I understand the question I believe. Even though both states have a different maximally complex states (Where one is blue, and one is pink), it's being defined by a complexity metric, and in this example, the computation is evaluating each state only by the size of the bar. In essence the computer would compute the optimality by breaking down the components of necromancer and the components of warrior into straight up numerical entities, where their properties can be described numerically.Even though numbers are different from other numbers, they are still just numbers and these numerical values have no intrinsically different qualities. So 1 meta build at the end of the computation is just 1 meta build, and it doesn't matter what class that build actually belongs to, or what the skills actually do because in theory, a computer can break down any and all of these skills and abilities into numerical components.

This is where the complexity metric comes into play, because the more complex the skills are, the harder it gets to compute all the components numerically. By having to do more and more of these simple operations to determine a numeric value.

Imagine for a moment you were to take some skill and compare it to another skill, how would one break down this skill into numerical components to compute... If it's a very simple skill like "Does 10 damage" and the other skill is "Heal 5 Health" then the computation for this particular operation will be simple, and the comparison is going to be within the magnitude of the elements in the system to compare it to.

If the skill is something abstract like "Apply Vulnerability to foes around you every time you gain Health" This skill is much much harder to break down into a small number of numerical components, because it's dependent on the number of foes, how often you gain health, and how relevant the vulnerability is and probably a slew of other things i can't even begin to think about. It may take millions or billions of comparison operations to determine the numerical value of this skill, but a computer should in theory be able to do it. It's just hard to do. And this is really the bread and butter behind the idea of complexity...that highly complex things are harder to judge optimality on because it's a difficult computation.

Hopefully this answers the question. Everything in Bold is basically a TLDR of what's being said here. But just another TLDR just incase i didn't answer the question, is that The point of the metric is to translate the qualities of a system into scaler numeric entities, that can be compared as one would with any other metric...like assigning the value of 1 dollar to an apple or orange. The apple and the orange look different, but the complexity metric gives these things a numeric value in terms of computation, and it's this numeric value that can define how two different states of a system can be equivalent.

[razaelll.8324]

@JusticeRetroHunter.7684 said:

@"razaelll.8324" said:I understand and agree with all said above, just 1 think i still fail to understand. Lets use your chart for the example.Lets assume that Pin bar is warrior and Blue bar is Necro. So in state A from your char the Warrior(pink) is the optimal choice and in State B the necro is the optimal choice (blue bar) , which means that if you are in State A then you have the best chances for wining if you play war, but in state B you have the best chance of wining if you play Necro do we agree here? So the number of optimal choices (classes) is equal in State A and in State B and it is 1, if that's what you are saying i completely agree.But State A is not equivalent to state B in respect to optimal choice since the optimal choice in this 2 states are different to each other, which can be used to say that state A is different from State B.

"and therefor all states of the system are equivalent in this regard." this can be interpreted in the both ways i mentioned above that's why i asking what specifically you mean by all states of the system are equivalent. Or maybe the right question is they are equivalent in terms of what specifically?

Okay, ya I think I understand the question I believe. Even though both states have a different maximally complex states (Where one is blue, and one is pink), it's being defined by a complexity metric, and in this example, the computation is evaluating each state only by the size of the bar.

In essence the computer would compute the optimality by breaking down the components of necromancer and the components of warrior into straight up numerical entities, where their properties can be described numerically.

Even though numbers are different from other numbers, they are still just numbers and these numerical values have no intrinsically different qualities. So 1 meta build at the end of the computation is just 1 meta build,

and it doesn't matter what class that build actually belongs to, or what the skills actually do

because in theory, a computer can break down any and all of these skills and abilities into numerical components.

This is where the complexity metric comes into play, because the more complex the skills are, the harder it gets to compute all the components numerically. By having to do more and more of these simple operations to determine a numeric value.

Imagine for a moment you were to take some skill and compare it to another skill, how would one break down this skill into numerical components to compute... If it's a very simple skill like "Does 10 damage" and the other skill is "Heal 5 Health" then the computation for this particular operation will be simple, and the comparison is going to be within the magnitude of the elements in the system to compare it to.

If the skill is something abstract like "Apply Vulnerability to foes around you every time you gain Health" This skill is much

much

harder to break down into a small number of numerical components, because it's dependent on the number of foes, how often you gain health, and how relevant the vulnerability is and probably a slew of other things i can't even begin to think about. It may take millions or billions of comparison operations to determine the numerical value of this skill, but a computer should in theory be able to do it. It's just hard to do. And this is really the bread and butter behind the idea of complexity...that highly complex things are harder to judge optimality on because it's a difficult computation.

