True Quantified Boolean Formula (TQBF)
True Quantified Boolean Formula (TQBF), also known as Quantified SAT (QSAT), is a concept from computational complexity theory involving quantified Boolean formulas where every variable is quantified with either existential (∃) or universal (∀) quantifiers. In these formulas, all variables are bound at the beginning, leaving no free variables, which means the formula can ultimately be evaluated as true or false.
The evaluation considers all possible variable assignments under the constraints imposed by the quantifiers. A formula is considered a TQBF if, after applying all quantifications, the resulting expression evaluates to true. TQBF is notable for its role in complexity theory, particularly as a PSPACE-complete problem, indicating that it captures the complexity of problems solvable using a polynomial amount of space but for which the time complexity may still be non-polynomial.
An application of TQBF in AI and machine learning can be seen in the domain of automated theorem proving and logic programming, where the goal is to verify the truth of statements based on a set of axioms and inference rules. For instance, in a game-playing AI, TQBF can be used to evaluate strategies that involve a sequence of moves with uncertain outcomes, where the existential quantifiers represent the AI's moves (choices) and the universal quantifiers represent the opponent's moves (responses).
By formulating the game strategy as a TQBF, the AI can determine if a winning strategy exists, irrespective of the opponent's moves. This approach can be applied in complex games like chess or Go, where the AI needs to consider a vast number of possible move sequences and outcomes.