Exhort

Beseech the maths to answer.

Overview

Exhort is an idomatic Elixir interface to the Google OR Tools.

Currently, there are C++ (native) Python, Java and C# interfaces to the Google OR tools.

Exhort is similar to the non-native interfaces to the tooling, but Exhort uses NIFs instead of SWIG to interface with the native libarary.

The goal of Exhort is to provide an idomatic Elixir interface to the Google OR Tools.

Setup

Because Exhort uses the Google OR tools, the first step is to install them on the target system.

MacOS

On MacOS, ensure the latest command line tools are installed.

pkgutil --pkg-info=com.apple.pkg.CLTools_Executables

Next, install the or-tools package from Homebrew:

brew install or-tools

Then leverage asdf for the required versions of Elixir and Elang:

asdf install

Finally, export the locations of Erlang and the OR Tools:

export ERLANG_HOME=$HOME/.asdf/installs/erlang/24.2.1
export ORTOOLS=/usr/local

Debian

Follow the instructions here and install from the appropriate archive. You will likely want to install them in a reasonable place like /usr/local/lib and perhaps link them to a consistent path.

For example:

wget https://github.com/google/or-tools/releases/download/v9.2/or-tools_amd64_debian-11_v9.2.9972.tar.gz
tar xf or-tools_amd64_debian-11_v9.2.9972.tar.gz -C /usr/local/lib
ln -s /usr/local/lib/or-tools_Debian-11-64bit_v9.2.9972 /usr/local/lib/ortools

Then export the locations of Erlang and the OR Tools:

export ERLANG_HOME=/usr/local/lib/erlang
export ORTOOLS=/usr/local/lib/ortools

Compiling

Exhort uses NIFs for interfacing with the Google OR tools. This means that Exhort NIFs must be compiled using a C compiler and Make. The Makefile contains these instructions. It just needs to know where you have installed both Erlang and the Google OR Tools. It will use the environment variables you exported above.

mix compile
mix test

Getting Started

The easiest way to get started is with the sample Livebook notebooks in the notebooks directory.

Start Livebook and open a notebook (use whatever method you like to start Livebook).

mix escript.install hex livebook
# if installed in `asdf` use `asdf reshim`
livebook server --name [email protected]

1. Use the link that is written to the console and browse the samples.
2. Open a sample in the notebooks directory
3. Run the notebook in the project by choosing the Mix standalone option in the left side of Livebook under "Runtime setteings"

The notebooks are mostly implementations of some of the samples that come with the Google OR Tools. That should provide a starting place for exploring the Exhort API and expression language. There is more about the Exhort API and expression language below, but the notebooks and tests are probably a good place to start.

As a dependency of a project

Add Exhort as a dependency to your project in the mix.exs:

  {:exhort, "~> 0.1.0"}

API

Exhort is in the early stages of development. As such, we are investigating a varity of API approaches. We may end up with more than one (a la Ecto), but in the short term will likely focus on a single approach.

The API is centered around the Builder and Expr modules. Those modules leverage Elixir macros to provide a DSL "expression language" for Exhort.

Builder

Building a model starts off with the Builder.

Builder has functions for defining variables, specifying constraints and creating a %Model{} using the build function.

By specifying use Exhort.SAT.Builder, all of the relevant modules will be aliased and the Exhort macros will be expanded.

  use Exhort.SAT.Builder
  ...

    builder =
      Builder.new()
      |> Builder.def_int_var("x", {0, 10})
      |> Builder.def_int_var("y", {0, 10})
      |> Builder.def_bool_var("b")
      |> Builder.constrain("x" >= 5, if: "b")
      |> Builder.constrain("x" < 5, unless: "b")
      |> Builder.constrain("x" + "y" == 10, if: "b")
      |> Builder.constrain("y" == 0, unless: "b")

    {response, acc} =
      builder
      |> Builder.build()
      |> Model.solve(fn
        _response, nil -> 1
        _response, acc -> acc + 1
      end)

    # 2 responses
    acc |> IO.inspect(label: "acc: ")
    response |> IO.inspect(label: "response: ")

    # :optimal
    response.status |> IO.inspect(label: "satus: ")
    # 10, 0, true
    SolverResponse.int_val(response, "x") |> IO.inspect(label: "x: ")
    SolverResponse.int_val(response, "y") |> IO.inspect(label: "y: ")
    SolverResponse.bool_val(response, "b") |> IO.inspect(label: "b: ")

See below for more about the expression language used in Exhort.