Hopefully this answers the question. Everything in

Bold

is basically a TLDR of what's being said here. But just another TLDR just incase i didn't answer the question, is that

The point of the metric is to translate the qualities of a system into scaler numeric entities, that can be compared as one would with any other metric...like assigning the value of 1 dollar to an apple or orange. The apple and the orange look different, but the complexity metric gives these things a numeric value in terms of computation, and it's this numeric value that can define how two different states of a system can be equivalent.

The point of the metric is to translate the qualities of a system into scaler numeric entities, that can be compared as one would with any other metric..

Exactly, and what i am saying is that number will be different when you compare apple to orange because for example the numerical value of the color of the orange is different than the numerical value of the color of apple, the numerical value of the taste of the apple will be different than the numerical value of the taste of orange and so on, and by comparing that numerical values you determine that apple is not orange.

So in the example which you give "like assigning the value of 1 dollar to an apple or orange" they are equivalent in terms of price, but the price is only one quality and in order to be objective us much as possible you need take in consideration all qualities (color , taste, price and so on), so when you assign a numerical value to all qualities and compare all of them you will see that apple and orange are different objects with the same price . And here again comes the question which are the important qualities which we search in terms of balance, is it the price or the color or it is a combination of all of them.

So if we go back in the example of warrior and necromancer, when you start comparing their qualities (their numerical value representation) they will be different and that help to determine that the optimal choice of state A is different than the one in State B. They can be equivalent in terms of one quality, but not equivalent in terms of another.

One of my professors worked on mathematical algorithm of how to find blood cells affected by leukemia he used ultrasonic waves and differential equations to determine the shape of "not affected blood cell" and then the shape of "affected blood cell" in this way he found that the numerical representation of the shape of not affected cell is different then the one of affected blood cells and that he can use that to design a "filter" for blood cells.

I hope you see what i am trying to say here.

[JusticeRetroHunter.7684]

@"razaelll.8324" said:So in the example which you give "like assigning the value of 1 dollar to an apple or orange" they are equivalent in terms of price, but the price is only one quality and in order to be objective us much as possible you need take in consideration all qualities (color , taste, price and so on), so when you assign a numerical value to all qualities and compare all of them you will see that apple and orange are different objects with the same price . And here again comes the question which are the important qualities which we search in terms of balance, is it the price or the color or it is a combination of all of them.

So this isn't the right conclusion to draw, and it's probably because I'm poorly explaining why the metric is so special.

but the price is only one quality and in order to be objective us much as possible you need take in consideration all qualities (color , taste, price and so on)

The metric is not just taking one quality. It's taking ALL qualities of an object and boiling down those qualities into simple operations. So color, taste, everything about the state is numerically expressed in the size of the bar. The more qualities that are encompassed into the computation, the more complex the object is, and therefor takes more computations to go from one state to another.

Also, there is a general lack of trying to express what 'different qualities' truly are. Example : Necromancers aren't made of necromancer atoms, and guardians aren't made of guardian-atoms. Both things, are made up of skills, and these skills are made up of numbers that add up or negate each other. It follows that necromancer, and guardians are both made up of numbers. So what makes Guardian different from Necromancer? Well it's the difference between these numbers. Counting up these numbers is essentially what the complexity metric is, and it's expressed as process of relative computation.

[razaelll.8324]

@JusticeRetroHunter.7684 said:

@"razaelll.8324" said:So in the example which you give "like assigning the value of 1 dollar to an apple or orange" they are equivalent in terms of price,

but the price is only one quality and in order to be objective us much as possible you need take in consideration all qualities (color , taste, price and so on)

, so when you assign a numerical value to all qualities and compare all of them you will see that apple and orange are different objects with the same price . And here again comes the question which are the important qualities which we search in terms of balance, is it the price or the color or it is a combination of all of them.

So this isn't the right conclusion to draw, and it's probably because I'm poorly explaining why the metric is so special.

but the price is only one quality and in order to be objective us much as possible you need take in consideration all qualities (color , taste, price and so on)

The metric is not just taking one quality. It's taking ALL qualities of an object and boiling down those qualities into simple operations. So color, taste, everything about the state is numerically expressed in the size of the bar. The more qualities that are encompassed into the computation, the more complex the object is, and therefor takes more computations to go from one state to another.

