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# true quantified Boolean formula

### the tl;dr

A true quantified Boolean formula (QBF) is a formula in which all variables are quantified, and in which the truth of the formula can be determined by evaluating the formula for all possible combinations of truth values for the quantified variables.

## What is a quantified Boolean formula?

A quantified Boolean formula (QBF) is a formula in which variables are quantified by existential (there exists) or universal (for all) quantifiers. QBF is a generalization of propositional logic, which does not allow variables to be quantified.

QBF can be used to represent problems in many areas, including mathematics, computer science, and artificial intelligence (AI). For example, a QBF formula can be used to represent a Sudoku puzzle, and AI techniques can be used to solve the formula and find a solution to the puzzle.

In AI, QBF can be used to represent problems that are difficult or impossible to solve using traditional propositional logic. For example, the problem of determining whether a given graph is colorable can be represented as a QBF formula, and AI techniques can be used to solve the formula and find a solution to the problem.

## What is the satisfiability problem?

The satisfiability problem, also known as the Boolean satisfiability problem, is the problem of determining whether there exists an interpretation that makes a given Boolean formula true. In other words, it asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called satisfiable. On the other hand, if no such assignment exists, the formula is unsatisfiable. For example, the formula "a AND NOT b" is satisfiable because we can set a to TRUE and b to FALSE, and the formula evaluates to TRUE. On the other hand, "a AND b" is unsatisfiable because no matter how we assign the values of a and b, the formula will always evaluate to FALSE.

## What is the Boolean satisfiability problem?

The Boolean satisfiability problem, also known as SAT, is a problem in AI that is used to determine whether or not a given Boolean formula can be satisfied by a set of truth values. A Boolean formula is a mathematical formula that consists of a set of variables, each of which can take on one of two values, true or false. The problem is to determine whether there exists a set of truth values for the variables that makes the formula true.

The Boolean satisfiability problem is of great importance in AI because many problems in AI can be expressed as Boolean formulas. For example, the problem of determining whether a given graph is colorable can be expressed as a Boolean formula. If the graph is not colorable, then the formula is not satisfiable.

The Boolean satisfiability problem is also of great importance in computer science. Many problems in computer science can be expressed as Boolean formulas, and the problem of determining whether a given formula is satisfiable is known as the satisfiability problem. The satisfiability problem is one of the most studied problems in computer science, and it is known to be NP-complete.

## What is the difference between a quantified Boolean formula and a Boolean formula?

A Boolean formula is a mathematical formula used to describe a logical relationship between two values, usually denoted as 0 and 1. A quantified Boolean formula (QBF) is a generalization of a Boolean formula in which variables can be quantified.

## What is the difference between the satisfiability problem and the Boolean satisfiability problem?

The satisfiability problem is the problem of determining whether a given Boolean formula is satisfiable, that is, whether there is an interpretation of the variables of the formula that makes the formula true.

The Boolean satisfiability problem is the problem of determining whether a given Boolean formula is satisfiable, that is, whether there is an interpretation of the variables of the formula that makes the formula true. The Boolean satisfiability problem is a special case of the satisfiability problem, and is of particular interest in the field of artificial intelligence because it is a NP-complete problem.