Physics Derivation Graph: 2017

Monday, August 28, 2017

validating equivalent and transformed expressions

To check whether a step was implemented correctly, the question is, "given the input expressions (and feeds), do the output expressions conform to the transform prescribed by the inference rule?"

For example, suppose we start with "A=B" and operate on this with the inference rule "add X to both sides," where X=2. If the output is "A+2 = B+2," was the step implemented correctly?

The inference rule can be described by an abstract syntax tree (AST):

= =
/ \ --> / \
LHS RHS + +
/ \ / \
LHS x RHS x

https://blog.plover.com/math/24-puzzle-2.html
https://news.ycombinator.com/item?id=15075110

Saturday, July 8, 2017

anatomy of the "file per expression" data format in the Physics Derivation Graph

Each derivation gets a folder. The folder typically contains four CSV files:

expression_identifiers.csv: two columns of integers (7 digits, 10 digits) where each row looks like "1432042,1029039904"
feeds.csv: a single column of integers (7 digits) where each row looks like "2342425"
inference_rule_identifiers.csv: two columns, the first is an integer (7 digits). An example row looks like 1294844,declareInitialExpression
derivation_edge_list.csv: two columns of integers (7 digits each) where each row looks like "3934948,3499522"

Elsewhere in this documentation I refer to the 7 digit integers as "temporary IDs" and the digit integers as "permanent IDs".

The graph visualization is built from the content of derivation_edge_list.csv, but additional decorations are need to renders a meaningful picture. That's where the other three CSVs come into play -- they indicate what the relevant decoration is (either a feed, expression, or inference rule).

Minimum viable product for the Physics Derivation Graph

I've been looking for ways to pare down the tasks that I face in building the Physics Derivation Graph. Defining a minimum viable product for delivery would help prioritize what I work on.

Although the d3js-based interactive graphs are neat, they don't provide much value. The static PNGs of derivations are not as sexy, but they are easier to navigate. The PNGs are the minimum viable product.

As far as storage formating, I think the "file per expression" approach is still the best option, compared to CSV, XML, and MathML.

Thus, the MVP needs to be able to generate PNGs of a graph from derivations stored in the "file per expression" method.

Generating a file per expression is a tedious task that both limits scalability and could introduce errors if done manually. The command-line based interactive user prompt is a useful tool that addresses both concerns.

The interactive user prompt is a potential time sink. Minimum functionality is user should be able to enter a new derivation and write results to file.
Do not include ability to edit existing derivations, or to link existing derivations by deconflicting expressions with different identifiers.

Task 1: generate graph PNG from file per expressions
Task 2: generate file per expression from interactive prompt
Task 3: convert from interactive command-line prompt to web-based prompt

interactive derivation input via the command line

Friday, July 7, 2017

MathML is probably better than LaTeX, but I'm lazy

There's a good overview of MathML provided by Wolfram. MathML interacts with Mathematica, both for import and export.

"MathML comes in two types: Presentation MathML, which describes what an equation looks like, and Content MathML, which describes what an equation means. By default, MathJax works with Presentation MathML and offers an extension for Content MathML."

There are online tools that enable conversion of LaTeX (mathematics) to Presentation MathML.

MathML is probably the better format to store data in the Physics Derivation Graph compared to Latex. However, Latex is more convenient for me as a data entry method.

proprietary code and data -- interacting with commercial projects

There are two aspects to a project like the Physics Derivation Graph that could be proprietary: the data, and the software which manipulates the data.

One motivation for protecting these aspects of a project are to make money. I'm not seeking money.
Another motivation for protection is to ensure citation when the project is used. I apply the Creative Commons Attribution 4.0 International License to the Physics Derivation Graph project.

I want to enable other people, including commercial efforts, to benefit from use of both the data and software I create. I also share the syntax of the data format so that future contributions from others can adhere to the data I already have. A published syntax standard also enables writing parsers for translation of the data format.

Wednesday, July 5, 2017

finding cranks using the Physics Derivation Graph

One of the applications of the Physics Derivation Graph would be to detect cranks. Existing methods include crackpot scoring, proposed by John Baez. Gerard 't Hooft provides an outline and description of bad theorists.