Also, there is a general lack of trying to express what 'different qualities' truly are. Example : Necromancers aren't made of necromancer atoms, and guardians aren't made of guardian-atoms. Both things, are made up of skills, and these skills are made up of numbers that add up or negate each other. It follows that necromancer, and guardians are both made up of numbers. So what makes Guardian different from Necromancer? Well it's the difference between these numbers. Counting up these numbers is essentially what the complexity metric is, and it's expressed as process of relative computation.

So what makes Guardian different from Necromancer? Well it's the difference between these numbersExactly! Thats what i said too.

I believe that means that State A (warrior meta) will have different number(for that metric) than State B (necromancer Meta), because Necromancer has different qualities than warrior which means different numbers. (seems wrong to represent different qualities with same numerical value right?)

So if that metric is equal this means that you are comparing objects with same qualities, but in our example we are comparing objects with different qualities.

Am i missing some kind of possibility which 2 objects with different qualities can give same result? if thats the case can you please point out that possibility since i fail to see such.

[JusticeRetroHunter.7684]

@"razaelll.8324" said:

So i spent some time thinking about your response today, and I believe I get what your saying, and I think the reason is because I'm poorly explaining exactly what's equivalent and what isn't...Because it is indeed the case, that comparing 3 objects that are different will definitely have 2 different relative complexity's.

My mistake in the explanation here I think is that I'm improperly addressing the difference between Relative Complexity and Maximal Complex State.

So what I'm going to do in the next response is essentially follow the rigorous proof that Susskind does in this particular lecture below, and apply this to our example. Hopefully you haven't watched the previous video yet because this lecture IMO is a much better one that is fully dedicated to just complexity, and it's a full mathematical explanation.

&

So ya in a few hours I'll come through with a 2nd response that perhaps cleans up any mis-explanations and errors I could have made in any of the examples so we can get on the same page on your question.

[razaelll.8324]

@JusticeRetroHunter.7684 said:

@razaelll.8324 said:

So i spent some time thinking about your response today, and I believe I get what your saying, and I think the reason is because I'm poorly explaining exactly what's equivalent and what isn't...Because it is indeed the case, that comparing 3 objects that are different will definitely have 2 different relative complexity's.

My mistake in the explanation here I think is that I'm improperly addressing the difference between Relative Complexity and Maximal Complex State.

Exactly. Thats my point. I am glad i could present it in understandable way this time.

So ya in a few hours I'll come through with a 2nd response that perhaps cleans up any mis-explanations and errors I could have made in any of the examples so we can get on the same page on your question.

sounds perfect mate. Here is already 00:00AM so i will read it tomorrow also i will watch the lecture which you posted.

Have a good day.

[JusticeRetroHunter.7684]

@"razaelll.8324"So the first step here, is that we need to define a couple of things in a very concise manner.

OperationsThe first thing to address is what exactly is an operation. In the Lecture, an operation is a simple flip of a Qubit...which is just a quantum mechanic bit of information, like flipping a 0 to a 1 or vice versa. For our example, instead of a qubit, we will define our most simplest operation as either a +1 or a -1 in terms of numerical change of some quantity.

The SpaceThe 2nd thing that's needed, is to define the space in which the problem is taking place. In the lecture, they use Hilbert Space, which is a vector space. This is cool, but we don't really need it as far as I know, and what we can use instead is a generic Euclidian space, since we are just dealing in the realm of numbers rather then angles and vectors. What needs to be clarified, is just like he does in Hilbert Space, we need to posit that the space is not a continuum and can be described in discrete integers. For example;

n₁, n₂,n₃...nₙ

SystemNow that we've described these two things, we can create a system, and 2 states in this system, which we can call State A and State B. State A. For the sake of our mental sanity, we will just say that these two states are different attributes of a single skill (or rather 2 different skills...where one skill will be changed into the other), rather then Classes, and that the attributes of this skill can be expressed in the form of some integer number.

The state described in the lecture is the state |ψ| = ∑1->2ᵏα which is basically describing the number of possible arrangement of things in the system, and the dimensionality of each thing. So the expression m²^ᵏ is a description of how large the space is for all possible states of this system. We don't really need this expression right now because we kind of know that the dimensionality of our example is going to be just 1, since we are dealing with arithmetical numbers on a proverbial number line. so our example will only scale with the number of elements. This will come into play later however, because the dimensionality of the system is gonna be extremely important.

The state of our entire system will just be described as |AB| = m^K where K equals 1 , which I believe is just a way of saying that the complexity is going to scale linearly with the number of elements in our system.

