Solving QSAT in Sublinear Depth

Abstract

Among PSPACE-complete problems, QSAT, or quantified SAT, is one of the most used to show that the class of problems solvable in polynomial time by families of a given variant of P systems includes the whole PSPACE. However, most solutions require a membrane nesting depth that is linear with respect to the number of variables of the QSAT instance under consideration. While a system of a certain depth is needed, since depth 1 systems only allows to solve problems in P #P , it was until now unclear if a linear depth was, in fact, necessary. Here we use P systems with active membranes with charges, and we provide a construction that proves that QSAT can be solved with a sublinear nesting depth of order n log n , where n is the number of variables in the quantified formula given as input.

Bounding the space in P systems with active membranes

“…Recently, it has been shown in [18] that such a structure is not strictly necessary; in fact, a solution for QSAT (in 3-CNF variant, see [27]) can be obtained using a membrane structure with a sublinear nesting depth of order n/log(n), where n is the number of variables in the quantified formula given as input.…”

“…Each level of the membrane structure is dedicate to verify a single quantifier. We have a quantification tree [18], where each level acts as an OR gate (it checks whether or not at least one needed object is present) or as an AND gate (all needed objects are present).…”

“…Even if this is the most intuitive way to approach the problem, it turned out recently [18] that such a structure is not strictly necessary: QSAT can be solved with a sublinear nesting depth of order n/log(n) (where n is the number of variables in the quantified formula given as input).…”

P systems attacking hard problems beyond NP: a survey

“…Specifically, P systems with active membranes and membrane structure of a constant depth d > 1 are able to solve problems in the complexity class , where is the dth level of the counting hierarchy [23], on one hand. On the other hand, only sub-linear depth of membrane structure is enough to reach the class PSPACE [26].…”

A new method to simulate restricted variants of polarizationless P systems with active membranes

“…It is also widely investigated how certain restrictions on P systems with active membrane affect the computation power of these systems (see for example [6,8,9,11,13,14,16,17,19,20,25]). Probably, the most investigated question in this research line is whether these P systems are still powerful enough to solve hard problems in polynomial time when the polarizations of the membranes are not used.…”