There's plenty of content which smells of crackpots, ie http://atomicprecision.com/OmniaOpera/omnia-opera-655.pdf

The Physics Derivation Graph enables a more rigorous approach for identifying both mathematical steps and assumptions.

Monday, July 3, 2017

finding major edges of the Physics Derivation Graph

I've decided to focus on building content for the Physics Derivation Graph. Since I don't think it's reasonable to enter all the mathematical physics content manually on my own, in this post I'll define some bounds.

Step 1: identify major fields in Physics

Electromagnetism
relativity

astrophysics

quantum mechanics
classical mechanics

thermodynamics

Step 2: identify top derivations associated with each area

Electromagnetism: Maxwell's equations
Relativity: Lorentz (time dilation, length contraction)

astrophysics:

Quantum mechanics: Schrodinger, Uncertainty
Classical mechanics: F=ma, conservation of energy and momentum

thermodynamics: entropy

Step 3:

There are expressions, domains, and notations that need to be included to demonstrate the comprehensive nature of the project and the capability of the framework.

Expressions	Domains	Notation
Schrodinger's equation	Quantum	Dirac/bra-ket

Maxwell's equations	Electromagnetics	differentiation, vectors
contravariant/covariant	Electrodynamics	Einstein notation
Navier Stokes	Fluid Mechanics	differentiation, vectors
mass-energy relation
	Special Relativity
	Thermodynamics
Uncertainty relation	Quantum Mechanics
energy conservation	classical mechanics
	string theory
wave equation
	statistical mechanics
	aerodynamics
simple harmonic oscillator

Other comprehensive lists of "things to know" in Physics

https://www.staff.science.uu.nl/~gadda001/goodtheorist/

https://news.ycombinator.com/item?id=10714048

http://www.physics.uoguelph.ca/poisson/research/notes.html

https://www.physics.uoguelph.ca/poisson/research/mech.pdf

http://theoreticalminimum.com/courses

https://github.com/allofphysicsgraph/proofofconcept/wiki/All-Branches-of-Physics

The Physics Derivation Graph has a few pertinent scales:

topics within physics, ie thermodynamics, quantum mechanics, classical mechanics

major (often named) expressions within a domain, ie "E=mc^2", Schrodinger's equation, Maxwell's equations

derivations linking the major expressions

abstract syntax trees of an expression

In this post, I'll focus on finding examples of derivations linking the major expressions

