Z3.Linq

.NET 8.0 LINQ bindings for the Z3 theorem prover from Microsoft Research.

Based on the proof of concept by Bart De Smet which was curated into Z3.LinqBinding by Ricardo Niepel.

Examples

A number of examples are included in this solution, which you can run from .NET Interactive (requires Visual Studio Code) or from Visual Studio.

Problem - 1st Order Propositional Logic

Provide a solution where either X is true or Y is true, but not both (using a ValueTuple).

using (var ctx = new Z3Context())
{
    var theorem = from t in ctx.NewTheorem<(bool x, bool y)>()
                  where t.x ^ t.y
                  select t;

    var result = theorem.Solve();

    Console.WriteLine(result);
}

Problem - Linear Algebra

Solve the following system with 3 variables, with linear equalities and inequalities:

$$ x_1 - x_2 \ge 1 \\ x_1 - x_2 \le 3 \\ x_1 = 2x_3 + x_2 $$

using (var ctx = new Z3Context())
{
    var theorem = from t in ctx.NewTheorem<Symbols<int, int, int>>()
                  where t.X1 - t.X2 >= 1
                  where t.X1 - t.X2 <= 3
                  where t.X1 == (2 * t.X3) + t.X2
                  select t;

    var result = theorem.Solve();

    Console.WriteLine(result);
}

Problem - Price Optimised Oil Purchasing

In this example, we have two countries that produce crude oil which we refine into three end-products: gasoline, jet fuel, and lubricant. The crude oil from each country yields different quantities of end-products once the oil is refined:

	Saudi Arabia	Venezuela
Cost	$20 / barrel	$15 / barrel
Max Order	9000 barrels	6000 barrels
Refining %	30% gasolene	40% gasolene
	40% jet fuel	20% jet fuel
	20% lubricant	30% lubricant
	10% waste	10% waste

Given we need to produce the following volume of refined end-product:

Product	Amount (barrels)
Gasolene	1900
Jet Fuel	1500
Lubricant	500

What is the most cost efficient purchase strategy of crude oil from Saudi Arabia and Venezuela?

using (var ctx = new Z3Context())
{
    var theorem = from t in ctx.NewTheorem<(double sa, double vz)>()
                  where 0.3 * t.sa + 0.4 * t.vz >= 1900 // Gasolene
                  where 0.4 * t.sa + 0.2 * t.vz >= 1500 // Jet fuel
                  where 0.2 * t.sa + 0.3 * t.vz >= 500  // Lubricant
                  where 0 <= t.sa && t.sa <= 9000 // Max # barrels we can purchase
                  where 0 <= t.vz && t.vz <= 6000 // Max # barrels we can purchase
                  orderby (20.0 * t.sa) + (15.0 * t.vz) // Optimize for cost
                  select t;

    var result = theorem.Solve();
    
    Console.WriteLine($"SA: {result.sa} barrels (${result.sa * 20}), VZ: {result.vz} barrels (${result.vz * 15})");
}

Problem - Minimizing Shipping Costs

In this example, you want to minimize the cost of shipping goods from 2 different warehouses to 4 different customers. Each warehouse has a limited supply and each customer has a certain demand.

Cost of shipping ($ per product):

	Customer 1	Customer 2	Customer 3	Customer 4
Warehouse 1	$1.00	$3.00	$0.50	$4.00
Warehouse 2	$2.50	$5.00	$1.50	$2.50

Number of products shipped:

	Customer 1	Customer 2	Customer 3	Customer 4	Total shipped		Available
Warehouse 1	0	13,000	15,000	32,000	60,000	<=	60,000
Warehouse 2	30,000	10,000	0	0	40,000	<=	80,000
Total received	30,000	23,000	15,000	32,000
Ordered	35,000	22,000	18,000	30,000
Total Shipping Cost		$299,500.00

1. The objective is to minimize the cost (Total Shipping Cost).
2. The variables are the number of products to ship from each warehouse to each customer.
3. The constraints are the number of products ordered and the number of products available in each warehouse.