Relative ComplexityFirst thing we do here is calculate the relative complexity. This is essentially the minimum number of these simple operations to go from State A to State B. In our case here, adding +5 from the first bar, -2 from the second bar, and +5 to the third bar is the minimum number of computations it would take to transform State A into State B, which if expressed as these simple operations would be (+1+1+1+1+1)+(-1-1)+(+1+1+1+1+1) for a total of 13 operations.

So in equation form this would look something like Ͼ|A->B| = 13 where the funny looking C with the dot in the middle is the relative complexity.

In the lecture, he states that you can describe the relative complexity as the shortest distance between the 2 states, which can be described as points in the space. This in Hilbert space is a geodesic connecting "epsilon balls" but for us in Euclidian Space, this is just a straight line connecting two points on a plane.

This idea of using geometry to describe the complexity framework is extremely fundamental to the framework. From here Susskind can attribute how Relative Complexity now has a set of axioms that define it as being a metric that also satisfy geometric properties.

• Ͼ|A->B| = 0 then A is equal to B.
• It' has symmetry, in that Ͼ|A->B| = Ͼ|B->A|
• it satisfied the triangle inequality, which is that Ͼ|A->B| ≤ Ͼ|A->C| + Ͼ|C->B|. It's basically just line segment and triangle stuff.
• He also talks about how it's Right Invariant. I'll be honest, I think this is extremely important but I don't truly know how to extrapolate this part to our example. I would need some more time to look into this particular part and brush up on my matrix math.

Maximal Complex State

So now, the notion that complexity can be described in terms of geometry becomes super important, There is a lot of information here at this point in the lecture, but essentially the most important part is that if Relative Complexity describes a geodesic between two points in the space, then one can analyze the evolution of possibilities (denoted as d) in the space as a state goes from an origin state (denoted I) and evolves toward other states, where a geodesic in that space can be considered the shortest path between one point on it to any other point. An important take away here is that as the state performs computations from One state to the next state, D goes to D-1, D-2 until it hits D-6, where all possible computations have been performed. So D-6 is the maximally complex state of the system, and here at D-6, the evolution of the system stops...or rather it cycles towards the other states at d-6.

For our example, since we are in a Euclidean space, and geodesics are described by straight lines, then what happens is that the decision tree would not be a fractal like it is in the higher dimensionality Hilbert Space. This is where the dimensionality we mentioned being 1 from earlier comes in, that because k grows exponentially in Leonard's lecture, while in our space it just grows (since k is 1) and that's about it.

This means that in a system of 2 skills, the evolution goes from A to B, and the system becomes maximally complex, reaching maximal complexity at 13 computations as well as becoming homogenous. This is why the Maximally Complex state, and the relative complexity in the example previously mentionedfrom earlier in the thread is the same as each other.

Now let's pretend we added another skill C, with some other set of attributes and the relative complexity between A and C is say, 20. The computation goes forward 13 computations to reach A->B, and then the computation goes an additional 7 computations to reach State C and the system is now maximally complex at state C. The system is now again, homogenous, where A = B = C.

Now I'm running out of time to finish this post, but I will edit this post tomorrow to include what happens when we start adding dimensionality to the system. Also I didn't exactly proof read this post, so i could be missing stuff, i could also be making mistakes. This is a good time to probably insert any questions about what i've said so far.

[razaelll.8324]

@JusticeRetroHunter.7684 said:@"razaelll.8324"So the first step here, is that we need to define a couple of things in a very concise manner.

OperationsThe first thing to address is what exactly is an operation. In the Lecture, an operation is a simple flip of a Qubit...which is just a quantum mechanic bit of information, like flipping a 0 to a 1 or vice versa. For our example, instead of a qubit, we will define our most simplest operation as either a +1 or a -1 in terms of numerical change of some quantity.

The SpaceThe 2nd thing that's needed, is to define the space in which the problem is taking place. In the lecture, they use Hilbert Space, which is a vector space. This is cool, but we don't really need it as far as I know, and what we can use instead is a generic Euclidian space, since we are just dealing in the realm of numbers rather then angles and vectors. What needs to be clarified, is just like he does in Hilbert Space, we need to posit that the space is not a continuum and can be described in discrete integers. For example;

n₁, n₂,n₃...nₙ

SystemNow that we've described these two things, we can create a system, and 2 states in this system, which we can call State A and State B. State A. For the sake of our mental sanity, we will just say that these two states are different attributes of a single skill (or rather 2 different skills...where one skill will be changed into the other), rather then Classes, and that the attributes of this skill can be expressed in the form of some integer number.

