formula game

A formula game is an artificial game represented by a fully quantified Boolean formula such as $\backslash exists\; x\_1\; \backslash forall\; x\_2\; \backslash exists\; x\_3\; \backslash ldots\; \backslash psi$.

One player (E) has the goal of choosing values so as to make the formula $\backslash psi$ true, and selects values for the variables that are existentially quantified with $\backslash exists$. The opposing player (A) has the goal of making the formula $\backslash psi$ false, and selects values for the variables that are universally quantified with $\backslash forall$. 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 $\backslash Phi$ such that Player E has a winning strategy in the game represented by $\backslash Phi$. 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 $\backslash psi$ true, no matter what choice Player A makes.