One-way communication complexity and the Neciporuk lower bound on formula size

@article{Klauck2007OnewayCC,
  title={One-way communication complexity and the Neciporuk lower bound on formula size},
  author={Hartmut Klauck},
  journal={SIAM J. Comput.},
  year={2007},
  volume={37},
  pages={552-583}
}
  • H. Klauck
  • Published 29 November 2001
  • Computer Science
  • SIAM J. Comput.
In this paper the Neciporuk method for proving lower bounds on the size of Boolean formulas is reformulated in terms of one-way communication complexity. We investigate the settings of probabilistic formulas, nondeterministic formulas, and quantum formulas. In all cases we can use results about one-way communication complexity to prove lower bounds on formula size. The main results regarding formula size are as follows: We show a polynomial size gap between probabilistic/quantum and… 
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References

SHOWING 1-10 OF 68 REFERENCES
On the Size of Probabilistic Formulae
  • H. Klauck
  • Mathematics, Computer Science
    ISAAC
  • 1997
TLDR
A lower bound method for probabilistic formulae based on the VC-dimension and the Neciporuk lower bound is devised and lower bounds on the size of 2-sided bounded error, Monte Carlo, and Las Vegas formULae over an arbitrary basis are given.
Lower Bounds for Quantum Communication Complexity
TLDR
This work generalizes a method for proving classical communication complexity lower bounds to the quantum case, and develops several other Fourier based lower bound methods, notably showing that /spl radic/(s~(f)/log n) n, for the average sensitivity of a function f, yields a lower bound on the bounded error quantum communication complexity.
Quantum Formulas: A Lower Bound and Simulation
TLDR
It is shown that Nechiporuk's method for proving lower bounds for Boolean formulas can be extended to the quantum case, leading to an $\Omega(n^2/\log^2 n)$ lower bound for quantum formulas computing an explicit function.
Tradeoffs between Nondeterminism and Complexity for Communication Protocols and Branching Programs
TLDR
A general technique is described here which allows to prove explicit lower bounds on the size of oblivious branching programs in an easy and transparent way and yields new insights into the power of nondeterminism and randomness for the considered models.
Lower bounds for quantum computation and communication
TLDR
This dissertation establishes limitations on the ways in which the exponentially many degrees of freedom hidden in quantum states may be accessed, and derives nearly optimal lower bounds for several problems in this model, including that of approximating the median.
An Almost-Quadratic Lower Bound for Quantum Formula Size
We show that Nechiporuk's method for proving lower bound for Boolean formulas can be extended to the quantum case. This leads to an n^2 / log^2 n lower bound for quantum formulas computing an
On quantum and probabilistic communication: Las Vegas and one-way protocols
TLDR
It is proved that for oneway protocols computing total functions quantum Las Vegas communication is asymptotical ly as efficient as exact quantum communication, which is exactly asefficient as determinist ic communication.
Quantum vs. classical communication and computation
TLDR
A simple and general simulation technique is presented that transforms any black-box quantum algorithm to a quantum communication protocol for a related problem, in a way that fully exploits the quantum parallelism, to obtain new positive and negative results.
Lower bounds for computation with limited nondeterminism
  • H. Klauck
  • Computer Science
    Proceedings. Thirteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat. No.98CB36247)
  • 1998
TLDR
A rounds-communication hierarchy for communication complexity with limited nondeterministic bits solving an open problem and the effect of limited nondeterminism on monotone circuit depth is investigated.
Communication complexity lower bounds by polynomials
  • H. Buhrman, R. D. Wolf
  • Computer Science
    Proceedings 16th Annual IEEE Conference on Computational Complexity
  • 2001
TLDR
The "log rank" lower bound extends to the strongest variant of quantum communication complexity (qubit communication+unlimited prior entanglement) and the polynomial equivalence of quantum and classical communication complexity for various classes of functions is proved.
...
...