Expr

Sometimes it may be more convenient to build up expressions separately and then add them to a %Builer{} all at once. This is often the case when more complex data sets are invovled in generating many variables and constraints for the model.

Instead of having to maintain the builder through an Enum.reduce/3 construct like this:

    builder =
      Enum.reduce(all_days, builder, fn day, builder ->
        Enum.reduce(all_shifts, builder, fn shift, builder ->
          shift_options = Enum.filter(shifts, fn {_n, d, s} -> d == day and s == shift end)
          shift_option_vars = Enum.map(shift_options, fn {n, d, s} -> "shift_#{n}_#{d}_#{s}" end)

          Builder.constrain(builder, sum(shift_option_vars) == 1)
        end)
      end)

Exhort allows the generation of lists of variables or constraint, maybe using Enum.map/2:

    shift_nurses_per_period =
      Enum.map(all_days, fn day ->
        Enum.map(all_shifts, fn shift ->
          shift_options = Enum.filter(shifts, fn {_n, d, s} -> d == day and s == shift end)
          shift_option_vars = Enum.map(shift_options, fn {n, d, s} -> "shift_#{n}_#{d}_#{s}" end)

          Expr.new(sum(shift_option_vars) == 1)
        end)
      end)
      |> List.flatten()

These may then be added to the builder as a list:

    builder
    |> Builder.add(shift_nurses_per_period)
...

Variables

Model variables in the expression language are symbolic, represented as strings or atoms, and so don't interfere to the surrounding Elixir context. This allows the variables to be consistently referenced through a builder pipeline, for example, without having to capture an intermediate result.

Elixir variables may be used "as is" in expressions, allowing variables to be generated from enumerable collections.

In the following expression, "x" is a model variable, while y is an Elixir variable:

"x" < y + 3

Variables may be defined in a few ways. It's often convenient to just focus on the Expr and Builder modules, which each have functions like def_int_var and def_bool_var.

    all_bins
    |> Enum.map(fn bin ->
      Expr.def_bool_var("slack_#{bin}")
    end)

However, BoolVar.new/1 and IntVar.new/1 may also be used:

    all_bins
    |> Enum.map(fn bin ->
      BoolVar.new("slack_#{bin}")
    end)

Of course, such names are still usable in expressions:

    Expr.new("slack_#{bin}" <= bin_total)

Note that any variables or expressions created outside of the Builder still need to be added to a %Builder{} struct for them to be part of the model resulting from build/1. There's no magic here, these are still Elixir immutable data structures.

    variables = ...
    expressions = ...

    Builder.new()
    |> Builder.add(variables)
    |> Builder.add(expressions)
    |> Builder.build()

Expressions

Exhort supports a limited set of expressions. Expressions may use the binary operators +, - and *, with their traditional mathematical meaning. They may also use comparison operators <, <=, ==, >=, >, the sum function and even the for comprehension.

    all_bins
    |> Enum.map(fn bin ->
      vars = Enum.map(items, &{elem(&1, 0), "x_#{elem(&1, 0)}_#{bin}"})
      load_bin = "load_#{bin}"

      Expr.constrain(sum(for {item, x} <- vars, do: item * x) == load_bin)
    end)

Model

The model is the result of finalizing the builder, created through the Builder.build/1 function.

The model may then be solved with Model.solve/1 or Model.solve/2.

The latter function allows for a function to be passed to receive intermediate solutions from the solver.

Implementation

Exhort relies on the underlying native C++ implementation of the Google OR Tools.

Exhort interacts with the Google OR Tools library when the model is built using Builder.build/1 and when solved using Model.solve/1 or Model.solve/2.

References to the native objects are returned via NIF resources to the Elixir runtime as %Reference{} values. These are often stored in corresponding Exhort structs under the res key.

The native code is compiled to a single nif.so library and loaded via the Exhort.NIF.Nif module.