Algorithms and Nature Language

A set of rules to follow form an algorithm. When we ask for directions to some destination from a given point, a set of directions to follow are given. This set of directions forms an example of an algorithm. Since Algorithms are usually composed of a discrete set of rules, continuous set of rules such as flows along vector fields tend to be ignored as algorithms. Nature itself displays both discrete set of rules (number of leaves on a branch of a plant for example) and continuous set of rules (heat flow for example). I shall try to show that Disccrete and Continuous can be identified together as forms of each other. Discrete objects are connected by a continuous flow. Looking at it the other way round, Continous flows of different kinds are identified by discrete objects.

Nature Language is Symmetric Algorithm

Identfiying the set of rules that Nature follows will help us build algorithms that are “Optimal”. In a previous post (Calculus Group ), I described how Calculus is all about symmetry. In this post, I will show through Nature and Algorithms how symmetry is synonymous to “Optimal”. This is seen by looking at “Optimal points” as dualities which compose together to a symmetry, which when reversed, reveals each duality as a symmetry in itself. Every symmetry is an mass (measure) balance (see Symmetry as Mass Balance) or an Equilibrium, so Nature Language is full of symmetry.

  • We see that electricity and magnetism behave in very particular ways. We see that arrows along electric fields lines show how to go from one charge to another in a particular symmetric way. Similarly, Fluids flow from a higher to lower potential in a symmetric way. Gravity takes objects from higher potential to lower potential along a symmetric path. These flows in Nature Language are to be seen as symmetries based on which we can build Symmetric Algorithms.

A Symmetry, Asymmetry

When anything and everything is allowed to happen, we choose to observe something particular by capturing it in a fixed Language (structure, that is, boundaries), which forms the constant. This fixed constant Language becomes the Identity (constant background) about which everything else can emerge. We included everything that is fixed to be captured in Identity, so whatever emerges cannot independently be another fixed Identity in itself (In mathematical language, this is seen as uniqueness of identity in a group). This argument captures the asymmetry needed in observation: we must assign a fixed identity to be able to separate (or bound or give a name to) what is not fixed. This is like assigning \(-\) and \(+\) in empty space to be able to talk about something. We need to have asymmetry to have change (time). This is why asymmetry or dual forms (\(-\) and \(+\)) alternating with each other form the basis of all structures in Time. The boundary between the two duals does not favour either asymmetry and hence forms the Identity (Equilibrium). So, we see that any observable is a particular form which has its corresponding dual (asymmetry). Once we start from observing one particular asymmetry, the next thing we can see is its dual asymmetry. On identifying the two asymmetries to be the same forms of a unique thing, we stop oscillating between the two asymmetries and identify them together to find our attention at the boundary which is the symmetry (Identity). Identifying them together requires spending “same measure” in either asymmetry. This was a simple case of two asymmetries. If there are multiple asymmetries, all of them compose (come together) to give identity. The emerging forms (such as \(-\) and \(+\)) can be seen as relative Identities in themselves about which other forms can emerge. The asymmetries on all sides of a fixed asymmetry naturally compose with each other to give identity which is the given fixed asymmetry. The inherent asymmetry in a Language (structure) is expressed in detail in Empirical Observations on Language

  • Integrals of a Language are the symmetry of that Language. They identify together asymmetries of a Language. The flow of this identification is the Integral.

    • Measure of a Space is a symmetry of the space

    • Connection is a symmetry

    • Flow is a symmetry

    • Communication is a symmetry

    • Meeting is a symmetry

    • Mapping is a symmetry

    • Group is a symmetry

    • Distance is a symmetry

    • Space is a symmetry

    • Time is a symmetry

    • Feynman Path Integral is a symmetry

    • Lebesgue Measure is a symmetry

    • Riemann Integral is a symmetry

    • Summation is a symmetry

    • Diagonal is a symmetry

Differential Equations in Physics identify together asymmetries. Hence the soultions are Integral flows (symmetries).