using (var ctx = new Z3Context())
{
    var theorem =
        from t in ctx.NewTheorem<(double w1c1, double w1c2, double w1c3, double w1c4, double w2c1, double w2c2, double w2c3, double w2c4)>()
        where t.w1c1 + t.w1c2 + t.w1c3 + t.w1c4 <= 60_000 // Warehouse 1 Product Availability
        where t.w2c1 + t.w2c2 + t.w2c3 + t.w2c4 <= 80_000 // Warehouse 2 Product Availability
        where t.w1c1 + t.w2c1 == 35_000 && (t.w1c1 >= 0 && t.w2c1 >= 0) // Customer 1 Orders
        where t.w1c2 + t.w2c2 == 22_000 && (t.w1c2 >= 0 && t.w2c2 >= 0) // Customer 2 Orders
        where t.w1c3 + t.w2c3 == 18_000 && (t.w1c3 >= 0 && t.w2c3 >= 0) // Customer 3 Orders
        where t.w1c4 + t.w2c4 == 30_000 && (t.w1c4 >= 0 && t.w2c4 >= 0) // Customer 4 Orders
        orderby (1.00 * t.w1c1) + (3.00 * t.w1c2) + (0.50 * t.w1c3) + (4.00 * t.w1c4) +
                (2.50 * t.w2c1) + (5.00 * t.w2c2) + (1.50 * t.w2c3) + (2.50 * t.w2c4) // Optimize for Total Shipping Cost
        select t;

    var result = theorem.Solve();

    Console.WriteLine($"|                     | Customer 1 | Customer 2  | Customer 3 | Customer 4 |");
    Console.WriteLine($"|---------------------|------------|-------------|------------|------------|");
    Console.WriteLine($"| Warehouse 1         | {result.w1c1}      | {result.w1c2}       |  {result.w1c3}      | {result.w1c4}          |");
    Console.WriteLine($"| Warehouse 2         | {result.w2c1}          | {result.w2c2}           | {result.w2c3}      | {result.w2c4}      |");
    Console.WriteLine();
    Console.WriteLine(string.Create(CultureInfo.CreateSpecificCulture("en-US"), $"Total Cost: {1.00 * result.w1c1 + 3.00 * result.w1c2 + 0.50 * result.w1c3 + 4.00 * result.w1c4 + 2.50 * result.w2c1 + 5.00 * result.w2c2 + 1.50 * result.w2c3 + 2.50 * result.w2c4:C}"));
}

Getting Started

You can install the Z3.Linq NuGet Package.

For Polyglot Notebooks

Add the package:

#r "nuget:Z3.Linq"

Then add the following using statements:

using System;
using Z3.Linq;

Then you can copy any of the above samples.

For Visual Studio

Add the Z3.Linq package. Configure your application to target x64 platform. This is a requirement as Z3.Linq uses the Microsoft.Z3 package.

Contributing

There are a number of ways in which you could contribute to this project:

• Create new examples!
• Improve the documentation.
• Report / fix bugs.
• Suggest any implementation improvements or optimizations.
• Blog about the project!

All PRs are welcome.

History

2009: Bart De Smet describes a prototype LINQ to Z3 binding in three blog posts:

• LINQ to Z3 - Part 1 – Exploring The Z3 Theorem Prover
• LINQ to Z3 - Part 2 – LINQ to the Unexpected
• LINQ to Z3 - Part 3 – Theorem Solving On Steroids

2010: Bart was interviewed on Channel 9 about the LINQ to Z3 concept:

2012: Bart presented LINQ to Everything at TechEd Europe 2012:

2015: Z3 was open sourced under the MIT license and the source code was moved to GitHub, where it is actively maintained.

2015: Ricardo Niepel (Microsoft) publishes the sample as Z3.LinqBinding using .NET 4.5 and Z3 binaries 4.4.0

2018: Jean-Sylvain Boige (My Intelligence Agency) adds Missionaries And Cannibals sample.

2020: Karel Frajtak adds support for fractions.

2021: Howard van Rooijen and Ian Griffiths (endjin) upgrade the project to .NET 6.0, added Optimize support via LINQ's OrderBy, ValueTuple support, demonstrate using record types, and fix nullability issues. They upgraded the solution to use Z3 NuGet package, merged in features from Jean-Sylvain Boige and Karel Frajtak forks, created archives of Bart's original blog posts and talks. They republished the project as Z3.Linq, created a new Polyglot Notebook of samples, and published a NuGet package Z3.Linq.

2023: Whit Waldo upgrades the project to .NET 8.0

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