Making Better Decisions with SAT Solvers

Every day, organizations and researchers face tough logic-based decisions:

How should we schedule classes efficiently?
How can we verify if a circuit design is error-free?
Can we automatically detect software bugs?

These problems require handling constraints and logical conditions. Boolean Satisfiability (SAT) solvers provide a powerful way to find solutions efficiently.

In this post, we'll introduce the SAT problem, explain how it works, and show a practical example of scheduling study groups using SAT solvers.

What Is SAT?

SAT (Boolean Satisfiability) is a problem where the goal is to determine whether there is an assignment of true (1) or false (0) values to a set of Boolean variables that satisfies all given logical conditions. These conditions are written as clauses, and for the problem to be solvable, all clauses must be satisfied.

Each clause is a logical condition made up of literals, which are either a variable (X) or its negation (NOT X). A clause is satisfied if at least one of its literals is true.

For example:

Clause1: (X OR NOT Y OR Z) means "X is true OR Y is false OR Z is true"
Clause2: (X OR NOT Z)

To satisfy the SAT problem, you need to find values for all the variables such that all the clauses are satisfied. If you can find such an assignment, the problem is solvable.

Example: Scheduling Study Groups

Imagine a university scheduling study groups for three courses:

Math (M)
Physics (P)
Computer Science (CS)

There are three available time slots:

Slot 1 (Morning)
Slot 2 (Afternoon)
Slot 3 (Evening)

Constraints:

Each course must be scheduled in exactly one time slot.
Math and Physics cannot be in the same slot. (They share a professor)
Physics and CS cannot be in the same slot. (They need to use the same classroom)
CS cannot be in the morning. (The professor is unavailable at this time)

Variable Definitions:

We define Boolean variables for each course-time assignment:

Variable	Meaning
M₁	Math in Morning
M₂	Math in Afternoon
M₃	Math in Evening
P₁	Physics in Morning
P₂	Physics in Afternoon
P₃	Physics in Evening
CS₁	CS in Morning
CS₂	CS in Afternoon
CS₃	CS in Evening

Each course should be assigned to exactly one of these slots.

CNF Encoding of the Problem

We now express the constraints in CNF format and explain why each one is needed.

1️⃣ Each course must be in at least one time slot:

2️⃣ Each course must be in at most one time slot:

3️⃣ Math and Physics cannot be in the same slot:

Explanation: This enforces the constraint that Math and Physics cannot both be scheduled in the same time slot. For example, if Math is scheduled in the Morning, then Physics cannot also be scheduled in the Morning.

4️⃣ Physics and CS cannot be in the same slot:

Explanation: This ensures that Physics and Computer Science (CS) are not scheduled in the same time slot. For example, Physics cannot be in the Morning if CS is also in the Morning.

5️⃣ CS cannot be in the morning:

Explanation: This explicitly states that CS cannot be scheduled in the Morning. It restricts CS to the Afternoon or Evening.

Final CNF Representation:

Solution from a SAT Solver

A SAT solver returns an assignment of variables that satisfies all constraints. Here’s one possible solution:

Interpreting the Solution:

Variable	Meaning	Value
M₁	Math in Morning	False
M₂	Math in Afternoon	True ✅
M₃	Math in Evening	False
P₁	Physics in Morning	True ✅
P₂	Physics in Afternoon	False
P₃	Physics in Evening	False
CS₁	CS in Morning	False
CS₂	CS in Afternoon	True ✅
CS₃	CS in Evening	False

Final Schedule:

📌 Math → Afternoon

📌 Physics → Morning

📌 Computer Science → Afternoon

✅ The solution satisfies all constraints!

How to Solve SAT Problems?

1️⃣ Use an Open-Source SAT Solver (Free but Requires Setup)

Run the SAT solver with your .cnf file, and it will return whether a valid solution exists.

2️⃣ Use NPCompleter to Solve It for You (Easiest Option)

✅ Just post your problem—someone else solves it for you.

✅ All solutions are verified to ensure correctness.

✅ Free for small problems (like those solvable by open-source solvers).

✅ Paid options for complex problems in the future.

🔗 Try NPCompleter Here!

Where Are SAT Solvers Used?

💡 Scheduling – Assigning employees, classrooms, or events.

💡 Circuit Verification – Ensuring a circuit functions correctly.

💡 Software Testing – Finding input combinations that break a program.

💡 Cryptography – Checking if a system is secure.

Conclusion

SAT solvers are powerful tools for decision-making problems involving yes/no constraints. Whether you're scheduling classes, verifying circuits, or solving logic puzzles, SAT solvers can help.

🚀 Start solving SAT problems today! 🎯