The state described in the lecture is the state |ψ| = ∑1->2ᵏα which is basically describing the number of possible arrangement of things in the system, and the dimensionality of each thing. So the expression m²^ᵏ is a description of how large the space is for all possible states of this system. We don't really need this expression right now because we kind of know that the dimensionality of our example is going to be just 1, since we are dealing with arithmetical numbers on a proverbial number line. so our example will only scale with the number of elements. This will come into play later however, because the dimensionality of the system is gonna be extremely important.

The state of our entire system will just be described as |AB| = m^K where K equals 1 , which I believe is just a way of saying that the complexity is going to scale linearly with the number of elements in our system.

Relative ComplexityFirst thing we do here is calculate the relative complexity. This is essentially the minimum number of these simple operations to go from State A to State B. In our case here, adding +5 from the first bar, -2 from the second bar, and +5 to the third bar is the minimum number of computations it would take to transform State A into State B, which if expressed as these simple operations would be (+1+1+1+1+1)+(-1-1)+(+1+1+1+1+1) for a total of 13 operations.

So in equation form this would look something like Ͼ|A->B| = 13 where the funny looking C with the dot in the middle is the relative complexity.

In the lecture, he states that you can describe the relative complexity as the shortest distance between the 2 states, which can be described as points in the space. This in Hilbert space is a geodesic connecting "epsilon balls" but for us in Euclidian Space, this is just a straight line connecting two points on a plane.

This idea of using geometry to describe the complexity framework is extremely fundamental to the framework. From here Susskind can attribute how Relative Complexity now has a set of axioms that define it as being a metric that also satisfy geometric properties.

• Ͼ|A->B| = 0 then A is equal to B.
• It' has symmetry, in that Ͼ|A->B| = Ͼ|B->A|
• it satisfied the triangle inequality, which is that Ͼ|A->B| ≤ Ͼ|A->C| + Ͼ|C->B|. It's basically just line segment and triangle stuff.
• He also talks about how it's Right Invariant. I'll be honest, I think this is extremely important but I don't truly know how to extrapolate this part to our example. I would need some more time to look into this particular part and brush up on my matrix math.

Maximal Complex State

So now, the notion that complexity can be described in terms of geometry becomes super important, There is a lot of information here at this point in the lecture, but essentially the most important part is that if Relative Complexity describes a geodesic between two points in the space, then one can analyze the evolution of possibilities (denoted as d) in the space as a state goes from an origin state (denoted I) and evolves toward other states, where a geodesic in that space can be considered the shortest path between one point on it to any other point. An important take away here is that as the state performs computations from One state to the next state, D goes to D-1, D-2 until it hits D-6, where all possible computations have been performed. So D-6 is the maximally complex state of the system, and here at D-6, the evolution of the system stops...or rather it cycles towards the other states at d-6.

For our example, since we are in a Euclidean space, and geodesics are described by straight lines, then what happens is that the decision tree would not be a fractal like it is in the higher dimensionality Hilbert Space. This is where the dimensionality we mentioned being 1 from earlier comes in, that because k grows exponentially in Leonard's lecture, while in our space it just grows (since k is 1) and that's about it.

This means that in a system of 2 skills, the evolution goes from A to B, and the system becomes maximally complex, reaching maximal complexity at 13 computations as well as becoming homogenous. This is why the Maximally Complex state, and the relative complexity in the example previously mentioned from earlier in the thread is the same as each other.

Now let's pretend we added another skill C, with some other set of attributes and the relative complexity between A and C is say, 20. The computation goes forward 13 computations to reach A->B, and then the computation goes an additional 7 computations to reach State C and the system is now maximally complex at state C. The system is now again, homogenous, where A = B = C.

Perfect , now from all of this we can conclude few things.

1. When the relative complexity of 2 states is 0 that means this 2 states are same(equal). Ͼ|A->B| = 0 then A is equal to B.
2. It' has symmetry, in that Ͼ|A->B| = Ͼ|B->A|
3. The more options we have the more complex the system becomes.

Now let's pretend we added another skill C, with some other set of attributes and the relative complexity between A and C is say, 20. The computation goes forward 13 computations to reach A->B, and then the computation goes an additional 7 computations to reach State C and the system is now maximally complex at state C. The system is now again, homogenous, where A = B = C

Here i fail to understand why making all 3 states equal to each other makes the system homogenous. As far as i know a system is homogenous if all the objects in it has the same qualities State C has 3 different skills (objects with different qualities) which means it is not a homogenous state.

Let me give a simple example please.