Useful places to search

Google Scholar
Arxiv
journals for Physics teachers

Useful search string: derivation of equation

Unsorted results

https://en.wikipedia.org/wiki/ADM_formalism
https://en.wikipedia.org/wiki/Abraham–Lorentz_force
https://en.wikipedia.org/wiki/Acoustic_wave_equation
https://en.wikipedia.org/wiki/Action-angle_coordinates
https://en.wikipedia.org/wiki/Adiabatic_process
https://en.wikipedia.org/wiki/Angular_momentum
https://en.wikipedia.org/wiki/Angular_momentum_operator
https://en.wikipedia.org/wiki/Appell's_equation_of_motion
https://en.wikipedia.org/wiki/Area_of_a_circle
https://en.wikipedia.org/wiki/Available_energy_(particle_collision)
https://en.wikipedia.org/wiki/Avrami_equation
https://en.wikipedia.org/wiki/Azimuthal_quantum_number
https://en.wikipedia.org/wiki/BET_theory
https://en.wikipedia.org/wiki/Backpropagation
https://en.wikipedia.org/wiki/Ballistic_pendulum
https://en.wikipedia.org/wiki/Beer–Lambert_law
https://en.wikipedia.org/wiki/Bell's_theorem
https://en.wikipedia.org/wiki/Bethe–Salpeter_equation
https://en.wikipedia.org/wiki/Bivector
https://en.wikipedia.org/wiki/Black–Scholes_model
https://en.wikipedia.org/wiki/Bloch_oscillations
https://en.wikipedia.org/wiki/Boltzmann_relation
https://en.wikipedia.org/wiki/Bose–Einstein_statistics
https://en.wikipedia.org/wiki/Braking_distance
https://en.wikipedia.org/wiki/CHSH_inequality
https://en.wikipedia.org/wiki/Capillary_wave
https://en.wikipedia.org/wiki/Catenary
https://en.wikipedia.org/wiki/Causal_dynamical_triangulation
https://en.wikipedia.org/wiki/Causality
https://en.wikipedia.org/wiki/Centimetre–gram–second_system_of_units
https://en.wikipedia.org/wiki/Centripetal_force
https://en.wikipedia.org/wiki/Charge_conservation
https://en.wikipedia.org/wiki/Classical_electron_radius
https://en.wikipedia.org/wiki/Clausius–Clapeyron_relation
https://en.wikipedia.org/wiki/Coefficient_of_restitution
https://en.wikipedia.org/wiki/Collective_noun
https://en.wikipedia.org/wiki/Collision_theory
https://en.wikipedia.org/wiki/Compton_scattering
https://en.wikipedia.org/wiki/Conductance_quantum
https://en.wikipedia.org/wiki/Confidence_interval
https://en.wikipedia.org/wiki/Convolution
https://en.wikipedia.org/wiki/Darcy's_law
https://en.wikipedia.org/wiki/Debye_model
https://en.wikipedia.org/wiki/Derivative
https://en.wikipedia.org/wiki/Diffusion_equation
https://en.wikipedia.org/wiki/Doppler_broadening
https://en.wikipedia.org/wiki/Drag_equation
https://en.wikipedia.org/wiki/Dynamic_mechanical_analysis
https://en.wikipedia.org/wiki/Eddington_luminosity
https://en.wikipedia.org/wiki/Effective_medium_approximations
https://en.wikipedia.org/wiki/Elastic_collision
https://en.wikipedia.org/wiki/Electrodynamic_tether
https://en.wikipedia.org/wiki/Equilibrium_constant
https://en.wikipedia.org/wiki/Euler–Bernoulli_beam_theory
https://en.wikipedia.org/wiki/Euler–Maclaurin_formula
https://en.wikipedia.org/wiki/Ewald_summation
https://en.wikipedia.org/wiki/Exponential_decay
https://en.wikipedia.org/wiki/Fermat's_principle
https://en.wikipedia.org/wiki/Feynman_parametrization
https://en.wikipedia.org/wiki/Fick's_laws_of_diffusion
https://en.wikipedia.org/wiki/Fictitious_force
https://en.wikipedia.org/wiki/Fisher_information
https://en.wikipedia.org/wiki/Flory–Huggins_solution_theory
https://en.wikipedia.org/wiki/Fluctuation-dissipation_theorem
https://en.wikipedia.org/wiki/Four-gradient
https://en.wikipedia.org/wiki/Four-momentum
https://en.wikipedia.org/wiki/Fourier_optics
https://en.wikipedia.org/wiki/Freund–Rubin_compactification
https://en.wikipedia.org/wiki/Gaussian_gravitational_constant
https://en.wikipedia.org/wiki/Geiger–Nuttall_law
https://en.wikipedia.org/wiki/Generalizations_of_the_derivative
https://en.wikipedia.org/wiki/Gibbs_free_energy
https://en.wikipedia.org/wiki/Gibbs–Duhem_equation
https://en.wikipedia.org/wiki/Giovanni_Semerano
https://en.wikipedia.org/wiki/Golden_ratio
https://en.wikipedia.org/wiki/Group_velocity
https://en.wikipedia.org/wiki/Group_velocity_dispersion
https://en.wikipedia.org/wiki/Gullstrand–Painlevé_coordinates
https://en.wikipedia.org/wiki/Gyroradius
https://en.wikipedia.org/wiki/Hamilton–Jacobi_equation
https://en.wikipedia.org/wiki/Harmonic_coordinate_condition
https://en.wikipedia.org/wiki/Haynes–Shockley_experiment
https://en.wikipedia.org/wiki/Heat_equation
https://en.wikipedia.org/wiki/Helmholtz_coil
https://en.wikipedia.org/wiki/Helmholtz_decomposition
https://en.wikipedia.org/wiki/Hyperfocal_distance
