On the Generalization and Possible Transformations of the Byzantine Generals Problem
We are going to consider the The Byzantine Generals Problem:
We imagine that several divisions of the Byzantine army are camped outside an enemy city, each division commanded by its own general. The generals can communicate with one another only by messenger. After observing the enemy, they must decide upon a common plan of action. However, some of the generals may be traitors, trying to prevent the loyal generals from reaching agreement.
However, we are going to generalize it and begin to transfer it in order to prepare it to fit into our mapping of Bohmiam Hidden Variables to our concept of The (fluxable) Consensus Field.
The BGP is a specifically formulated philosophical inquiry into the limitations of untrustworthy network communication. It is usually formed for computer science and network theory types problems. But for example the concept of ‘untrustworthiness’ perfectly parallels the idea of a faulty communication line or perhaps a variable representing the ‘correctness’ of a message that is passed from player to player or node to node etc.
In order to generalize the BGP such that we might fit it to a more universal observation we can consider a type of base case:
Let us first examine the scenario pictured in Figure 1 in which the commander is loyal and sends an "attack" order, but Lieutenant 2 is a traitor and reports to Lieutenant 1 that he received a "retreat" order. For IC2 to be satisfied, Lieutenant 1 must obey the order to attack.
Now consider another scenario, shown in Figure 2, in which the commander is a traitor and sends an "attack" order to Lieutenant 1 and a "retreat" order to Lieutenant 2. Lieutenant 1 does not know who the traitor is, and he cannot tell what message the commander actually sent to Lieutenant 2. Hence, the scenarios in these two pictures appear exactly the same to Lieutenant 1. If the traitor lies consistently, then there is no way for Lieutenant 1 to distinguish between these two situations, so he must obey the "attack" order in both of them. Hence, whenever Lieutenant 1 receives an "attack" order from the commander, he must obey it.
However, a similar argument shows that if Lieutenant 2 receives a "retreat" order from the commander then he must obey it even if Lieutenant 1 tells him that the commander said "attack". Therefore, in the scenario of Figure 2, Lieutenant 2 must obey the "retreat" order while Lieutenant 1 obeys the "attack" order, thereby violating condition IC1.
Here the problem is reduced to 3 players to which one of the players (commander in the BGP paper) is the ‘coordinator’ and the other two receive the coordination signal. In the BGP it is suggested and shown a defector cannot be detected with this scenario.
Therefore, the assumption or lesson is, there needs to be GREATER than ⅔ honest generals and so:
Hence, no solution exists for three generals that works in the presence of a single traitor.
If we are thinking in a universal context we are already thinking of how this problem maps to a three body problem, however, we have some necessary characteristics and so our transformations will seem to lead us elsewhere at least temporarily.
If we have one coordinator that propagates a signal (here we haven’t defined the characteristics of that signal ie who receives and how trustworthy it is etc.) and we are concerned with the validity of communication with regard to the context of meaningful consensus, we are in a sense considering the closing of the loop of consistent information.
Thus, if the coordinator sends a signal (i.e. either a 0 or 1) to L2 (L for lieutenant) we are considering whether L2 reliably receives the correct signal and passes that correct signal to L1 who then passes that correct signal back to the coordinator.
We don’t need to consider the reverse. If we begin to solve and model the observation of a consistently arranged cycle of transferred information the consideration of the same in a counter cycle is not different. If we solve this problem clockwise then we have the same solution as applied counter-clockwise.
In the BGP there is the idea of traitors, players or agents. They don’t reliably pass messages on nor do they reliably act in coordination. This provides the antagonist for the problem. In the base case of the BGP with 3 players it is shown you can’t deduce reliable coordination with a single traitor.
We want to consider the traitor to be of a different color.
Among other things this allows us to observe from the outside of the system, a visual that represents the concept of a faulty or nefarious node, but also brings the concept of color-blindness of nodes, so that we can identify traitors from our observational standpoint (as the observers of a hypothetical BGP world) but not necessarily implying this information is available to the players of the BGP world.
