Defining Systems

Systems Theory: Part I

Defining systems, looking at a football (soccer) game, computation.

Rules and Start Conditions

Rules:

• no player shall carry the ball with their hands.
• no player shall carry out undue violence upon any other player.
• Possession of the ball switches after the ball leaves the field
• if the ball leaves the field by way of the long side, it shall be thrown in. If the ball leaves the field by way of the short side, it shall be kicked in.
• Every time a team pushes the ball between the two shoe-goal-posts, their counter shall be increased by one.
• the game shall start when Bob whistles very loudly, at which point he will start his chronometer on the really cool watch that his father gave him.
• the game shall end after 90 minutes, after which the team with the highest counter wins.
• After 45 minutes, the passage of play shall stop and the ball shall switch possession

Start Conditions:

• twenty players shall start on the field, ten on each team
• the field shall be 80 yards long and 50 yards wide
• two shoes shall be set up as goal posts 10 steps apart, in the middle of both short ends
• both score counters are set to zero
• Bob sets his chronometer to zero
• the ball starts in possession of the starting team, with the opposing team 10 yards back

These probably seem familiar as describing the rules of football (soccer). Well, a simple version of the rules, one that me and my friends would use since we had little concept for the details of offside, or how to define what constituted a foul.

From this setup, you are probably able to imagine what the start conditions look like: 20 little boys on a big field, with 2 pairs of shoes at each end, the ball standing still, waiting for Bob’s whistle.

You can probably imagine the very next moment - a shrill whistle, the ball is put in play, and the boy closest to the ball start running towards it, while boys farther away start yelling for it.

While you cannot predict the outcome, or even predict what the field will look like at any given moment in time, you can make inferences based on the rules of what will not happen. No boy will pick up the ball and carry it across the field in his hands (or if he did, the “game” would stop so his actions would not count towards the final score). No boy will punch another boy in the face in order to take possession of the ball (or if he did, the game would stop, the boy would be forced to leave, and the ball would be returned to the first boy who was punched).

In this sense, the “rules” don’t determine exactly where the ball will go, or which boy will run where. But they do form constraints through which the game must “pass.” I will call these rules “prescriptive,” because they detail how things “should be.”

But let’s pick a specific moment, where one boy - we’ll call him Adam - “has” the ball, which is to say the ball is at his feet and there is no other boy within 5 feet of him. How do we know what will happen next? The prescriptive rules are mute here.

Put another way, if Adam had never played before, but fully understood the rules of the game as defined above, he might still have no idea what to do next. Likely the other boys would yell various commands at him, and he would pick one, until he subconsciously picked up what he should be doing at any given point in time.

So how do we go about defining what happens, what goes on in this game?

Now, if Adam kicked the ball, we could determine what would go on next, because we could calculate its trajectory through the air according to Newtonian physics, with an added factor for the friction of the air on the ball. Just as the rules of physics would describe what wouldn’t happen - a boy magically flying across the field, or the ball stopping in mid-air - they also describe what will happen next, after a given action (the ball being kicked). But all this is saying is that the game follows the rules of physics. Perhaps we can then say that the prescriptive rules of my childhood soccer are nested within the prescriptive rules of physics. We’ll get back to nested systems later, though, and assume the “rules” of physics for all that goes on in this game.

But this only explains what happens to the ball after it is kicked. What about before? What determines who Adam will kick the ball to, or if he’ll kick it at all? We cannot make this inference based on the prescriptive rules of the game, nor the rules of physics. But there are certainly “patterns” that will emerge that are not contained in either of those sets of rules.

For example, Adam will probably not kick the ball straight to an opponent. We know this intuitively, because we understand that losing possession of the ball is bad for trying to score a goal. Furthermore, scoring is a necessary condition to winning, and every boy on the field wants to win. The desire to win, then, is a central component of the game, but it is not outlined in the prescriptive rules of the game. What, then, is this desire with regard to the game?

Well, clearly, it was implicit in the starting conditions: twenty players shall be on the field, all of them trying to win the game.

But even knowing that Adam is trying to win, does not make it perfectly clear what Adam will do next. We as viewers, however, intuit that there are really only so many actions Adam can take at this point. Adam can kick the ball, either towards a teammate or towards the other team’s shoe-goal-posts. Adam can move forward with the ball, to get to a better position for himself or for his teammates vis-a-vis the opposing teammates. Or he can stand still, and wait for a better position of his teammates (or a worse one of the opposing teammates).