The maximally complex state shows which is the most complex state in the system in our case that is State C because it contains 3 different skills.

So with the example above i claim that maximally complex state does not always mean homogenous state, because State A = State B = State C =/= Skill1 = Skill 2 = Skill 3

[JusticeRetroHunter.7684]

@"razaelll.8324" said:Here i fail to understand why making all 3 states equal to each other makes the system homogenous. As far as i know a system is homogenous if all the objects in it has the same qualities State C has 3 different skills (objects with different qualities) which means it is not a homogenous state.

Let me give a simple example please.

The maximally complex state shows which is the most complex state in the system in our case that is State C because it contains 3 different skills.

So with the example above i claim that maximally complex state does not always mean homogenous state, because State A = State B = State C =/= Skill1 = Skill 2 = Skill 3

It's because the system is both homogenous and heterogenous, at the same time. This can even be shown using this particular example, where if one were to imagine there being a smaller subsystem of components that exist within each attribute, that these components are just homogenous arrangements.

The reason for this behavior is because Homogeneous and heterogenous notions are scale invariant. That systems can contain homogenous components that at a larger scale are heterogenous...and these heterogenous components are homogenous at an even bigger scale, and those homogenous components are heterogenous at an even bigger bigger scale, so on and so forth ad infinitum at any and all scales.

So in the particular example, the state of the system of skills is homogenous, while the state of the system of attributes is heterogenous...and this could go on and on forever based on the number of scales that we are dealing with. So in Gw2 it could be that we could have a system of homogenous Attributes->heterogenous pool of Skills -> among a homogenous set of Builds-> that make up some number of heterogenous classes. Collapsing more and more systems toward homogenous states, implies that the system at all scales (as a whole) is Becoming more homogenous, and a collapse of more and more states towards a heterogenous state at all scales, implies the system is becoming more and more heterogenous. The universe is said to be homogenous in totality, but we see heterogenous systems all around us...and when we zoom in on these things that are heterogenous and find that they are made up of homogenous atoms, where every electron 'seems' to be no more different than any other electron.

Notice how I say "seems" is because homogeneity and heterogeneity are local and approximate notions, of some larger more complicated mechanism at play. It could be that electrons are made up of a zoo heterogenous strings...or it could be that electrons are made up of a singular homogenous combination of components (binary information)

[razaelll.8324]

@JusticeRetroHunter.7684 said:

@"razaelll.8324" said:Here i fail to understand why making all 3 states equal to each other makes the system homogenous. As far as i know a

system is homogenous if all the objects in it has the same qualities

State C has 3 different skills (objects with different qualities) which means it is not a homogenous state.

Let me give a simple example please.

The maximally complex state shows

which is the most complex state in the system

in our case that is State C because it contains 3 different skills.

So with the example above i claim that maximally complex state does not always mean homogenous state, because State A = State B = State C =/= Skill1 = Skill 2 = Skill 3

It's because the system is both homogenous and heterogenous, at the same time. This can even be shown using this particular example, where if one were to imagine there being a smaller subsystem of components that exist within each attribute, that these components are just homogenous arrangements.

The reason for this behavior is because Homogeneous and heterogenous notions are scale invariant. That systems can contain homogenous components that at a larger scale are heterogenous...and these heterogenous components are homogenous at an even bigger scale, and those homogenous components are heterogenous at an even bigger bigger scale, so on and so forth ad infinitum at any and all scales.

So in the particular example, the state of the system of skills is homogenous, while the state of the system of attributes is heterogenous...and this could go on and on forever based on the number of scales that we are dealing with. So in Gw2 it could be that we could have a system of homogenous Attributes->heterogenous pool of Skills -> among a homogenous set of Builds-> that make up some number of heterogenous classes. Collapsing more and more systems toward homogenous states, implies that the system at all scales (as a whole) is Becoming more homogenous, and a collapse of more and more states towards a heterogenous state at all scales, implies the system is becoming more and more heterogenous. The universe is said to be homogenous in totality, but we see heterogenous systems all around us...and when we zoom in on these things that are heterogenous and find that they are made up of homogenous atoms, where every electron 'seems' to be no more different than any other electron.

Notice how I say "seems" is because homogeneity and heterogeneity are local and approximate notions, of some larger more complicated mechanism at play. It could be that electrons are made up of a zoo heterogenous strings...or it could be that electrons are made up of a singular homogenous combination of components (binary information)

I agree with this. But the example which i posted and we were discussing for simplicity we use 1 system which dont have sub systems and is not part of another system. Just 1 simple system which shows that equal states not always means homogenous system .