If the coordinator gives a signal to the next player, and the signal is correctly passed around the loop and returns to the coordinator then we can consider this as a consensus (this is coincidentally well captured by the game chinese whispers).
If a player is a traitor, they don’t pass messages reliably. We can consider this as if they ‘flip’ a message bit they receive and pass.
Of course if the next player KNOWS the sender is a traitor they can flip the bit back.
This is where colorblindness is a helpful metaphor.
Then in our model we can consider a player that receives a bit from a traitor, IF they can see they are the color of a traitor, they can compensate and reverse the ill effects of the traitor.
Then in this model we still can achieve a full circuit consensus of consistent information.
However if there is color-blindness it is as if we are back to the BGP base case and there is again no hope to find such a full circuit of consensus.
In this consideration where consensus is the goal the color of the players isn’t important.
If we consider the concept of a traitorous side that can affect an opposing team's ability to coordinate we can consider then if the traitor side has the ability to coordinate. Then we are asking if the system can find consensus by EITHER team and then considering as if the communication system or universe is a third player that may or may not negatively affect the ability of either of the two teams to find logically consistent consensus.
Then SUCCESS is achieved by consensus of EITHER team and color-blindness simply represents a team that is having communication problems. Sometimes these problems can be reasoned around if there is enough trustworthy communication just as if there is no color-blindness and faulty communication can simply be accounted for and fixed.
In this context then we have TWO possibilities of consensus, if there are two competing teams and we don’t prefer consensus from one team over the other.
We are preparing the concept of consensus for Bohimian Hidden Variable theory; the previous section gives us an example of one possible preparation. Bohm means to collect his variables from a concept of periodicity and so we can begin to consider the difference between consensus of team1 and the consensus of team2 as if periodic.
In such a case the consensus would have to be unstable and have some sort of chaotic principle that lends to it the force that might reverse its consensus completely. But here we just mean to note that if consensus was denoted by 1 or 2 for each of the teams we could still consider an interactive out come of consensuses such as: 111221211111121 as oscillation between 1 and 2 even though there might be re-consensus of 1 multiple times before there is consensus with team 2.
In other words we can CHOOSE to see the system as periodic even though its with some loose restrictions to do so.
In An Optimal Probabilistic Protocol to Synchronous Byzantine agreement there is proposed a protocol that produces an ‘ oblivious common coin’ flip. Without explaining the details of the protocol the idea is that every honest participant, or every potential consensus agent, is able to locally conclude either a 1 or a 0 to which their ‘teammates’ from their local knowledge can also reliably produce.
It is a curious necessary outcome of the solution that a random bit is produced.
This is another possible concept of periodicity we can consider conducive to the concept of deciding to ‘attack’ or ‘retreat’ in the general description of the problem.
We can also consider the rounds in the probabilistic solution as a possible phenomenon of periodicity. In the OPP solution each round doesn’t necessarily produce a consensus on the state of the common coin. If this is the case another round is played and the statistical likelihood of locally confirmable consensus on the state of the coin increases.
As a relevant side note the number of rounds is probabilistically predictable.
We can also consider the difference between a state of consensus, that is asserted by either team1 or team2, that is a coinsate of either 0 or 1, perhaps determined in a certain round, as one state versus the state where no consensus is achieved yet, as if there is a state of flux BETWEEN consensus states as an oscillation.
Thus we gather our periodic considerations here:
The protocol involves multiple rounds of communication, where each round can be seen as a period. In each round, nodes attempt to reach consensus through the mechanisms of gradecast, secret sharing, and consensus evaluation. This cyclical process, which repeats until consensus is achieved, exhibits periodicity in the sense that there is a recurrent sequence of steps aimed at driving the network towards agreement.
