Paths Through State Space and the Set of Reachable States (in a State-Based Systems Formalism)
Something can happen in many ways.
There are, for example, near-infinite ways of hammering a nail. Exactly how you do it varies with when you do it (on a Monday or a Tuesday), how you hold the nail, the angle of the hammer as you strike, the strength you use, where you look, what you think, what food you’re digesting at the time, and so on. Most models don’t take this immense complexity into account because doing so simply isn’t useful — but what if there is a way to formalize it in a way that doesn’t sacrifice utility?
This can be done using paths through state space.
Note: This text builds on the formalism introduced here: Introducing a State-Based Systems Formalism.
Paths through state-space as the ways something can happen.
Given a system, there are different ways to move between states.
Define a path (from state a to b) as an ordered sequence (or list) of states that starts with state a and ends with state b.
For example, each way of hammering a nail represents a unique path through state space — illustrated in the following image.
Formally, we can denote a path between two states, a and b, as (a, σ₀, σ₁, …, σₙ, b), where the state starts at a then changes sequentially to σ₀, σ₁, and so on — until it reaches state b.
Every variation on how an end state is reached — down to the microscopic level — changes the path. This includes seemingly trivial things, such as precisely how you move your body and what you think about as you hammer.
Actualizable paths — those that are actually possible.
All paths are not equal: some are possible — others are not.
The simplest path is (a, b), where we go directly from state a to state b without intermediate steps — e.g., teleporting from Earth to Mars. Nothing prevents us from defining such a path — even though it might be impossible for the actualized state to “travel through.”
To limit ourselves to what’s possible, we must bring in the laws of state change.
Denote the set of actualizable paths between states A and B as Λ(A, B, L) — which are the paths that the actualized state can “follow” from states A to B given rules L.¹ If, for example, A={a} represents a set of states where we are on Earth and B={b} where we are on Mars; then we can conclude that the teleport-path (a, b) is not in the set of actualizable states Λ(A, B, L) if L represents the known laws of physics.² This is because, in our universe, our position needs to change continuously and not instantaneously — which means there must be some intermediate states between a and b.
In other words, how we can achieve something depends on where we are A, where we are going B, and the rules and laws L that determine the way it can be achieved. Depending on the system, there is often a large — perhaps infinite — number of ways to achieve something — represented by the number of paths in Λ(A, B, L).
By putting criteria on the set of possible paths, we can extract useful information — for example, by finding the paths that take the least time or energy.
Or why not the paths where we have the most fun?
The set of what is possible (i.e., the set of actualizable states).
Given a starting point, it is reasonable to ask: where can we go from here?
The actualizable paths provide the answer, but the problem is that they don’t only include the possible destinations but also the details of how to get there. We might want to know, “Can we get to Mars?” without considering exactly how.
To fix this, we make the following definition.
Define the set of actualizable states as the set of all states in the set of actualizable paths from a starting set of states to the set of all states.
This is the set of all possible end states — i.e. destinations.
In our universe, for example, it is not possible to go backward in time — i.e., a state σ(t, …) cannot change into a state σ’ (t’, …) where time t’ <t. It is also impossible for objects with mass to travel faster than the speed of light — which is why we can’t teleport to Mars.
Formally, the actualizable (or “reachable”) states R are given by a starting set of states A and then checking which states can be “reached” given laws L, such that R(A, L) = { σ | σ ∈ Λ(A, Ω, L) }, where Ω is the set of all states.
If we imagine the set of states as a plane (even though it often consists of more than two parameters), the actualizable states can be seen as a “tunnel” that divides state space into states that can — and cannot — be “reached” from a given starting point.
Like the actualizable paths, what is achievable R(A, L) depends on where we are A and the laws of state change L — but not where we are going B (since that is now an open question).
Paths and actualizable states enable a way of representing immense complexity without losing utility. They allow us to, for example, formally include the state of every particle in the universe and still figure out if we can reach Mars in a week or with a certain amount of fuel or payload.
Footnotes
¹Notice that we no longer go limit ourselves to single states a and b — but between sets of states A and B. This is because we generally want to ask how we can go from different situations, e.g. the set of states where we are on Earth, and not specific states — e.g. where we are on Earth but where Obama is president.
²However, if we re-made the analysis based on some alternative and cooler laws of physics L’, then teleportation could be possible — such that (a, b) ∈ Λ(A, B, L’).
Disclaimer
These ideas are a work in progress and have yet to be peer-reviewed. Please feel free to engage with them and let me know your thoughts — whether you find ways to improve them or reasons to discard them.