Based on his goal of winning, Adam will at any point take whichever action brings him and his team closer to scoring a goal. Similarly, there are only a handful of options available to his teammates while he has the ball, and a handful available to the opposing teammates, who are then trying to minimize the chance of Adam’s team scoring a goal. We can simplify this by saying that players on the team in possession, are choosing based on whatever seems to bring the team closer to scoring a goal, and the other team based on whatever seems to bring the team further from scoring a goal.

And just like that, we now have a different set, which we can call a set of instructions that determine what will happen next. And using these instructions, we will re-define this match in a way that, perhaps, makes it easier to understand what is going on and what will happen next, on this little grass field. If Adam, then, had no logical faculty to derive these instructions from the rules of the game, we could feed him these very simple instructions to turn him into a functional team player on the field.

Instructions:

• When a player “has” the ball, choose to “wait,” move to another location “with the ball,” or kick the ball to a teammate or towards the opposing goal.
• When a player does not “have” the ball, if the ball is in their team’s possession, run to a different position, or wait.
• If the opposing team has the ball, the player should run to a different position, attempt a tackle (a committal effort to get the ball), or wait.
• Players on the team in possession, choose between options based on whichever one seems to bring their team closer to scoring a goal. Players on the team out of possession, choose between options based on whichever one seems to bring the team in possession further from scoring a goal.

And we can add a few more instructions to describe the ball going out, or being scored:

• If the ball moves through the shoe-goal-posts, all the players stop play until possession of the ball is switched. If the ball moves across the boundary of the field, possession of the ball is switched.
• If a player “has” the ball after the ball goes out the side, his only option is to throw it towards one of his teammates.
• If a player “has” the ball after the ball goes out the back end, or after a goal, his only option is to kick it towards one of his teammates.

Now, these instructions apply to individual agents, so let’s make sure we keep a description of the the elements to which these instructions apply:

• twenty players, ten on each team
• a ball
• two score counters
• Bobby’s watch chronometer
• two pairs of shoes

And we’ll use the start conditions to describe the initial configuration of these elements:

• the field shall be 80 yards long and 50 yards wide (so all the boys and the ball should stay within this boundary)
• two shoes shall be set up as goal posts 10 steps apart, in the middle of both short ends
• the score counters and Bobby’s chronometer are set to zero
• the ball starts in possession of one team, with the opposing team 10 yards back

With this breakdown of elements, instructions, and start conditions, we can start the game. We just need one more thing, an end condition, that stops it:

• When Bobby’s chronometer shows 45 minutes, all the players will reset and the ball switches possession.
• When Bobby’s chronometer shows 90 minutes, the game is over.

Now we have fully defined the system. Of course, there are small expansions we should probably make, like the fact that a player will expend energy in such a way as not to get gassed before the end of the 90 minutes. Or that the players will make certain decisions slightly differently based on the current score. Or that a team might use their time in possession as a way to keep possession, and thus decrease the other team’s chance of scoring, instead of focusing exclusively on trying to maximize their chance of scoring a goal (the innovative realization of the legendary football manager Pep Guardiola, who did this much to the chagrin of fans who actually preferred seeing “action” to just winning).

As you can see, it is no simple feat. If you go back and read through the original rules, and our new instructions, you will see that there is some bridge between them, but it is not a clear one. It depended heavily on me using words like “certain patterns will emerge,” “we understand that,” and “inference.” The process of deduction is a complicated one to pin down, and I won’t attempt that here, but already the first lesson here is clear: the instructions of a system are not always obvious to us, but once we pin them down, the behavior of the system becomes, in a way, “obvious according to them.”

There is something crucial you may have noticed in our switch from our prescriptive rules to the instructions. I’ll give you a second to go back and see if you caught it.

There is nothing in the new definition of the system that has to do with the final score. There is no rule about who the winning team is, and who the losers are (and no, that does not mean that “everyone is a winner” here, and on that field that was certainly not true). That is because the system, while in motion, did not depend on the final state. What mattered was perhaps who was winning at a given point in time, which is to say who had the higher score at a given minute of the match - but only insofar as changing the risk tolerance of players or the energy they were deciding to keep in store.

