# What is satisfiability?

by Stephen M. Walker II, Co-Founder / CEO

## What is the definition of satisfiability in AI?

In the context of artificial intelligence (AI) and computer science, satisfiability refers to the problem of determining if there exists an interpretation that satisfies a given Boolean formula. A Boolean formula, or propositional logic formula, is built from variables and operators such as AND, OR, NOT, and parentheses. A formula is said to be satisfiable if it can be made TRUE by assigning appropriate logical values (TRUE, FALSE) to its variables.

For instance, consider the formula "a AND NOT b". This formula is satisfiable because one can find the values a = TRUE and b = FALSE, which make (a AND NOT b) = TRUE. In contrast, "a AND NOT a" is unsatisfiable because there's no assignment of TRUE or FALSE to 'a' that would make the entire formula TRUE.

The Boolean satisfiability problem (SAT) is the task of checking whether a given formula is satisfiable. This problem is significant in computer science because it was the first problem proven to be NP-complete, meaning that all problems in the complexity class NP are at most as difficult to solve as SAT.

In a broader sense, a set of sentences is satisfiable if there exists an interpretation in which every sentence is true. Conversely, a formula is unsatisfiable if it's not possible to assign such values that make the formula true.

Satisfiability is a fundamental concept in various areas of computer science and AI, including hardware design, software analysis, planning, mathematics, security analysis, and more.

## Understanding Satisfiability in AI

Satisfiability in artificial intelligence (AI) refers to the capability of a system to find a solution that adheres to all given constraints or requirements. A satisfiable problem has at least one solution that fulfills all criteria, whereas an unsatisfiable problem lacks any such solutions.

### Types of Satisfiability Problems

AI encounters various satisfiability problems, including:

**Constraint Satisfaction Problems**— These require meeting specific constraints, like filling a Sudoku grid without repeating digits in any row, column, or sub-grid.**Optimization Problems**— The objective is to identify the optimal solution, such as determining the shortest path between two points.**Boolean Satisfiability Problems**— A solution is valid if a boolean formula with variables and operators evaluates to true.**Search Problems**— Solutions are found through searching, as in finding the sequence of moves to solve a Rubik’s Cube.

### Applications of Satisfiability

Satisfiability algorithms are pivotal in AI for various applications, such as navigating robots around obstacles, planning and scheduling, resource allocation, and system configuration. These algorithms efficiently explore possible solutions to find one that satisfies all constraints.

### Challenges in Satisfiability

Satisfiability in AI faces several challenges:

**The Frame Problem**— Difficulty in representing and relating information for automated reasoning, affecting the system's ability to determine achievable goals.**Scalability**— The exponential growth of potential solutions with the increase in variables and constraints, leading to longer solution times.**Uncertainty**— Handling incomplete or uncertain information, which can lead to errors in automated reasoning.

### Research Directions in Satisfiability

Current research in satisfiability focuses on developing new algorithms, improving problem representations for easier resolution, and enhancing solver efficiency to tackle larger and more complex problems.