The protocol's design to fluctuate between 1> seeking convergence on '0' and '1' as necessary conditions for progression also reflects a form of periodicity. This is evident in the sequential aim to achieve agreement first on one state and then on the other, creating a pattern of alternating states that could be seen as mirroring periodic behavior. Oblivious Common Coin Tosses: The use of the oblivious common coin within the protocol, especially when considering the protocol's attempt to reach a unanimous decision, can also be viewed under the concept of periodicity. Each coin toss represents a discrete event that occurs within the framework of each consensus round, contributing to the cyclic nature of decision-making in the network.
While not strictly periodic in a deterministic sense, the probabilistic aspect of the protocol's termination (that is, the protocol may require varying numbers of rounds to reach consensus based on random outcomes) introduces a form of periodic uncertainty. Each round presents another 'cycle' in the attempt to reach agreement, and the system's dynamics during these cycles exhibit a form of stochastic periodicity.
The process of moving from individual secret contributions to a unified agreement (either all 0's or all 1's) within the network reflects a binary oscillation that can be considered periodic. The network's state oscillates between periods of disagreement and agreement, aiming for a stable state where all honest nodes consistently report the same value.
On The Recursive Nature of the BGP and Mapping The Concept of Bohmian Hidden Variables
The BGP framework tolerates only up to ⅓ corruption this means you need just more than 3 times the efforts of the traitors:
3m + 1
In the BGP this concept is granulated with the concept of lieutenants so that we consider a system cannot achieve consensus when ⅓ participants are corrupt but that if 1 of the 3 participants was the commander of an army then the nature of his participation is determined by the consensus of his army.
In a general sense and thinking from a universal perspective of 3 bodies looking for equilibrium in motion (perhaps they do) we are saying at some equilibrium point of stability it only takes one atom of difference to disturb the equilibrium
This would put the system in a state of flux in our consensus model.
This is quite a classical mapping and both physicists (of the quantum and mechanical) and network scientists should easily be able to enter into it and find it agreeable.
We can think of a circuit of consistent consensus (where traitor nodes are able to be identified and thus their erroneous messages corrected) as a simple covenant based con-catentant co-ordination.
Here we are thinking of the initial coordination signal and that the concept of convergence or consensus is achieved when every consecutive player is well ordered ie with logical consistency where messages are correctly passed or compensated for.
When each message is correctly passed and compensated for we can call this catentation and when it's passed but not correctly irre-catantion etc.
We are considering the mapping of the computer science BGP and consensus studies to observations of the universe. There are classical considerations and quantum considerations perhaps in either the natural world and the computational. Through our inquiry and the bridging of these subjects we are considering the concepts of classical and quantum orientations of The Consensus Field we are defining (and we are defining this consensus field so as to be helpful and easy to map with Bohm’s hidden variable theory etc.)
For the BGP problem the concept of a message being either 0 or 1 is satisfactory for representing a piece of information that is unknown to anyone but a receiver. Without at least that choice then everyone knows the answer. But this simply means to represent ‘clear and proper signal received’. In that sense it's really just a unary phenomenon (It happens or not but whether its 0 or 1 is somewhat an extra fact).
So we are asking if we received the message properly, and we realize that message needs at least one bit of information, else the problem reduces to already solved.
A similar concept is explored, and this maps to the further extension of the BGP papers, if we consider some probabilistic or determined but unknown ratio or error in the messaging system or messages sent and received. And of course this is akin to having unreliable or untrustworthy messengers.
Consider the introduction to the BGP problem form the OPP solution of it:
Broadcasting guarantees the recipient of a message that everyone else has received the same message. This guarantee no longer exists in a setting in which all communication is person-to-person and some of the people involved are untrustworthy: though he may claim to send the same message to everyone. an untrustworthy sender may send different messages to different people. In such a setting, Byzantine agreement offers the "best alternative" to broadcasting.
In a network where messages can be broadcasted near simultaneously and reliably to everyone the byzantine considerations are unnecessary. Its only when there is the introduction of corruption of a sorts that there is the need to study the fault tolerance of the communication network.