Distance Measure is Symmetry in Algorithms

Distance is measure. Every action can be assigned a measure. Assign a measure for a given algorithm. Once a measure (distance) is assigned to a family of Algorithms, the Optimal Algorithm is one which has Optimal Measure (distance). In other words, an Optimal Algorithm is a Symmetry among the given Algorithms”. This is inspired from the corresponding Thermodynamics condition in Nature which identifies the actions of least measure as the symmetry or the optimal paths to proceed along (this was detailed in the post Equilibrium Thermodynamics). For an algorithm involving multiple steps, a measure assigned to each step must be composed with appropriate Measures (weights) to give the total measure of an Algorithm.

  • Triangle Inequality identifies straight line joining two points as a symmetry of the Euclidean Plane. Left and right are asymmetries in the Euclidean Plane. Starting from a given point, to reach a fixed point, join them by a line. Any point on either side of the line joining the two points has an “opposite” point which is the reflection of the point about the line. Hence, the line joining the two points is the symmetry (mirror). The opposite points cancel out due to opposite orientations (measure). This makes the line an optimal path. Let, \(a, b, c\) be the measures (distances) of the sides of a triangle opposite to vertices A, B and C respectively. An algorithm is needed for going from point A to point B then the triangle inequality identifies that increasing asymmetry (increasing degrees of freedom or increasing measure) increases the total measure of the action to go from point A to point B in Euclidean Plane:
\[c \,\, \le \,\,a\,\,+\,\,b\]

In other words, the optimal algorithm to go from a given point A to another given point B in the Euclidean Plane is to go along the straight line joining them. That is, the straight line joining two points is the symmetry connecting the two dual points.

  • Dijkstra’s algorithm is an Optimal Algorithm because it has the minimum measure (distance). At each step, it chooses the action with least measure and then discretely adds (composes) them up to make the total measure of the algorithm minimum.

  • Brachistochrone Problem gives Cycloid as the optimal Path because that is the path with least measure. This is seen by minimizing the Hamiltonian measure and identifying the Cycloidal path as the minimizer or looking at the correspinding analogue of the problem in Electricity and Magnetism where in a Uniform Magnetic field orthogonal to plane in which electric field exists, gives a cycloid motion as the optimal path for the charge.

Gravity identifies Geodesic Flow as Symmetry

In the case of Space-time asymmetry, the Measure of a path is given by the asymmetric metric \((1+,-1,-1,-1)\) hence instead of Euclidean optimal path, the correspinding Geodesics give the Optimal Flow.

Light Flow is Symmetric

Huygen’s Principle describing Flow of Light through spehrical Wavefronts is an optimal algorithm for travel of Light in Space since Euclidean metric has Spheres as the Optimal measures. Hence Huygen’s Principle is an Optimal Algorithm. This is also captured by Fermat’s Principle that Light flows along paths of extremal time. These both capture half the story by treating metrics of space and time separately. Einstein’s unification of Space-time allows bringing both together to give the Light cone as the optimal path. Hence flow of Light is an Optimal Algorithm.

Heat Flow Is Symmetric Flow

Heat flow from points of high temperature distribution to points of low temperature distributions occurs along symmetric paths. This flow is modeled by the Heat Equation:

\[\partial_t u = \partial_{space, space} u\]

This shows that when all heat flow has happened, at equilibrium with respect to time, the term \(u_t=0\) as there is no more time change. Resulting in \(\Delta u =0\) which is the equilibrium metric. Hence Heat flow is seen as an Optimal Algorithm.

Hamilton’s Ricci Flow is Symmetric Flow

Quantum Mechanics is Symmetry

The Young’s Double Slit Experiment provides a fascinating display of symmetry. In mathematics, there is a very close model of the same: The symmetry in Riemann Zeta function can be seen in the Young’s Double slit experiment. There’s two slits (\(1/2\)) composed with waves (\(i \,\,t\)) and the Screen composes alternate bright and dark bands (negative even integers as roots).