That is because one characteristic of this system was its end condition: that 90 minutes were up. As long as that hadn’t happened, no matter the score, the game played on. That is the second lesson: the instructions constitute everything that might happen next while the system is in motion.

We’ll note that a simple rule addition could have been made to tie the final state to the definition of the system. For example, the boys could have decided to end the game if any one team was ever up by 5 goals, a mercy rule. This would have done two things: it would have allowed for the teams to be scrambled into a more “fair” split, and it would have turned the game from a fixed length of 90 minutes to some variable length between 1 and 90 minutes (assuming an incredibly skilled team could score once every 12 seconds).

We’ll also note that a different rule subtraction would have radically changed the experience of my childhood: omitting the 90 minute end time. In that case, the game would have never ended. We would perhaps need breaks in order to recuperate our strength, and thus the game would have long pauses, and our score counters would have to learn to count really high, and also I would still be playing that game and would never have gotten the time to sit down and write all these things down for the pleasure of my reader.

This concept of “end condition” of a system is pretty important, which is why I dedicate a lot of chapter 8 to it. But this is our third lesson: a system does not need an end condition that stops the motion of the system, although many systems will have one.

There are two types of “end conditions:” one is defined by a length of time (here, a match of 90 minutes). Others are defined as a specific configuration (the mercy rule, a difference between the two scores).

Let’s dive a bit deeper on time, starting by way of these ‘updates’ we keep referencing.

A specific ‘update’ based on our instructions, is an exact configuration of the locations of all the players and the ball. Now, in this case, every player is continually following instructions, so the ‘updating’ is continuous, ever-flowing. But what if every boy stood still at a given configuration, waiting for Adam with the ball to make a move? What if this happened for a full 10 seconds? We see here that the ‘updates’ are separate from time - a transition occurs at every update, but that transition is not predicated on a unit of time. It is just predicated on the work being applied to the instruction (in this case, Adam’s energy being channeled to kick the ball or run with it). There is an important intuition here: the number of “steps” or “updates” in a system does not always depend on Time. The 90 minute match might then have 10,000 ‘updates’ or 1,000,000.

The end condition of time makes this interesting, because it separates the number of steps from the conditions required to end the game. But what if we returned to our mercy rule? Then the game could take 1,000 updates and only 5 minutes.

We could, of course, modify the game by altering the length of time between updates. For example, a new actor could blow a whistle every three seconds, and rule that the boys can only move when they hear the whistle. This starts to smell like turn-based strategy games where everyone decides what they’re going to do, then announce it at the same time. Turn-based strategy games are also interesting, because the end condition is either based on a specific ‘winning’ configuration, or a timer that is based on the number of turns.

Each turn corresponds to one “update” of the elements according to the instructions. These steps are what determine how much has happened in a given system. More steps = more has happened, and vice versa.

So, as a simplifying factor that we’ll then re-complexify later, we’ll divide time in capital-T-Time and step-based-time. Step-based-time is, in this case, the difference between a ‘regular’ game of football and a move-freeze-every-three-seconds match, so we’ll add it in the definition of the system as well.

Using this updated definition, we can then further break down end conditions, into three categories: defined by a length of Time (a match of 90 minutes), defined by a length of step-based-time (20 turns then the game ends), or defined as a specific configuration (the mercy rule). A system can have multiple end conditions.

We have set up all elements, held them in an initial configuration, defined how the elements can ‘update’ moment to moment, and even optionally defined when the elements will stop ‘updating.’ All we need now is that which sets the system in motion.

One might hastily chime in with the word “energy” to plug this gap, but then one might ashamedly remember that energy cannot be created nor destroyed, so the energy of a system should remain constant over time. But energy is the right hint at the answer: the “movement” of energy. This is also called “work” in physics - whatever thing it is that forces all the elements of the system to update, according to their instructions, moment to moment. In the case of our football game, the work isn’t hard to find - every boy comes with energy and a willingness to expend that energy in carrying out these instructions. Without either of those components, every boy would just stand still while the clock ticked by…

To summarize, here is all that is required to fully define a system:

• a list of all types and counts of elements
• a set of instructions that outline how each element can be updated from moment to moment
• start conditions outlining the initial configuration of all the elements
• one or more optional end conditions that “halt” the successive updates/work
• movement of energy through the system, which updates the elements according to the instructions, from moment to moment

Now I know that this smells like a computer program to some of you. You’re not wrong, but there are important distinctions:

When people think of programs, they typically mean the instructions and some part of the start state. This allows a user to provide an input - the final ‘piece’ of the start state - which then starts the instructions running. And in almost every case, the focus is on reaching the end condition, the ‘output,’ the ‘answer.’ This is what they mean by computation - and thus the entire field of study that is computability theory: the study of which programs produce an output, and which don’t. Another way this is framed is “which problems are solvable, and which aren’t.”