In this sense we can consider the communication network as a field and consider corruption of the communication lines, the messengers, the messengers, and the players or nodes (as well as the nodes being made up of a multiple of voting lieutenants etc.)
We can think of this field as a third player, or exogenous force, with regard to different considerations of the communication lines, the messengers, the messengers, and the players or nodes (as well as the nodes being made up of a multiple of voting lieutenants etc.) and with respect to the 3m +1 requirements of each.
On The Usefulness of A Third Player and Game Theory As a Reminder of Caution and Machievellian Communication Field Considerations
We want to consider the useful types of mappings and formalizations of the BGP and we think that using a 3rd player and the framing of a game is helpful to remind us that the network could be unpredictably corrupt in general or it could be quite predictably corrupt in a very organized manner.
This could be missed in certain fields that mean to only consider real world applications. We want to consider all possible variations of the corruption of a network, even one that is a function of an intelligent (perhaps all knowing!) malevolent attacker.
This means the flux of the ‘communication field’ could in theory create an impossible situation for any type of consensus to be reached for any period of time etc.
This could be something the other players who mean to find consensus either know of or not as if the third player was part of the rule maker or axiom definer of the game, or functioning from a god role, or simply a hidden ‘haunting’ type role.
In the simpler formalization the 3m + 1 rule can be upheld throughout all the transformations, however, if the 3rd player can make complex moves then perhaps there is a greater percent of trustworthiness needed from the communication field.
We should consider the concept of distance and the effect it could have on a ‘msg’. And we can do the same with regard to time. Messages that are further away perhaps are subject to a greater chance of corruption. Or perhaps this happens because they lose context over the time it takes to travel.
This concept then could be represented by color. And its not a coincidence that we mean to consider this concept with respect to light. Here we are thinking about the concept of the redshift of light with respect to gravitational waves.
As wonderful well fitting as this seems, we actually also fit this purposefully to highlight a paradox or uncertainty problem.
If we understand redshift and gravity well (if are superadvanced in our knowledge of these things etc.), then we aren’t so concerned about the distances of our senders and receivers because we can account for the variables and deconstruct the effects of the shift.
But if we didn’t have this type of confidence, and we didn't know the distance our sender, and probability was relevant, we wouldn’t have the decryption information necessary to know what ‘color’ the original message was.
Then this model doesn’t just port well, its also ports the uncertainties problems of the quantum mechanical observations that support the Heisenberg observations.
We have thus generalized the BGP problem slightly with examples of possibly useful transformations. While doing this we specifically and purposefully highlight some concepts from a context of periodicity.
One such examples we found we feel is a critical observation-that a GENERAL consensus, consensus of either the loyal or traitor side considered as success or state models well as periodicity.
This periodicity is missed or an unnecessary consideration when considering tele-communication fault tolerance (which generally considers only one side of consensus as consensus) and yet is very relevant to consider from a universal nature perspective.
This concept then also naturally illuminates the periodicity between the two possibilities of consensus between either the general or the traitor team and the state in which there is no consensus yet. That is to say, if we don’t favor a side, but only consensus, there are then concerned times where there is consensus and times when there is no consensus (flux).
Summary In Regard to the Curious and Undeterminable Determined Nature of the Common Coin Flip In An OPPBGP Framework
Another concept we wish to highlight in our summary is the implicit need to create a random variable for consensus in the OPP BGP framework. There is a curious nature in that the need for the variable to be random is only because if it's predictable then the defectors could use that knowledge, and yet the variable is only a byproduct of the process and not the basis for the consensus (it is somewhat paradoxical that it is an after the fact OUTPUT and not an INPUT but yet critical!).
Here we are noting that there is a necessarily probabilistic component effectively produced from a very determinant protocol. To watch this process from an external unknowing perspective one would need to be careful not to try to use the pattern of coinflips (the string of random zeros and ones) to reverse extrapolate the meaning behind them.
