Wolfram Models as Set Substitution Systems | Getting Started | Symbols and Functions | Physics | Acknowledgements

Wolfram Models as Set Substitution Systems

Set Substitution Systems

SetReplace is a Wolfram Language package for manipulating set substitution systems. To understand what a set substitution system does consider an unordered set of elements:

{1, 2, 5, 3, 6}

We can set up an operation on this set which would take any of the two elements and replace them with their sum:

{a_, b_} :> {a + b}

In SetReplace, this can be expressed as the following (the new element 1 + 2 -> 3 is put at the end)

In[] := SetReplace[{1, 2, 5, 3, 6}, {a_, b_} :> {a + b}]
Out[] = {5, 3, 6, 3}

Note that this is similar to SubsetReplace function of Wolfram Language (introduced in version 12.1, it replaces all non-overlapping subsets at once by default):

In[] := SubsetReplace[{1, 2, 5, 3, 6}, {a_, b_} :> a + b]
Out[] = {3, 8, 6}

Wolfram Models

A more interesting case (which we call a Wolfram model) is one where the set elements are related to each other. Specifically, we can consider a set of ordered lists of atomic vertices; in other words, an ordered hypergraph.

As an example consider a set:

{{1, 2, 3}, {2, 4, 5}, {4, 6, 7}}

We can render it as a collection of ordered hyperedges:

In[] := WolframModelPlot[{{1, 2, 3}, {2, 4, 5}, {4, 6, 7}},
 VertexLabels -> Automatic]

We can then have a rule which would pick a subset of these hyperedges related through common vertices (much like a join query) and replace them with something else:

{{v1_, v2_, v3_}, {v2_, v4_, v5_}} :>
 Module[{v6}, {{v5, v6, v1}, {v6, v4, v2}, {v4, v5, v3}}]

Note the Module on the right-hand side creates a new variable (vertex) which causes the hypergraph to grow. Due to optimizations, it's not always a Module that creates vertices, so its name may be different. After a single replacement we get this (the new vertex is v11):

In[] := WolframModelPlot[SetReplace[{{1, 2, 3}, {2, 4, 5}, {4, 6, 7}},
  {{v1_, v2_, v3_}, {v2_, v4_, v5_}} :>
   Module[{v6}, {{v5, v6, v1}, {v6, v4, v2}, {v4, v5, v3}}]],
 VertexLabels -> Automatic]

After 10 steps, we get a more complicated structure:

In[] := WolframModelPlot[SetReplace[{{1, 2, 3}, {2, 4, 5}, {4, 6, 7}},
  {{v1_, v2_, v3_}, {v2_, v4_, v5_}} :>
   Module[{v6}, {{v5, v6, v1}, {v6, v4, v2}, {v4, v5, v3}}], 10],
 VertexLabels -> Automatic]

And after 100 steps, it gets even more elaborate:

In[] := WolframModelPlot[SetReplace[{{1, 2, 3}, {2, 4, 5}, {4, 6, 7}},
  {{v1_, v2_, v3_}, {v2_, v4_, v5_}} :>
   Module[{v6}, {{v5, v6, v1}, {v6, v4, v2}, {v4, v5, v3}}], 100]]

Exploring the hypergraph models of this variety is the primary purpose of this package.

Getting Started

Dependencies

You only need three things to use SetReplace:

• Windows, macOS 10.12+, or Linux.
• Wolfram Language 12.1+ including WolframScript. A free version is available as Wolfram Engine.
• A C++17 compiler to build the low-level part of the package. Instructions on how to set up a compiler to use in WolframScript are here.

Build Instructions

To build:

1. cd to the root directory of the repository.

2. Run ./build.wls to create the paclet file. If you see an error message about c++17, make sure the C++ compiler you are using is up-to-date. If your default system compiler does not support c++17, you can choose a different one with environmental variables. The following, for instance, typically works on a Mac:

   COMPILER=CCompilerDriver\`ClangCompiler\`ClangCompiler COMPILER_INSTALLATION=/usr/bin ./build.wls

   Here ClangCompiler can be replaced with one of << CCompilerDriver`; "Compiler" /. CCompilerDriver`CCompilers[Full], and COMPILER_INSTALLATION is a directory in which the compiler binary can be found.

3. Run ./install.wls to install the paclet into your Wolfram system.

4. Evaluate PacletDataRebuild[] in all running Wolfram kernels.

5. Evaluate << SetReplace` every time before using the package.

A less frequently updated version is available through the Wolfram public paclet server and can be installed with PacletInstall["SetReplace"].

Contributing

Keep in mind that this is an active research project. While we try to keep the main functionality backward compatible, it might change in the future as we adjust our models and find better ways of analysis. Keep that in mind when building on top of SetReplace, and keep track of git SHAs as you go.

SetReplace is an open-source project, and everyone is welcome to contribute. Read our contributing guidelines to get started.

We have a Discord server. If you would like to contribute but have questions or don't know where to start, this is the perfect place! In addition to helping new contributors, we discuss feature and research ideas. So, if you are interested, please join!

Symbols and Functions

• SetReplace*
• ToPatternRules
• WolframModel and WolframModelEvolutionObject
  • Properties
    • States
    • Plots of States
    • Plots of Events
    • All Edges throughout Evolution
    • States as Edge Indices
    • Events
    • Events and States
    • Creator and Destroyer Events
    • Causal Graphs
    • Expression Separations
    • MultiwayQ
    • Rule Indices for Events
    • Edge and Event Generations
    • Termination Reason
    • Generation Counts
    • Event Counts
    • Element Count Lists
    • Final Element Counts
    • Total Element Counts
    • Rules
    • Version
  • Options
    • "VertexNamingFunction"
    • "IncludePartialGenerations"
    • "IncludeBoundaryEvents"
    • "EventOrderingFunction"
    • "EventSelectionFunction"
    • "EventDeduplication"
    • Method
    • Time Constraint
• WolframModelPlot
• RulePlot of WolframModel
• Utility Functions
  • IndexHypergraph
  • IsomorphicHypergraphQ
  • HypergraphToGraph
  • RandomHypergraph
  • Subhypergraph
  • WolframModelRuleValue
  • GeneralizedGridGraph
  • HypergraphAutomorphismGroup
  • HypergraphUnifications
  • WolframPhysicsProjectStyleData
  • Build Data

Physics

A hypothesis is that spacetime at small scales is a network, and the fundamental law of physics is a system similar to the one this package implements.

A slightly different version of this system was first introduced in Stephen Wolfram's A New Kind Of Science.

You can find many more details about our physics results in Stephen Wolfram's Technical Introduction, and Jonathan Gorard's papers on Relativity and Quantum Mechanics. And there is much more on wolframphysics.org.

Acknowledgements

In additional to commit authors and reviewers, Stephen Wolfram has contributed to the API design of most functions, and Jeremy Davis has contributed to the visual style of WolframModelPlot, RulePlot and "CausalGraph".