fixpoint

# Constraint Programming Solver

## The approach 
The implementation follows the ideas described in Chapter 12, "Concepts, Techniques, and Models
  of Computer Programming" by Peter Van Roy and Seif Haridi.

[An overview of CP implementation in Mozart/Oz.](http://mozart2.org/mozart-v1/doc-1.4.0/fdt/index.html)
## Status

Proof of concept. Not suitable for use in production. Significant API changes and core implementation rewrites are expected.

## Intro

[![Run in Livebook](https://livebook.dev/badge/v1/black.svg)](https://livebook.dev/run?url=https%3A%2F%2Fgithub.com%2Fbokner%2Ffixpoint%2Fblob%2Fmain%2Flivebooks%2Ffixpoint.livemd)


## Implemented constraints

- `equal`, `not_equal`, `less_or_equal`
- `absolute`
- `all_different`
- `inverse`
- `sum`, `modulo`
- `element`, `element2d`
- `circuit`
- `OR`

## Features
- views (linear combinations of variables in constraints)
- partial support for reified constraints  
- solving constraint satisfaction (CSP) and constrained optimization (COP) problems
- parallel search
- pluggable search strategies
- distributed solving  

## Installation
The package can be installed by adding `fixpoint` to your list of dependencies in `mix.exs`:

```elixir
def deps do
  [
    {:fixpoint, "~> 0.8.28"}
  ]
end
```

## Usage
  
### Getting started  

Let's solve the following *constraint satisfaction problem*:

***Given two sets of values***

 x = {1,2}, y = {0, 1}

***, find all solutions such that*** $x$ $\neq$ $y$

First step is to create a model that describes the problem we want to solve.
The model consists of *variables* and *constraints* over the variables.
In this example, we have 2 variables $x$ and $y$ and a single constraint $x$ $\neq$ $y$

```elixir
alias CPSolver.IntVariable
alias CPSolver.Constraint.NotEqual
alias CPSolver.Model
## Variable constructor takes a domain (i.e., set of values), and optional parameters, such as `name`
x = IntVariable.new([1, 2], name: "x")
y = IntVariable.new([0, 1], name: "y")
## Create NotEqual constraint
neq_constraint =  NotEqual.new(x, y)
```
Now create an instance of `CPSolver.Model`:
```elixir
model = Model.new([x, y], [neq_constraint])
```
Once we have a model, we pass it to `CPSolver.solve/1,2`.

We can either solve asynchronously:
```elixir
## Asynchronous solving doesn't block 
{:ok, solver} = CPSolver.solve_async(model)
Process.sleep(10)
## We can check for solutions and solver state and/or stats,
## for instance:
## There are 3 solutions: {x = 1, y = 0}, {x = 2,  y = 0}, {x = 2, y = 1} 
iex(46)> CPSolver.solutions(solver)
[[1, 0], [2, 0], [2, 1]]

## Solver reports it has found all solutions    
iex(47)> CPSolver.status(solver)
:all_solutions 

## Some stats
iex(48)> CPSolver.statistics(solver)
%{
  elapsed_time: 2472,
  solution_count: 3,
  active_node_count: 0,
  failure_count: 0,
  node_count: 5
}

```
, or use a blocking call:
```elixir
iex(49)> {:ok, results} = CPSolver.solve(model)
{:ok,
 %{
   status: :all_solutions,
   statistics: %{
     elapsed_time: 3910,
     solution_count: 3,
     active_node_count: 0,
     failure_count: 0,
     node_count: 5
   },
   variables: ["x", "y"],
   objective: nil,
   solutions: [[2, 1], [1, 0], [2, 0]]
 }}
```




### API
```elixir
#################
# Solving       
#################
# 
# Asynchronous solving.
# Takes CPSolver.Model instance and solver options as a Keyword. 
# Creates a solver process that runs asynchronously
# and could be controlled and queried for produced solutions and/or status as it runs.
# The solver process is alive even after the solving is completed.
# It's the responsibility of a caller to dispose of it when no longer needed.
# (by calling CPSolver.dispose/1)
  
{:ok, solver} = CPSolver.solve_async(model, solver_opts)

# Synchronous solving.
# Takes CPSolver.Model instance and solver options as a Keyword. 
# Starts the solver and gets the results (solutions and/or solver stats) once the solver finishes.
{:ok, solver_results} = CPSolver.solve(model, solver_opts)
```

, where 
- ```model``` - [specification of the model](#model-specification);
- ```solver_opts (optional)``` - [solver options](#configuring-solver).

### Model specification

#### For CSP (constraint satisfaction problem):

```elixir
model = CPSolver.Model.new(variables, constraints)
```
, where
-  `variables` is a list of [variables](lib/solver/variables/variable.ex) up to a concrete implementation.
  
  Currently, the only implementation supported is for [variables over integer finite domain](lib/solver/variables/int_variable.ex).
- `constraints` is a list of [constraints](lib/solver/core/constraint.ex). 
  
  [Available constraints](lib/solver/constraints/)

#### For COP (constraint optimization problem):
  
```elixir
model = CPSolver.Model.new(variables, constraints, objective: objective)
```
The same as for CSP, but with additional `:objective` option. The objective is constructed by using
`CPSolver.Objective.minimize/1` and `CPSolver.Objective.maximize/1`. 

[Example of COP model](lib/examples/knapsack.ex) 

### Configuring solver

Available options:

- solution_handler: function()

  A callback that gets called performed every time the solver finds a new solution. The single argument is a list of tuples

  `{variable_name, variable_value}`

- timeout: integer()
  
  Time to wait (in milliseconds) for terminating `CPSolver.solve_sync/2` call. Defaults to 30_000.
- stop_on: term() | condition_fun()
  
  Condition for stopping the solving. Currently, only `{:max_solutions, max_solutions}` condition is available.
  Defaults to `nil`.