There are two interesting points, the NECESSITY of a probabilistic BYPRODUCT as well as the INDETERMINABLE origin from an external perspective of a protocol that is meant to produce deterministic outcomes (and we haven’t yet inquired into the nature of the individual inputs of participants, ie do they have random sources etc).
For readers that might not have taken the time or effort to understand the nature of the OPP solution this section might be illuminating. In some sense each of the individual honest participants are able to do ‘local’ math to come up with a number that is either a 0 or a 1. This math is based on a predetermined convants. The necessity of the setup is that each participant, who has uniquely verifiable information (ie information only they possess the verifiability to), such that when they use the predetermined covenants and apply it to their local information each of them results with the same answer either 0 or 1.
This is the trick of the solution, HOW its setup to solve the problem like this is the explanation of the paper. That the group either lands on 0 or 1 but no one can predict which of the two was explained in the previous section.
Here we are noting something we wonder is also missed from a tele-communications application perceptive which is the simultaneity of the information produced and observed.
At first it might not seem simultaneous. The message might not be received at the same time, in fact laws of physics prevent it, and the rounds that are necessary would effectively prevent it etc. In other words whether there is consensus on either 0 or 1 is determined independently locally as the players do their computations. This act doesn’t happen at the same time…
HOWEVER…
It should be quite obvious to the reader following our work that the concept of localized knowledge of a shared piece of information, producible or observable at will, without problem of travel or distance, is VERY interesting.
We have noted what we feel are DRAMATICALLY significant observations when considered from a universal and natural perspective. There is a NECESSARY probabilistic byproduct when it comes to the consideration of periodic consensus in systems we describe from the BGP framework which is not at all necessarily able to be reverse engineered (to find the nature of the protocol that creates the output) and also we have modeled the concept of ‘manual’ simultaneity, accidentally, as a solution to indeterminable communication fields.
If the reader is still confused on the significance and simultaneity (of computations that certainly don’t finish at the same time) consider the end of a successful consensus round but before all of the players compute the common coinflip individually and locally.
We can say that at that point each of them simultaneously has the same information in their ‘box’ of inputs to be calculated. Even though we don't know the outputs of their box in this regard we can note they are in fact entangled.
We have now successfully mapped the generalization of the BGP so that it fits our concepts of Hayekian fields with regard to how prices can simultaneously transfer otherwise non-local information (something we used our deconstruction of Cantillon's observations to illustrate)
This mapping calls more into question the symmetrical aspects in regard to the number of players considered for consensus versus flux periods.
It is perhaps ALARMING this is not discussed in the bitcoin and wider crypto-communities.
Is the optimal scenario that the protocol should stabilize consensus to team1? Or should it oscillate between team1 and team2 (as the most basic example).
Such a seemingly innocent or pesky question really is more alarming when we consider whether or not a state of flux is or isn't and acceptable state.
In this sense the concept of 50% versus 50% as opposed to a super-majority ie 3m + 1 has significant and relevant considerations (In these cases games of 4 nodes are quite different than games of 5 as 4 doesn't have a tie-breaker).
These considerations are UNHEARD of in the Bitcoin community.
We have already noted the natural mapping of the 3 generals problem to one in which at least one of the generals has an army.
We consider then that there is only really the disruption of equilibrium that is necessary to not have a stable solution of consensus available.
It could be the shadow player's role or intent to hold the system in a non-consensus state, or a consensus state, or a periodic cycle etc.
The interesting consideration and concern however is that we have never heard of removing the concept of needing to overthrow the necessary '2/3s' limitation to achieve consensus by simply adding perturbation into the system
There are many consideration this brings to mind with the most critical simplest one being that in physical reality of probabilistic based matter (ie intrinsic vibration) if the players aren't quantum units the system moves in and out of consensus and thus possibility might still have fluxation at its defined solutions points etc.
We stumbled upon the idea that an achieved consistent consensus circuit can thus be called a decision of that system and then a sustained or iterative phenomenon of such can be called action.
This will help us bridge more with Mises and Smith as well as emerging AI phenomenon.