Abstract

The complexity class Θ2P, which is the class of languages recognizable by deterministic Turing machines in polynomial time with at most logarithmic many calls to an NP oracle, received extensive attention in the literature. Its complete problems can be characterized by different specific tasks, such as deciding whether the optimum solution of an NP problem is unique, or whether it is in some sense “odd” (e.g., whether its size is an odd number). In this paper, we introduce a new characterization of this class and its generalization ΘkP to the k-th level of the polynomial hierarchy. We show that problems in ΘkP are also those whose solution involves deciding, for two given sets A and B of instances of two Σk−1P-complete (or Πk−1P-complete) problems, whether the number of “yes”-instances in A is greater than those in B. Moreover, based on this new characterization, we provide a novel sufficient condition for ΘkP-hardness. We also define the general problem Comp-Validk, which is proven here Θk+1P-complete. Comp-Validk is the problem of deciding, given two sets A and B of quantified Boolean formulas with at most k alternating quantifiers, whether the number of valid formulas in A is greater than those in B. Notably, the problem Comp-Sat of deciding whether a set contains more satisfiable Boolean formulas than another set, which is a particular case of Comp-Valid1, demonstrates itself as a very intuitive Θ2P-complete problem. Nonetheless, to our knowledge, it eluded its formal definition to date. In fact, given its strict adherence to the count-and-compare semantics here introduced, Comp-Validk is among the most suitable tools to prove ΘkP-hardness of problems involving the counting and comparison of the number of “yes”-instances in two sets. We support this by showing that the Θ2P-hardness of the Max voting scheme over mCP-nets is easily obtained via the new characterization of ΘkP introduced in this paper.

Highlights

  • In the quest of characterizing the exact computational complexity of problems, many complexity classes have been defined, and hard problems for them have been, and are currently being, sought

  • We prove that the hardness of Comp-Validk holds even if |A| = |B|, all formulas in A, B are instances of QBFCk,N∃F, have the same number of clauses, and, for each 1 ≤ d ≤ k, quantifiers Qd of all formulas in A, B are defined on the very same set of variables

  • To prove that this restriction on the structure of the formulas does not influence the hardness of the problem, we show that a generic instance of Comp-Validk can always be rewritten in polynomial time in an instance fulfilling the required constraints

Read more

Summary

Introduction

In the quest of characterizing the exact computational complexity of problems, many complexity classes have been defined, and hard problems for them have been, and are currently being, sought. By exploiting the characterization given in this paper, we define the following general ΘPk+1-complete problem Comp-Validk (observe the different subscripts, k + 1 and k, respectively): given a pair A, B of sets of quantified Boolean formulas with at most k alternating quantifiers, decide whether the number of valid formulas in A is greater than those in B. To our knowledge, this is the first time that such a problem is proposed and shown to be ΘPk+1-complete.

Preliminaries
Decision problems and complexity classes
Prototypical hard problems
A previous characterization of ΘPk
Overview of results
Derivation of the general results
Complexity of Comp-Validk
Applications of the new characterization
Conclusion
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call