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Example text
The "alternation of quantifiers" in QBF leads to an interesting game-theoretic interpretation. SATISFIABILITY is the prototype of a solitaire game, where the player has to look for some configuration with a particular property or for a sequence of simple moves leading to some particular goal state. The alternating quantifiers turn QBF into a two-person game. Two players Alice and Emily in turn choose the truth values to be assigned to existentially or universally quantified variables in the order of their nesting inside the formula.
This situation is illustrated by the following recognition problem: It is not hard to see that on a RAM the above language can be recognized on-line by reading the words w; as addresses, and storing a 1 in the corresponding registers. , whereas it requires space O(m k) if the measure based on sizeb is used. The latter amount of space is a provable lower bound for Turing machine on-line recognition, since the Turing machine on-line acceptor must write a full description of its input on its worktapes before it can process the last word w0.
This example leaves the situation open for the case of off-line computations, but it is not difficult to see that also in this case O(n) nodes on an SMM are capable of storing as much information as O(n · log n) tape squares on a Turing machine. The basic technique consists of a A-structure which represents a cycle of O(n) nodes with in each node a pointer to some other node in the cycle. The latter pointer stores in this way O(log n) bits of information, and it is a matter of simple programming to show that this representation can be used and updated, using no more than O(log n) additional nodes.