https://en.wikipedia.org/wiki/Interference_(wave_propagation)
https://en.wikipedia.org/wiki/Ion_acoustic_wave
https://en.wikipedia.org/wiki/Isotropic_radiator
https://en.wikipedia.org/wiki/Kalman_filter
https://en.wikipedia.org/wiki/Killing_vector_field
https://en.wikipedia.org/wiki/Kinetic_energy
https://en.wikipedia.org/wiki/Klein–Gordon_equation
https://en.wikipedia.org/wiki/Kramers–Kronig_relations
https://en.wikipedia.org/wiki/Kramers–Wannier_duality
https://en.wikipedia.org/wiki/Kutta–Joukowski_theorem
https://en.wikipedia.org/wiki/Lamb_shift
https://en.wikipedia.org/wiki/Landau_quantization
https://en.wikipedia.org/wiki/Langmuir_(unit)
https://en.wikipedia.org/wiki/Laplace_expansion_(potential)
https://en.wikipedia.org/wiki/Large_eddy_simulation
https://en.wikipedia.org/wiki/Larmor_formula
https://en.wikipedia.org/wiki/Leibniz_integral_rule
https://en.wikipedia.org/wiki/Length_contraction
https://en.wikipedia.org/wiki/Lie_algebra
https://en.wikipedia.org/wiki/Liénard–Wiechert_potential
https://en.wikipedia.org/wiki/Linear_least_squares_(mathematics)
https://en.wikipedia.org/wiki/Linearized_gravity
https://en.wikipedia.org/wiki/Lippmann–Schwinger_equation
https://en.wikipedia.org/wiki/Magic_number_(physics)
https://en.wikipedia.org/wiki/Magnetic_quantum_number
https://en.wikipedia.org/wiki/Mass–energy_equivalence
https://en.wikipedia.org/wiki/Maxwell–Boltzmann_statistics
https://en.wikipedia.org/wiki/Mean_effective_pressure
https://en.wikipedia.org/wiki/Mean_free_path
https://en.wikipedia.org/wiki/Metalloid
https://en.wikipedia.org/wiki/Mooney–Rivlin_solid
https://en.wikipedia.org/wiki/Newton's_theorem_of_revolving_orbits
https://en.wikipedia.org/wiki/Noether's_theorem
https://en.wikipedia.org/wiki/Non-associative_algebra
https://en.wikipedia.org/wiki/Optical_flat
https://en.wikipedia.org/wiki/Optical_theorem
https://en.wikipedia.org/wiki/Orbital_state_vectors
https://en.wikipedia.org/wiki/Parabola
https://en.wikipedia.org/wiki/Parallax
https://en.wikipedia.org/wiki/Parametric_oscillator
https://en.wikipedia.org/wiki/Particle_in_a_spherically_symmetric_potential
https://en.wikipedia.org/wiki/Paschen's_law
https://en.wikipedia.org/wiki/Path_integral_formulation
https://en.wikipedia.org/wiki/Perpendicular_axis_theorem
https://en.wikipedia.org/wiki/Pilot_wave
https://en.wikipedia.org/wiki/Planck's_law
https://en.wikipedia.org/wiki/Planck_particle
https://en.wikipedia.org/wiki/Plasmon
https://en.wikipedia.org/wiki/Poisson–Boltzmann_equation
https://en.wikipedia.org/wiki/Polarization_identity
https://en.wikipedia.org/wiki/Ponderomotive_energy
https://en.wikipedia.org/wiki/Ponderomotive_force
https://en.wikipedia.org/wiki/Potential_energy
https://en.wikipedia.org/wiki/Potential_temperature
https://en.wikipedia.org/wiki/Poynting's_theorem
https://en.wikipedia.org/wiki/Price_equation
https://en.wikipedia.org/wiki/Principal_quantum_number
https://en.wikipedia.org/wiki/Principle_of_maximum_entropy
https://en.wikipedia.org/wiki/Quadratic_equation
https://en.wikipedia.org/wiki/Quartz_crystal_microbalance
https://en.wikipedia.org/wiki/RRKM_theory
https://en.wikipedia.org/wiki/Radian
https://en.wikipedia.org/wiki/Radius_of_gyration
https://en.wikipedia.org/wiki/Rainbow
https://en.wikipedia.org/wiki/Read_(surname)
https://en.wikipedia.org/wiki/Redshift
https://en.wikipedia.org/wiki/Reduced_mass
https://en.wikipedia.org/wiki/Relational_quantum_mechanics
https://en.wikipedia.org/wiki/Richards_equation
https://en.wikipedia.org/wiki/Roche_limit
https://en.wikipedia.org/wiki/Rodrigues'_rotation_formula
https://en.wikipedia.org/wiki/Rotating_wave_approximation
https://en.wikipedia.org/wiki/Rutherford_scattering
https://en.wikipedia.org/wiki/Saha_ionization_equation
https://en.wikipedia.org/wiki/Scale_relativity
https://en.wikipedia.org/wiki/Schrödinger_equation
https://en.wikipedia.org/wiki/Schwinger–Dyson_equation
https://en.wikipedia.org/wiki/Second_law_of_thermodynamics
https://en.wikipedia.org/wiki/Shallow_water_equations
https://en.wikipedia.org/wiki/Shockley_diode_equation
https://en.wikipedia.org/wiki/Spin_connection
https://en.wikipedia.org/wiki/Spin_quantum_number
https://en.wikipedia.org/wiki/Stefan–Boltzmann_law
https://en.wikipedia.org/wiki/Stretch_rule
https://en.wikipedia.org/wiki/String_theory
https://en.wikipedia.org/wiki/Sverdrup_balance
https://en.wikipedia.org/wiki/Taylor's_theorem
https://en.wikipedia.org/wiki/Terminal_velocity
https://en.wikipedia.org/wiki/Tetum_language
https://en.wikipedia.org/wiki/Theorem
https://en.wikipedia.org/wiki/Torque

