The language TQBF is a formal language in computer science that contains True Quantified Boolean Formulas. A fully quantified boolean formula is a formula in firstorder logic where every variable is quantified (or bound), using either existential or universal quantifiers, at the beginning of the sentence. Any such formula is always either true or false (since there are no free variables). If such a formula evaluates to true, then that formula is in the language TQBF. It is also known as QSAT (Quantified SAT).
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In computational complexity theory, the quantified Boolean formula problem (QBF) is a generalization of the Boolean satisfiability problem in which both existential quantifiers and universal quantifiers can be applied to each variable. Put another way, it asks whether a firstorder logic sentential form over a set of Boolean variables is true or false. For example, the following is an instance of QBF:
QBF is the canonical complete problem for PSPACE, the class of problems solvable by a deterministic or nondeterministic Turing machine in polynomial space and unlimited time.^{[1]} Given the formula in the form of an abstract syntax tree, the problem can be solved easily by a set of mutually recursive procedures which evaluate the formula. Such an algorithm uses space proportional to the height of the tree, which is linear in the worst case, but uses time exponential in the number of quantifiers.
Provided that MA ⊂ PSPACE, which is widely believed, QBF cannot be solved, nor can a given solution even be verified, in either deterministic or probabilistic polynomial time (in fact, unlike the satisfiability problem, there's no known way to specify a solution succinctly). It is trivial to solve using an alternating Turing machine in linear time, which is no surprise since in fact AP = PSPACE, where AP is the class of problems alternating machines can solve in polynomial time.^{[2]}
When the seminal result IP = PSPACE was shown (see interactive proof system), it was done by exhibiting an interactive proof system that could solve QBF by solving a particular arithmetization of the problem.^{[3]}
QBF formulas have a number of useful canonical forms. For example, it can be shown that there is a polynomialtime manyone reduction that will move all quantifiers to the front of the formula and make them alternate between universal and existential quantifiers. There is another reduction that proved useful in the IP = PSPACE proof where no more than one universal quantifier is placed between each variable's use and the quantifier binding that variable. This was critical in limiting the number of products in certain subexpressions of the arithmetization.
A fully quantified Boolean formula can be assumed to have a very specific form, called prenex normal form. It has two basic parts: a portion containing only quantifiers and a portion containing an unquantified Boolean formula usually denoted as φ. If there are n Boolean variables, the entire formula can be written as
where every variable falls within the scope of some quantifier. By introducing dummy variables, any formula in prenex normal form can be converted into a sentence where existential and universal quantifiers alternate. Using the dummy variable y_{1},
The second sentence has the same truth value but follows the restricted syntax. Assuming fully quantified Boolean formulas to be in prenex normal form is a frequent feature of proofs.
There is a simple recursive algorithm for determining whether a TQBF is true. Given some QBF
If the formula contains no quantifiers, we can just return the formula. Otherwise, we take off the first quantifier and check both possible values for the first variable:
If , then return . If , then return .
How fast does this algorithm run? For every quantifier in the initial QBF, the algorithm makes two recursive calls on only a linearly smaller subproblem. This gives the algorithm an exponential runtime O(2^n).
How much space does this algorithm use? Within each invocation of the algorithm, it needs to store the intermediate results of computing A and B. Every recursive call takes off one quantifier, so the total recursive depth is linear in the number of quantifiers. Formulas that lack quantifiers can be evaluated in space logarithmic in the number of variables. The initial QBF was fully quantified, so there are at least as many quantifiers as variables. Thus, this algorithm uses O(n + log n) = O(n) space. This makes the TQBF language part of the PSPACE complexity class.
The TQBF language serves in complexity theory as the canonical PSPACEcomplete problem. Being PSPACEcomplete means that a language is in PSPACE and that the language is also PSPACEhard. The algorithm above shows that TQBF is in PSPACE. Showing that TQBF is PSPACEhard requires showing that any language in the complexity class PSPACE can be reduced to TQBF in polynomial time. I.e.,
This means that, for a PSPACE language L, whether an input x is in L can be decided by checking whether f(x) is in TQBF, for some function f that is required to run in polynomial time (relative to the length of the input) Symbolically,
Proving that TQBF is PSPACEhard, requires specification of f.
So, suppose that L is a PSPACE language. This means that L can be decided by a polynomial space deterministic Turing machine (DTM). This is very important for the reduction of L to TQBF, because the configurations of any such Turing Machine can be represented as Boolean formulas, with Boolean variables representing the state of the machine as well as the contents of each cell on the Turing Machine tape, with the position of the Turing Machine head encoded in the formula by the formula's ordering. In particular, our reduction will use the variables c_{1} and c_{2}, which represent two possible configurations of the DTM for L, and a natural number t, in constructing a QBF which is true if and only if the DTM for L can go from the configuration encoded in c_{1} to the configuration encoded in c_{2} in no more than t steps. The function f, then, will construct from the DTM for L a QBF , where c_{star t} is the DTM's starting configuration, c_{acce pt} is the DTM's accepting configuration, and T is the maximum number of steps the DTM could need to move from one configuration to the other. We know that T = O(exp(n)), where n is the length of the input, because this bounds the total number of possible configurations of the relevant DTM. Of course, it cannot take the DTM more steps than there are possible configurations to reach c_{accept} unless it enters a loop, in which case it will never reach c_{accept} anyway.
At this stage of the proof, we have already reduced the question of whether an input formula w (encoded, of course, in c_{star t}) is in L to the question of whether the QBF , i.e., f(w), is in TQBF. The remainder of this proof proves that f can be computed in polynomial time.
For t = 1, computation of is straightforwardeither one of the configurations changes to the other in one step or it does not. Since the Turing Machine that our formula represents is deterministic, this presents no problem.
For t > 1, computation of involves a recursive evaluation, looking for a socalled "middle point" m_{1}. In this case, we rewrite the formula as follows:
This converts the question of whether c_{1} can reach c_{2} in t steps to the question of whether c_{1} reaches a middle point m_{1} in t / 2 steps, which itself reaches c_{2} in t / 2 steps. The answer to the latter question of course gives the answer to the former.
Now, t is only bounded by T, which is exponential (and so not polynomial) in the length of the input. Additionally, each recursive layer virtually doubles the length of the formula. (The variable m_{1} is only one midpointfor greater t, there are more stops along the way, so to speak.) So the time required to recursively evaluate in this manner could be exponential as well, simply because the formula could become exponentially large. This problem is solved by universally quantifying using variables c_{3} and c_{4} over the configuration pairs (e.g., {(c_{1},m_{1}),( m_{1},c_{2})}), which prevents the length of the formula from expanding due to recursive layers. This yields the following interpretation of :
This version of the formula can indeed be computed in polynomial time, since any one instance of it can be computed in polynomial time. The universally quantified ordered pair simply tells us that whichever choice of (c_{3},c_{4}) is made, .
Thus, , so TQBF is PSPACEhard. Together with the above result that TQBF is in PSPACE, this completes the proof that TQBF is a PSPACEcomplete language.
(This proof follows Sipser 2006 pp. 310313 in all essentials. Papadimitriou 1994 also includes a proof.)
