Laplacean Machine

Out of the fecund mind of my friend Bryce Laliberte has sprung an idea that I find quite fascinating.  The idea is of machines capable of calculating everything that occurs within a universe at any given time, which he calls Laplacean machines, or L-machines.  I think this is a very fertile idea for speculation, so in this post I will focus on creating a typology for different kinds of L-machines and on exploring some results concerning possible L-machines and the universes they model.

Bryce works from the premise of an L-machine as an entity distinct from the universe that it models.  While this is certainly a possibility, I want to classify it as a special case.  We will call an L-machine that models a universe different from the one in which it resides an exterior L-machine and an L-machine that models the universe in which it resides as an interior L-machine.

We can also think of L-machines in terms of constant versus occasional calculation.  That is, we can have a machine which could calculate the state of a universe at any time, or we can have a machine that is constantly calculating.  To help make this distinction clearer, consider an L-machine which can calculate what happens in a universe at any time but requires a specified time at which to calculate the result and contrast this with a machine which actually does calculate what occurs at every time.  We will call the first type of machine, which requires a specified time, a specific L-machine and the second type which calculates for all times a general L-machine.

A distinction that applies uniquely to general L-machines concerns the relationship between when it finishes calculations to model a universe and when the state it “predicts” actually occurs.  The calculations can occur before, after, or simultaneously with the events in the universe.  We will call these prior, posterior, and simultaneous L-machines respectively.

If we wish, we might also oppose L-machines which have inputs and actually engage in calculation to those which have not received inputs and so are not calculating.  I don’t see much profit in exploring this distinction, so we will simply call L-machines awaiting instructions inert and not bother specifying L-machines that actually are calculating.

Finally, we can classify L-machines with respect to the number of calculations they perform at any given time.  We will call L-machines that perform finitely many calculations at any given time finite and L-machines that perform a countably infinite number of calculations infinite.  I have difficulty imagining how a machine could perform uncountably many calculations, so while I won’t exclude the possibility altogether, we will simply lump such machines under the blanket heading of uncountable L-machines.

I believe we now have a typology sufficient to allow discussion of some interesting results regarding possible L-machines and their respective universes.  Unfortunately, we don’t have a fully worked out system for this analysis, so these proofs will be a little sloppy.  I’m fairly sure they’re correct as far as they go.

Proposition 1: In discrete time, a deterministic universe is a prior L-machine.

Proof: In a deterministic universe, all events at time t are determined by events at times prior to t, so the events in a deterministic universe are the same as the calculations of an L-machine.  In a universe with discrete time, the supremum of possible times prior to t is t-1.  Thus, the calculations modeling a state of the universe are finished at a definite time before that state comes about, making that universe a prior L-machine.

Proposition 2: In continuous time, a deterministic universe is a prior or simultaneous L-machine.

Proof: As proof for Proposition 1, except that the supremum for possible times prior to t is t itself, allowing for the possibility of simultaneous L-machines.

I suspect a fair bit of work can be done in trying to determine under what conditions a deterministic universe can be a simultaneous L-machine.  For example,

Theorem 1: Let U be a deterministic universe with continuous time consisting of point-objects in an algebraic vector space with pointwise-continuous position functions with respect to time that are not constant on any small time interval.  U is a simultaneous L-machine.

Proof: By Proposition 2, U is either prior or simultaneous.  For an object with position function s(t) and a specified time t_o, the limit as t goes to t_o is s(t_o).  If s(t) is constant on a neighborhood of t_o, then s(t_o) could be determined at some time prior to t_o.  Since s(t) is constant on no neighborhood of t_o, the supremum of the time by which calculations are completed to determine s(t_o) is t_o.  Thus, U is simultaneous.

Another way in which this discussion gets interesting is when we try to determine characteristics of universes by studying L-machines corresponding to them.

Axiom of Irreducible Simplicity: The simplest, complete representation of a system is itself.

This should seem fairly intuitive.  If we have a system we are trying to model, we need some way to produce all the data of the system in addition to the data itself.  Thus, the model must have more going on in it than the system itself does.  This is equivalent to the statement “Any single event requires at least one calculation to completely model it.”  If you have a problem with accepting one of these principles as an axiom, then take the other and prove it as a theorem.

Proposition 3: For any deterministic universe, there are infinitely many possible L-machines that model it.

Proof: Let U be a deterministic universe.  U is an L-machine by Propositions 1 and 2.  Let O₁ be an object not in U with two possible behaviors which alternate according to a fixed pattern.  Then the universe     U U O₁ is deterministic and so an L-machine modeling U.  This procedure can be repeated for U U O₁, producing a new L-machine which also models U, and so on indefinitely.

We can also try to think of the smallest possible L-machine of a given type for a given universe.  This leads to an interesting result.  (Note: Hereafter, “universe” denotes “deterministic universe.”)

Lemma 1: Let U be a finite universe.  The smallest L-machine of any type for U is finite.

Proof: Obvious from Axiom of Irreducible Simplicity.

Definition: A proper L-machine for a universe is an interior L-machine that is different from the universe itself.

Lemma 2: Let U be a finite universe with a lower bound on time.  There is no proper, general, prior L-machine for U at all times.

Proof: If time in a universe has a lower bound, then no proper, general, prior L-machine for that universe could calculate what it would do on a neighborhood of time zero.

Lemma 3: Let U be a finite universe with an upper bound on time.  There is no proper, general, posterior L-machine for U at all times.

Proof: As for Lemma 2, mutatis mutandis.

Lemma 4: Let U be a finite universe with a lower bound on time.  There is no proper, general, simultaneous L-machine for U.

Proof: If time is discrete for U, then obvious.  If time is continuous and L is a general, simultaneous L-machine for U, then at time 0 L must perform as many calculations as there are events in U, which means that all of the events in U are calculations of L.  Thus for any subsequent time, all calculations of L must be performed at that time and so be all the events in U at that time.

Lemma 5: Let U be a finite universe with an upper bound on time.  There is no proper, specific L-machine for U at all times.

Proof: Any proper, specific L-machine in U has a minimum number of calculations it must perform to calculate the events at any given time.  Let n be that minimum time for calculation for a specified time and t be the supremum of time in U.  For times greater than t-n, such a machine will not be able to calculate the events at the time in question.

Theorem 2: A finite universe with bounded time is its smallest L-machine for all times.

Proof: Follows from Lemmas 1-5.

I’m sure plenty more results could be proved, and I may be proving some more in the future.  For now, however, I want to close with a few general comments.  There are several fields of inquiry that I see opening up from L-machines based on what I’ve already worked out:  (1) when a universe is a simultaneous L-machine; (2) what we can learn about universes by studying their L-machines; and (3) properties of proper L-machines.  Other areas that I haven’t dealt with include infinite universes and L-machines, differences between general and specific L-machines, and probabilistic universes.  With a probabilistic universe, the universe’s L-machine merely calculates the probability of events occurring instead of whether they actually occur or not.  A small point I can see even now is that if a probabilistic universe has a proper, general L-machine, then it must be infinite.  All of these lines of inquiry could be very fertile, and I eagerly await any new developments deriving from them.