https://en.wikipedia.org/wiki/Torricelli%27s_equation#Derivation

https://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm#Derivation
tag: math

https://en.wikipedia.org/wiki/Tropical_cyclone#Maximum_potential_intensity
tag: weather

https://en.wikipedia.org/wiki/Twomey_effect#Derivation

https://en.wikipedia.org/wiki/Van_der_Waals_equation#Derivation

https://en.wikipedia.org/wiki/Variable-range_hopping#Derivation

https://en.wikipedia.org/wiki/Visibility#Derivation

https://en.wikipedia.org/wiki/Wallis_product#Derivation
tag: math

https://en.wikipedia.org/wiki/Washburn%27s_equation#Derivation

https://en.wikipedia.org/wiki/Wheeler%E2%80%93DeWitt_equation#Derivation_from_path_integral

https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_equation#Derivation

https://en.wikipedia.org/wiki/Optical_theorem
https://farside.ph.utexas.edu/teaching/qm/lectures/node87.html

Seismic Wave Equation
http://www.ees.nmt.edu/outside/courses/GEOP523/Docs/waveeq.pdf
tag: PDF

Noether's theorem
https://en.wikipedia.org/wiki/Noether%27s_theorem#Derivations
https://www.youtube.com/watch?v=Rqfj7n5aSwY

https://en.wikipedia.org/wiki/Kalman_filter#Derivations

E=mc2 and F=ma
Ives, Herbert E. (1952), "Derivation of the mass–energy relation", Journal of the Optical Society of America, 42 (8): 540–543, doi:10.1364/JOSA.42.000540 | tag: paywall
On the page http://www.emc2-explained.info/Emc2/Deriving.htm
there's a step which involves https://en.wikipedia.org/wiki/Integration_by_substitution
--> I don't understand the third line of equations. The infinitesimal switches from ds to d(mv), and the integration limit changes.

