Exponential Separation of Quantum and Classical Non-Interactive Multi-Party Communication Complexity

We give the first exponential separation between quantum and classical multi-party communication complexity in the (non-interactive) one-way and simultaneous message passing settings. For every k, we demonstrate a relational communication problem between k parties that can be solved exactly by a quantum simultaneous message passing protocol of cost O(log n) and requires protocols of cost n^{c/k^2}, where c>0 is a constant, in the classical non-interactive one-way message passing model with shared randomness and bounded error. Thus our separation of corresponding communication classes is superpolynomial as long as k=o(\sqrt{\log n / \log\log n}) and exponential for k=O(1)

Min-Rank Conjecture for Log-Depth Circuits

A completion of an m-by-n matrix A with entries in {0,1,*} is obtained by setting all *-entries to constants 0 or 1. A system of semi-linear equations over GF(2) has the form Mx=f(x), where M is a completion of A and f:{0,1}^n --> {0,1}^m is an operator, the i-th coordinate of which can only depend on variables corresponding to *-entries in the i-th row of A. We conjecture that no such system can have more than 2^{n-c\cdot mr(A)} solutions, where c>0 is an absolute constant and mr(A) is the smallest rank over GF(2) of a completion of A. The conjecture is related to an old problem of proving super-linear lower bounds on the size of log-depth boolean circuits computing linear operators x --> Mx. The conjecture is also a generalization of a classical question about how much larger can non-linear codes be than linear ones. We prove some special cases of the conjecture and establish some structural properties of solution sets.Comment: 22 pages, to appear in: J. Comput.Syst.Sci

A lower bound for the pigeonhole principle in tree-like Resolution by asymmetric Prover-Delayer games

In this note we show that the asymmetric Prover–Delayer game developed in Beyersdorff et al. (2010) [2] for Parameterized Resolution is also applicable to other tree-like proof systems. In particular, we use this asymmetric Prover–Delayer game to show a lower bound of the form 2Ω(nlogn) for the pigeonhole principle in tree-like Resolution. This gives a new and simpler proof of the same lower bound established by Iwama and Miyazaki (1999) [7] and Dantchev and Riis (2001) [5]

A note on applicability of the incompleteness theorem to human mind

AbstractWe shall present some relations between consistency and reflection principles which explain why is Gödel's incompleteness theorem wrongly used to argue that thinking machines are impossible