A formula game is an artificial game represented by a fully quantified Boolean formula such as ${\displaystyle \exists x_{1}\forall x_{2}\exists x_{3}\ldots \psi }$.

One player (E) has the goal of choosing values so as to make the formula ${\displaystyle \psi }$ true, and selects values for the variables that are existentially quantified with ${\displaystyle \exists }$. The opposing player (A) has the goal of making the formula ${\displaystyle \psi }$ false, and selects values for the variables that are universally quantified with ${\displaystyle \forall }$. The players take turns according to the order of the quantifiers, each assigning a value to the next bound variable in the original formula. Once all variables have been assigned values, Player E wins if the resulting expression is true.

In computational complexity theory, the language FORMULA-GAME is defined as all formulas ${\displaystyle \Phi }$ such that Player E has a winning strategy in the game represented by ${\displaystyle \Phi }$. FORMULA-GAME is PSPACE-complete because it is exactly the same decision problem as True quantified Boolean formula. Player E has a winning strategy exactly when every choice they must make in a game has a truth assignment that makes ${\displaystyle \psi }$ true, no matter what choice Player A makes.

• Sipser, Michael. (2006). Introduction to the Theory of Computation. Boston: Thomson Course Technology.

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