https://en.wikipedia.org/wiki/Maxwell_relations#Derivation

Derivation of Electromagnetic Waves from Maxwell’s Equations
tag: PDF
Maxwell's equations as presented in Feynman's lectures
Feynman's derivation of Maxwell equations and extra dimensions: https://arxiv.org/abs/hep-ph/0106235

derivation of the time‐dependent convective‐diffusion equation for surfactant transport along a deforming interface
tag: paywall

Derivation of the Continuous-Time Random-Walk Equation
tag: paywall

derivation of the chemical master equation
tag: paywall

Derivation of Kramer's equation, friction coefficient, and macroscopic laws for physisorption
tag: paywall

QUANTUM-MECHANICAL DERIVATION OF AN EQUATION OF STATE OF IRON
tag: PDF

derivation of an equation for predicting minimum spouting velocity - DOI: 10.1002/aic.690040423
tag: paywall

Derivation of the Biot-Savart Law from Ampere's Law Using the Displacement Current
tag: paywall
https://physics.stackexchange.com/questions/67445/is-biot-savart-law-obtained-empirically-or-can-it-be-derived
http://www.wikihow.com/Derive-the-Biot%E2%80%90Savart-Law

Classical mechanics

Entropy

http://micro.stanford.edu/~caiwei/me334/Chap7_Entropy_v04.pdf

tag: PDF

https://en.wikipedia.org/wiki/Fundamental_thermodynamic_relation

Derivation of the Ideal Gas Law
DOI: 10.1021/ed084p1832
http://homepage.smc.edu/gallogly_ethan/files/ideal%20gas%20law%20derivation.pdf
tag: PDF
http://quantumfreak.com/derivation-of-pvnrt-the-equation-of-ideal-gas/
https://en.wikipedia.org/wiki/Ideal_gas_law#Theoretical

Quantum Mechanics

https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation#Derivation
Deriving time dependent Schrödinger equation from Wave-Mechanics
tag: PDF

Derivation of Schrodinger's equation
tag: arxiv

Derivation of Schrödinger's equation from stochastic electrodynamics -DOI: 10.1007/BF00670387
tag: paywall

Derivation of the Schrödinger Equation from Newtonian Mechanics
tag: paywall, PDF

Covariant Feynman derivation of Schrodinger's equation in a riemannian space
tag: paywall, PDF
http://fermatslibrary.com/s/feynmans-derivation-of-the-schrodinger-equation

Derivation of the Dirac equation from a relativistic representation of spin
tag: paywall, PDF
Physical meaning and derivation of Schrodinger and Dirac equations
tag: arxiv

The Dirac Equation
tag: PDF

Particle in a 1D box
https://chem.libretexts.org/Core/Physical_and_Theoretical_Chemistry/Quantum_Mechanics/05.5%3A_Particle_in_Boxes/Particle_in_a_1-dimensional_box
http://quantummechanics.ucsd.edu/ph130a/130_notes/node136.html
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/pbox.html
https://en.wikipedia.org/wiki/Particle_in_a_box

Plasma

Thermodynamic derivation of Saha's equation for a multi-temperature plasma
tag: paywall, PDF

The Einstein and the Navier-Stokes Equations: Connecting the Two by Bredberg, Irene, Ph.D., HARVARD UNIVERSITY, 2012, 132 pages; 3513905
Abstract: This thesis establishes a precise mathematical connection between the Einstein equations of general relativity and the incompressible Navier-Stokes equation of fluid dynamics.
see also http://arxiv.org/abs/1101.2451

Relation between uncertainty and the quantum harmonic oscillator
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc4.html
https://www.eng.fsu.edu/~dommelen/quantum/style_a/nt_uprl.html
http://hitoshi.berkeley.edu/221a/coherentstate.pdf

Schrodinger equation and quantum harmonic oscillator
http://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_5.pdf

Schrodinger equation and wave equation
http://physics.stackexchange.com/questions/75363/how-is-the-schroedinger-equation-a-wave-equation
http://www.tcm.phy.cam.ac.uk/~bds10/aqp/handout_foundations.pdf

Wednesday, June 28, 2017

Realizing my value for the Physics Derivation Graph

I previously attempted to catalog the various domains of activities associated with the Physics Derivation Graph. The relevant skill sets include logic, computer science, programming, mathematics, and physics. I don't have the time to learn all these domains to sufficient degree, but I do think I have some value to provide.