[razaelll.8324]

please correct me if i am wrong but in your prevous post you stated

Now let's pretend we added another skill C, with some other set of attributes and the relative complexity between A and C is say, 20. The computation goes forward 13 computations to reach A->B, and then the computation goes an additional 7 computations to reach State C and the system is now maximally complex at state C. The system is now again, homogenous, where A = B = C

Thats why i made the example above which shows that that's not necessarily true in all cases, so the point which i am trying to make is it cannot be judged is the system heterogenous or homogenous by equalizing the different states

[JusticeRetroHunter.7684]

@razaelll.8324 said:I agree with this. But the example which i posted and we were discussing for simplicity we use 1 system which dont have sub systems and is not part of another system. Just 1 simple system which shows that equal states not always means homogenous system .

That's the thing, is that you can't separate them. The collection of attributes is a description of a system of skills (in our case, a system consisting 2 skills, each with 3 attributes). So in the maximally complex state, that Cmax state is a description of the state of skills, where each computation is transforming a set of attributes (lower scale components) from one skill into another skill.

The system simply can't be reduced by just isolating the attributes from their collective state (the skill) otherwise we can't explain the evolution from one state (skill) to another state (skill) via a computation of their attributes.

[razaelll.8324]

@JusticeRetroHunter.7684 said:

@"razaelll.8324" said:I agree with this. But the example which i posted and we were discussing for simplicity we use 1 system which dont have sub systems and is not part of another system. Just 1 simple system which shows that

equal states not always means homogenous system

.

That's the thing, is that you can't separate them. The collection of attributes is a description of a system of skills (in our case, a system consisting 2 skills, each with 3 attributes). So in the maximally complex state, that Cmax state is a description of the state of skills, each computation is pushing a set of attributes (lower scale components) from one skill into another skill.

The system simply

can't

be reduced by just isolating attributes from their collective state (the skill) otherwise we can't explain the evolution from one state (skill) to another state (skill) via a computation of their attributes.

I didn't , maybe i had to clarify more my example , excuse me for that.

In the example i posted i have in State A and State B 2 skills each with 1 different attribute in state C i am adding 3rd skill with again 1 different attribute , so we have system of 3 skills each with just 1 attribute which is also different than the attribute of the other skills. So i am not isolating any attributes.

To be more clear lets say:Skill 1 healing - attribute heals for x HPSkill 2 damage - attribute does x dmgSKill3 stun - stuns for x seconds

And from the picture you can see we first made State A to become State B then State C after that we made STate B to become state C but the attributes in state C are still different to each other so the system is not yet homogenous because of that

otherwise we can't explain the evolution from one state (skill) to another state (skill) via a computation of their attributes.Exactly!

In order to equalize the attributes of 2 skills this skills need to be the same (int terms of their "effect") and in our case they are not thats why we cannot equalize them, that same thing apply for classes.

[Paradoxoglanis.1904]

@Psycoprophet.8107 said:U guys ^ know ur posting these in a gw2 forum thread right? Just checking lol.Must be fun at parties.

Math is cool. I wish more people talked about math at parties.

[JusticeRetroHunter.7684]

@razaelll.8324 said:

@JusticeRetroHunter.7684 said:

@razaelll.8324 said:I agree with this. But the example which i posted and we were discussing for simplicity we use 1 system which dont have sub systems and is not part of another system. Just 1 simple system which shows that

equal states not always means homogenous system

.

That's the thing, is that you can't separate them. The collection of attributes is a description of a system of skills (in our case, a system consisting 2 skills, each with 3 attributes). So in the maximally complex state, that Cmax state is a description of the state of skills, each computation is pushing a set of attributes (lower scale components) from one skill into another skill.

The system simply

can't

be reduced by just isolating attributes from their collective state (the skill) otherwise we can't explain the evolution from one state (skill) to another state (skill) via a computation of their attributes.

I didn't maybe i had to clarify more my example. In the example i posted i have in State A and State B 2 skills each with 1 different attribute in state C i am adding 3rd skill with again 1 different attribute , so we have system of 3 skills each with 1 attribute different than the attribute of the other skills. So i am not isolating any attributes.

I mean I understand what you are getting at, Because the state of the skill isn't in complete homogeneity, it's just some imperfect degree of heterogenous because it's components (attributes) consist of heterogenous components... But when talking about the Cmax we aren't talking about the attributes, we are talking about the state of the skill from A to B. The state of the skill A to B is homogenous, and this notion is approximate and local only to the scale of skills, not their attributes. Essentially, all levels of the system, even if there is one singular heterogenous component among an entire ocean of homogenous components, will be imperfectly heterogenous.