But computation is a much broader concept: computing is the act of successively updating the elements of a system according to the instructions, and doing so due to the “work” that is working through the system. So, in effect, the study of computing is the study of systems, as I am defining them. And within that vein, the study of systems is not the study of calculation, nor of computer programs, but it is the study of computation in motion.

This is what I think a lot of people miss when they think about “computers” - they only think about those programs that end, and in doing so, they miss a lot of the interesting-ness that happens in between. And I pretty firmly believe that most of the interestingness of the universe actually is that in-between.

Another layer: formal systems

A formal system is a mathematical setup with axioms, rules of inference/production, and a decision procedure.

I won’t go into too much depth here, but an example might give you most of the intuition needed here. I’ll draw on Hofstadter’s work in GEB to give you the MIU-puzzle.

In the MIU-system, there are only three letters of the alphabet: M, I, and U. Any combination of letters is just called a “string.”

There are 4 rules:

1. if the last letter of your string is I, you can add a U to the end of it
2. if you have a string in the form Mx, you can add Mxx to the end of it.
3. If there is an III anywhere in your string, you can replace it with U.
4. If there is a UU anywhere in your string, you can remove it.

Now, the puzzle is this: starting with the string MI, can you produce the string MU?

Let’s add some language about formal systems:

We call every string that is “reachable” from the starting string, a theorem. In that sense, the theorems of the MIU-universe don’t need to be proven, just produced. The starting string is an axiom, from which you can then produce other theorems. (There is one more element to formal systems, which is a decision procedure, namely some finite method of determining if any given string is a theorem, but this isn’t as relevant to us now.)

So, another way to ask the question of the MU puzzle is, within the MIU-system, given the axiom MI, can you produce MU as a theorem?

First, I encourage you to try and solve this puzzle for a few minutes, as it’ll give you some intuition for systems as a whole.

…

As you try to solve the puzzle, you probably took out a piece of paper and started applying rules at random to the initial MI string. If you were really organized, you drew it out as a tree, something like this:

But, similar to the boys standing still on the field, there is nothing to “push” or “propel” the string from one place to the next. There is no ‘work’ to do the computing here, except for a passionate person trying to reach MU and scribbling furiously on a piece of paper. Also there is no specific direction at every branching. For example, when reaching MII at step 1 - should the system go left to MIIU or right to MIIII?

Well, in this case, the only ‘agent’ is us, the passionate scribblers, and we are using this tree to explore every possible path. So we don’t have to make any decision at a fork in the road, we can just draw out both.

So this tree is the possible paths of a system. If we did this for the boys playing football, this tree would rapidly become so huge that it would take up more paper than could cover the Earth (I don’t know that scientifically, but it’s definitely true). And every node or leaf in the tree is some configuration that is ‘reachable,’ so in that sense every configuration of the boys on the field is a theorem of that formal system.

But the football game itself was a specific journey, a choice at every fork, that led to one specific future.

Within this image, computation is the act of moving through this tree (and always going down, never back up, because the rules of instruction don’t allow that). The entire football game, then, was the entire sequence of ‘events’ - the full string of the decisions made. Not MIIIIU as an end state, but the sequence of MI-MII-MIIII-MIIIIU. So when those specific 20 boys all got on the field and agreed to follow the same rules (meaning they’d each follow certain shared instructions), the system was defined. And as they played, the full 90-minute match was the computation of that specific system.

The ‘output’ of the game might have been a specific score - maybe Adam’s team won 3 to 1. But the entire computation was the full sequence of configurations, just as the computation undertaken in the image above was a given path of the sequence MI-MII-MIIII-MIIIIU (note: the dashes represent step-based-time).

Hopefully you’re starting to see the concept of computation as something that extends far beyond the realm of computer science. In a way, physics is an attempt to understand the “system of physics,” as is biology, chemistry, sociology, economics.

Which is just as well, because we’re about to start with one of the first big claims here: the universe as computation.