I could do tasks in each of these domains, but the implementation would be poor and there's an opportunity cost for me. Alternatively, I can focus on tasks I have expertise on.

What I think I may be good at is gathering the relevant equations and identifying how they are related. This would first be done on paper rather than attempting to find the appropriate method for computer entry.

Task 1: enumerate potential equations for inclusion in the Physics Derivation Graph

Task 2: given a list of relevant equations, enumerate all pairs

Task 3: for each pair, is there a derivation relation?

Thursday, June 15, 2017

not getting caught in the details -- identifying priorities

There are lots of relevant topics with the Physics Derivation Graph:

representing expressions prettily -- ie Latex, MathJax, MathML (presentation)
storing expressions meaningfully -- ie an Abstract Syntax Tree, MathML (content)
converting between presentation and content
converting between non-propriatry content representation and a Computer Algebra System
finding a Computer Algebra System capable of handling most of mathematical physics
manually creating content (ie actual derivations)
entering content manually into the Physics Derivation Graph
searching for content (ie PDFs of derivations)
converting found content (ie PDFs) to input for the Physics Derivation Graph
checking the consistency of content in the Physics Derivation Graph
rendering the content of the Physics Derivation Graph, ie using d3js or GraphViz

There are dependencies among these tasks. Also, there is an order for the tasks. Below I group and order the tasks.

Getting content is vital to the Physics Derivation Graph being useful.

manually creating content (ie actual derivations)

entering content manually into the Physics Derivation Graph

searching for content (ie PDFs of derivations)

converting found content (ie PDFs) to input for the Physics Derivation Graph

Before entering content into a computer, it makes sense to choose how to store the data

representing expressions prettily -- ie Latex, MathJax, MathML (presentation)
storing expressions meaningfully -- ie an Abstract Syntax Tree, MathML (content)

converting between presentation and content

Once content exists, it would be useful to validate

checking the consistency of content in the Physics Derivation Graph
finding a Computer Algebra System capable of handling most of mathematical physics

converting between non-proprietary content representation and a Computer Algebra System

Lastly, but no less important, how will consumers interact with the data?

rendering the content of the Physics Derivation Graph, ie using d3js or GraphViz

Wednesday, June 7, 2017

representing inference rules as both LaTeX and Abstract Syntax Trees

All inference rules in the Physics Derivation Graph are written in LaTeX. See the full list at
https://github.com/allofphysicsgraph/proofofconcept/tree/gh-pages/v4_file_per_expression/inference_rules
For example, the inference rule "add X to both sides" in LaTeX is
Add $#1$ to both sides of Eq.~\ref{eq:#2}.

AST representation in plain text

https://calculem.us/abstract-binding-trees-1/
Inference rules are transformations to the abstract syntax trees that represent expressions.
For example, the "add X to both sides" (addition property of equality) does the following transform:

input:expression
op
LHS
RHS

input:feed
x

output:expression
op
+
LHS
x
+
RHS
x

Here I'm using a two space indent to show the tree structure of the AST.
The "LHS" and "RHS" are sides of the expression. The "op" is the operator relating LHS and RHS.
I wanted a format that is visually accessible and not to verbose, while capable of being converted to some other format.

Order matters

My AST representation needs to include order. The expression "a-b" is distinct from "b-a" even though a tree doesn't specify the order:

input:expression:1
op
c
-
a
b

which is distinct from
input:expression:2
op
c
-
b
a

This also applies to cross product since it's also non-commutative.
To provide clarification, I'll assume the "top-to-bottom" order in the above format corresponds to "left-to-right." With that specification, the top AST corresponds to "c=a-b" and the bottom AST is "c=b-a".

