Complexity classes between $\Theta _k^P$ and $\Delta _k^P$

Lanthanides Final Amended With corrections _KP,MD_

3 Relations Between Randomized Complexity Classes and Other Complexity Classes

Proof Fix a satisfying assignment a. Define the distance of r to a, denoted dist(r, a), as the number of coordinates where ri 6= ai (this is called Hamming distance). Let r = r, r, r, · · · be the assignments generated by the algorithm, and let di = dist(r , a). Note that as we only change one bit of the assignment, di+1 = di + ∆i where ∆i ∈ {−1, 1}. We claim that Pr[∆i = −1] ≥ 1/2. Once this i...

Complexity Classes between Θ

We give different characterizations of the classes Plogi(NP), which are located between Θ2 and ∆2 . First we show that these classes are equal to the classes AC i−1(NP). Second we prove that they are also equivalent to certain binary search classes over NP. Third we show that they can be characterized as the classes defined by circuits of size O(log n) with NP oracle gates. All the proofs given...

Interpolating Between Quantum and Classical Complexity Classes

We reveal a natural algebraic problem whose complexity appears to interpolate between the well-known complexity classes BQP and NP: ⋆ Decide whether a univariate polynomial with exactly m monomial terms has a p-adic rational root. In particular, we show that while (⋆) is doable in quantum randomized polynomial time when m=2 (and no classical randomized polynomial time algorithm is known), (⋆) i...

On relations between counting communication complexity classes

We develop upper and lower bound arguments for counting acceptance modes of communication protocols. A number of separation results for counting communication complexity classes is established. This extends the investigation of the complexity of communication between two processors in terms of complexity classes initiated by Babai, Frankl, and Simon [Proc. 27th IEEE FOCS 1986, pp. 337{347] and ...