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v1 Oct.28.2017
Satoshi Egi
Rakuten Institute of Technology
Rakuten, Inc.

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My aim is to create a language that can represent directly all algorithms that can be
discovered.
Currently, my biggest challenge is to improve Egison in order to represent directly
calculations that appear in mathematical physics.

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Pattern-matching against the wider range of data types.
Customizable symbolic computation using Egison pattern-matching.
Tensor index notation in programming.

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https://www.egison.org/math

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https://www.egison.org/math

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The theory for investigating a curved space.
It is easy for us in the 3 dimensional space to recognize a 2 dimensional curved
surface is curved.
How about our world? Is it curved for who can recognize the higher dimensional
space?
https://commons.wikimedia.org/wiki/File:Trian
gles_(spherical_geometry).jpg

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The theory explains the gravity as the curve of 4 dimensional time-space.
https://commons.wikimedia.org/wiki/File:Spacetime_lattice_analogy.svg

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In our recognition, the space-time is flat.

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But, according to the theory, the space-time is curved.

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And, things move straight in this space-time.

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In our recognition, it looks like things are falling.

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Formulas in Riemannian geometry are represented with partial derivative
operator and tensor index notation.
We can represent both of them concisely in Egison!

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Egison program that represents the above formula
Formula of Riemann curvature tensor
~: superscript
_: subscript

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Pattern-matching against the wider range of data types.
Customizable symbolic computation using Egison pattern-matching.
Tensor index notation in programming.

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Pattern-matching against the wider range of data types.
Customizable symbolic computation using Egison pattern-matching.
Tensor index notation in programming.

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Pattern Body
Matcher
Target

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are patterns.

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Pattern-matching against the wider range of data types.
Customizable symbolic computation using Egison pattern-matching.
Tensor index notation in programming.

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We can apply Egison pattern-matching
against math expressions.
Math expressions are a multiset of terms.
Terms are a multiset of factors.
Therefore, Egison pattern-matching is
very useful to handle them.

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Pattern-matching against the wider range of data types.
Customizable symbolic computation using Egison pattern-matching.
Tensor index notation in programming.

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Tensor index notation is a popular notation in the field of mathematics and physics.

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We can control the way for multiplying vectors by indices.

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We can use tensor index notation to multiply any order of tensors.
The tensor index notation is necessary to represent the multiplication of order
tensors higher than matrices.

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In Egison method, we can apply directly both “∂/∂” and “.” functions to tensors.
Egison program that represents the above formula
Formula of Riemann curvature tensor
~: superscript
_: subscript

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Wolfram program that represents the above formula
Egison program that represents the above formula
Formula of Riemann curvature tensor

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28https://arxiv.org/abs/1702.06343

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https://commons.wikimedia.org/wiki/File:T
riangles_(spherical_geometry).jpg

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https://commons.wikimedia.org/wiki/File:Spacetime_lattice_analogy.svg

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Currently, I am working to represent differential forms, exterior derivative, and
Hodge operator directly in Egison.
If we realize that, there are the wide range of application, e.g. mathematics,
physics, and computer simulation.