AST for integrals and derivatives

Shown here: https://tug.org/TUGboat/tb12-3-4/tb33arnon.pdf

Mentioned here (http://www.math.wpi.edu/IQP/BVCalcHist/calc5.html) but not explored explicitly.

A definite integral in Latex
\int_{low}^{high} LHS d(x) = \int_{low}^{high} RHS d(x)
can be written as an AST:

input:expression
op
\int
low
high
LHS
x
\int
low
high
RHS
x

Similarly, a differential equation in Latex
\frac{d}{d(x)} LHS = \frac{d}{d(x)} RHS
can be written as an AST:
input:expression
op
dif
LHS
x
dif
RHS
x

AST for Dirac notation

Distinguishing input and output expressions

Some inference rules act on multiple expressions, and some inference rules produce multiple expressions (ie the taking the square root). Here's the AST for "add Eq1 to Eq2":

input:expression:1
op
LHS:1
RHS:1

input:expression:2
op
LHS:2
RHS:2

output:expression
op
+
LHS:1
LHS:2
+
RHS:1
RHS:2

Complicated expressions as ASTs

Some expressions are more complicated than simply "LHS = RHS". Suppose we have an expression
y = { x^2 for x>0
{ 0 for x<=0

I don't know how to represent this as an AST. Here's an attempt:

op
y
set
domain
^
x
2
>
x
0
domain
0
<=
x
0

I needed to introduce two new symbols: "set" and "domain"

Related work

Mathlex thesis

http://mathlex.org/doc/how-mathlex-works

Monday, May 29, 2017

abstract syntax trees and inference rules

The Physics Derivation Graph is composed of inference rules and mathematical expressions. Inference rules describe how to get from one expression to another.

In this post I first show that expressions can be represented as abstract syntax trees. Then I show that inference rules are effectively transformations applied to abstract syntax trees.

First I'll illustrate that every expression in the Physics Derivation Graph can be represented as an abstract syntax tree. For example, the expression

a+b = c

would be represented as

An example of an inference rules in the Physics Derivation Graph is, "add 2 to both sides of the expression." In this example the value 2 is a "feed" to the inference rule and the inference rule operates against a single expression.

Writing the inference rule as a function,

<output_expression> = add_x_to_both_sides( <feed value>, <input_expression> )

Applying the inference rule "add 2 to both sides of the expression" yields

The "add __ to both sides of the expression" inference rule essentially means transform input

LHS = RHS

LHS + __ = RHS + __

or, in terms of transforming an abstract syntax tree,

Claim: every inference rule can be written as a transform from one abstract syntax tree to another.

Saturday, February 11, 2017

editing Physics Derivation Graph on AWS EC2 Ubuntu

I wanted to play around with EC2, and AWS offers limited compute for 1 year for free.

Start here: https://aws.amazon.com/free/faqs/?ft=nf

I turned on alerts for billing to prevent charges.

$ chmod 600 filename.pem
$ ssh -i filename.pem ubuntu@<IP address>

PDG on AWS

$ sudo apt-get install git
$ git config --global user.name "<my name>"
$ git config --global user.email <my email address>
$ git config --list
$ ssh-keygen -t rsa -b 4096 -C "<my email address"
go to https://github.com/settings/keys

$ ssh-add -l
Could not open a connection to your authentication agent.
$ eval "$(ssh-agent -s)"
Agent pid 2690
$ ssh-add -l
The agent has no identities.
$ ssh-add ~/.ssh/id_rsa
Identity added: /home/ubuntu/.ssh/id_rsa (/home/ubuntu/.ssh/id_rsa)
$ ssh-add -l
4096 64:84:9f:91:c7:8c:7d:14:98:2a:db:05:6d:39:c8:0f /home/ubuntu/.ssh/id_rsa (RSA)
Validate setup:
$ ssh -T git@github.com

On the page https://github.com/allofphysicsgraph/proofofconcept
switch from HTTPS to SSH to find the address
git clone git@github.com:allofphysicsgraph/proofofconcept.git