- search: {variable_choice(), value_choice()}
  
  [Search strategy](#search). 

- space_threads: integer()

  Defines the number of processes for parallel search. Defaults to 8.

- distributed: boolean() | [Node.t()]
  
  If `true`, all connected nodes will participate in distributed solving.
  Alternatively, one can specify the sublist of connected nodes.
  Defaults to `false`.
    

### Distributed solving

*Fixpoint* allows to solve an instance of CSP/COP problem using multiple cluster nodes.

Note: *Fixpoint* **will not configure the cluster nodes!** 
It's assumed that each node has the cluster membership and the `fixpoint` dependency is installed on it.
The solving starts on a 'leader' node, and then the work is distributed across participating nodes.
The 'leader' node coordinates the process of solving through shared solver state.

Let's collect all solutions for 8-Queens problem using distributed solving.

For demonstration purposes, we will spawn peer nodes like so:

```zsh
iex --name leader --cookie solver -S mix
```

```elixir
### Let's spawn 2 worker nodes...
worker_nodes = Enum.map(["node1", "node2"], fn node -> 
  {:ok, _pid, node_name} = :peer.start(%{name: node, longnames: true, args: ['-setcookie', 'solver']})
  :erpc.call(node_name, :code, :add_paths, [:code.get_path()])
  node_name
end)
```

Then we'll pass spawned worker nodes to the solver: 

```elixir
## To convince ourselves that the solving runs on worker nodes, we'll use a solution handler:
solution_handler = fn solution -> IO.puts("#{inspect Enum.map(solution, fn {_name, solution} -> solution end)} <- #{inspect Node.self()}") end 
{:ok, _solver} = CPSolver.solve_async(CPSolver.Examples.Queens.model(8), 
  distributed: worker_nodes, 
  solution_handler: solution_handler)
``` 


### Search

*Fixpoint* allows to specify strategies for searching for feasible and/or optimal solutions.

This is controlled by `:search` option, which is a tuple {`variable_choice`, `value_choice`}.

Generally, `variable_choice` is either an implementation of [variable selector](lib/solver/search/variable_selector.ex), or an identificator of out-of-box implementation that fronts such an implementation. 

Likewise, `value_choice` is either an implementation of [value partition](lib/solver/search/value_partition.ex), or an identificator of out-of-box implementation.

Available standard search strategies:

- For `variable_choice`: 
  - `:first_fail` : choose the unfixed variable with smallest domain size
  - `:input_order` : choose the first unfixed variable in the order defined by the model 

- For `value_choice`
  - `:indomain_min, :indomain_max, :indomain_random` : choose minimal, maximal and random value from the variable domain, respectively

Default search strategy is `{:first_fail, :indomain_min}`


The choice of search strategy may significantly affect the performance of solving,

as the following example shows: 

#### Let's use some out-of-box strategies for solving an instance of Knapsack problem,


```elixir
alias CPSolver.Examples.Knapsack
## First, use the default strategy
{:ok, results} = CPSolver.solve(Knapsack.tourist_knapsack_model())
results.statistics

%{
  elapsed_time: 689543,
  solution_count: 114,
  active_node_count: 0,
  failure_count: 1614,
  node_count: 3455
}

## Now, use the :indomain_max for the value choice. 
## Decision variables for items have {0,1} domain, where 1 means that the item will be packed.
## Hence, :indomain_max tells the solver to try to include the items first (i.e. choose 1 over 0).
## 
##
{:ok, results} = CPSolver.solve(Knapsack.tourist_knapsack_model(), search: {:first_fail, :indomain_max})

iex(main@zephyr.local)21> results.statistics
%{
  elapsed_time: 301501,
  solution_count: 14,
  active_node_count: 0,
  failure_count: 693,
  node_count: 1413
}
```
The solution time for :indomain_max is more than twice less compared to the default value choice strategy

## [Examples](lib/examples)

#### [Reindeer Ordering](lib/examples/reindeers.ex)

Shows how to put together a model that solves a simple riddle.

#### [N-Queens](lib/examples/queens.ex)

Classical N-Queens problem

#### [Sudoku](lib/examples/sudoku.ex)

No explanation needed :-)

#### [SEND+MORE=MONEY](lib/examples/send_more_money.ex)

Cryptoarithmetics problem - a riddle that involves arithmetics.

#### [Knapsack](lib/examples/knapsack.ex)

Constraint Optimization Problem - packing items so they fit the knapsack ***and*** maximize the total value. Think Indiana Jones trying to fill his backpack with treasures in the best way possible :-)

#### [Quadratic Assignment](lib/examples/qap.ex)

Constraint Optimization Problem - assign facilities to locations so the cost of moving goods between facilities is minimized.

#### [Travelling Salesman problem](lib/examples/tsp.ex)

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

#### [SAT Solver](lib/examples/sat_solver.ex)

#### [Stable Marriage problem](lib/examples/stable_marriage)

https://en.wikipedia.org/wiki/Stable_marriage_problem
 
#### [`xkcd` comic](livebooks/xkcd_np.livemd)

Two combinatorial problems from https://xkcd.com/287/

#### [Fixpoint models created by Håkan Kjellerstrand](http://hakank.org/elixir/)

#### [Zebra puzzle](https://en.wikipedia.org/wiki/Zebra_Puzzle)