In essence, With Relative Complexity the higher we go up in scale, if at the highest scale of the system it equals 0, then the relative complexity of all it's components must also equal 0, I don't disagree with that. But when we are describing the system's state of the skill, the Cmax is going to be homogenous because its a descriptor of the state of the skill, not a descriptor of the state of the attributes. This is I believe is the heart this issue, is that Cmax describes the evolution of a state at scales above the computation.

[razaelll.8324]

@JusticeRetroHunter.7684 said:

@razaelll.8324 said:

@JusticeRetroHunter.7684 said:

@razaelll.8324 said:I agree with this. But the example which i posted and we were discussing for simplicity we use 1 system which dont have sub systems and is not part of another system. Just 1 simple system which shows that

equal states not always means homogenous system

.

That's the thing, is that you can't separate them. The collection of attributes is a description of a system of skills (in our case, a system consisting 2 skills, each with 3 attributes). So in the maximally complex state, that Cmax state is a description of the state of skills, each computation is pushing a set of attributes (lower scale components) from one skill into another skill.

The system simply

can't

be reduced by just isolating attributes from their collective state (the skill) otherwise we can't explain the evolution from one state (skill) to another state (skill) via a computation of their attributes.

I didn't maybe i had to clarify more my example. In the example i posted i have in State A and State B 2 skills each with 1 different attribute in state C i am adding 3rd skill with again 1 different attribute , so we have system of 3 skills each with 1 attribute different than the attribute of the other skills. So i am not isolating any attributes.

I mean I understand what you are getting at, Because the state of the skill isn't in complete homogeneity, it's just some imperfect degree of heterogenous because it's components (attributes) consist of heterogenous components... But when talking about the Cmax we aren't talking about the attributes, we are talking about the state of the skill from A to B. The state of the skill A to B is homogenous, and this notion is approximate and local only to the scale of skills, not their attributes. Essentially, all levels of the system, even if there is one singular heterogenous component among an entire ocean of homogenous components, will be imperfectly heterogenous.

In essence, With Relative Complexity the higher we go up in scale, if at the highest scale of the system it equals 0, then the relative complexity of all it's components must also equal 0, I don't disagree with that. But when we are describing the system's state of the skill, the Cmax is going to be homogenous because its a descriptor of the state of the skill, not a descriptor of the state of the attributes. This is I believe is the heart this issue, is that Cmax describes the evolution of a state at scales above the computation.

Okay i think i understand your point now.

[JusticeRetroHunter.7684]

@"razaelll.8324" said:

Right, it's because the notion of homogenous and heterogenous are absolutely approximate concepts. This is also why it's so hard to discuss it because in essence your basically right, that in reality everything is heterogenous to some degree, and nothing is every completely homogenous unless everything is, on top of which we have homogenous systems nested in heterogenous systems nested inside of homogenous systems...and so on.

For me, because these notions are approximate, it's almost like talking about balance...we shouldn't even be using these terms because they might not mean anything and behind it might be some larger more elegant architecture that explains what's going on in a precise notion rather than an approximate one. just my opinion heh :pensive:

Edit: so i was going to edit my comment from earlier that i didn't get to finish, but i think right now we are on the same page on the issue, and really that's my fault for not being concise enough with the detail between Maximal Complexity and the Relative Complexity notions.

Also i think the OP abandoned the thread so i was gonna just insert in my edit, a more in depth example that explains rather than using computation for balance changes, one that describes meta evolution (how states of the game collapse to a single metagame) but for now i'm kind of just exhausted thinking on the subject, so I'm just gonna say this... is that given everything said above so far, one can describe the meta evolution in the same way (by just changing the meaning of the computation) and that in the end, it resembles essentially the inverse of backward Induction Where instead of starting at the end state, and working your way backward, it's instead just regular old induction, working your way forward toward the most optimal state.

Backward induction, is basically an optimality problem, where it gets exponentially more difficult to know what the most optimal decision was the further back in time you travel...The reverse of this is that starting from a diverse set of choices, one picks through each choice via an optimality computation, and it brings you to the Cmax state .In essence you can use the complexity metrics and the Cmax thingy to approach this problem, and what you get is essentially meta evolution.

The more complex the game, the harder the computation, and the longer it will take to reach meta evolution, so it's the same principle as backward induction except inverse