P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches

While the P vs NP problem is mainly approached form the point of view of discrete mathematics, this paper proposes reformulations into the field of abstract algebra, geometry, fourier analysis and of continuous global optimization - which advanced tools might bring new perspectives and approaches for this question. The first one is equivalence of satisfaction of 3-SAT problem with the question of reaching zero of a nonnegative degree 4 multivariate polynomial (sum of squares), what could be tested from the perspective of algebra by using discriminant. It could be also approached as a continuous global optimization problem inside $[0,1]^n$, for example in physical realizations like adiabatic quantum computers. However, the number of local minima usually grows exponentially. Reducing to degree 2 polynomial plus constraints of being in $\{0,1\}^n$, we get geometric formulations as the question if plane or sphere intersects with $\{0,1\}^n$. There will be also presented some non-standard perspectives for the Subset-Sum, like through convergence of a series, or zeroing of $\int_0^{2\pi} \prod_i \cos(\varphi k_i) d\varphi$ fourier-type integral for some natural $k_i$. The last discussed approach is using anti-commuting Grassmann numbers $\theta_i$, making $(A \cdot \textrm{diag}(\theta_i))^n$ nonzero only if $A$ has a Hamilton cycle. Hence, the P$\ne$NP assumption implies exponential growth of matrix representation of Grassmann numbers. There will be also discussed a looking promising algebraic/geometric approach to the graph isomorphism problem -- tested to successfully distinguish strongly regular graphs with up to 29 vertices.Comment: 19 pages, 8 figure

Retrieve-and-Read: Multi-task Learning of Information Retrieval and Reading Comprehension

This study considers the task of machine reading at scale (MRS) wherein, given a question, a system first performs the information retrieval (IR) task of finding relevant passages in a knowledge source and then carries out the reading comprehension (RC) task of extracting an answer span from the passages. Previous MRS studies, in which the IR component was trained without considering answer spans, struggled to accurately find a small number of relevant passages from a large set of passages. In this paper, we propose a simple and effective approach that incorporates the IR and RC tasks by using supervised multi-task learning in order that the IR component can be trained by considering answer spans. Experimental results on the standard benchmark, answering SQuAD questions using the full Wikipedia as the knowledge source, showed that our model achieved state-of-the-art performance. Moreover, we thoroughly evaluated the individual contributions of our model components with our new Japanese dataset and SQuAD. The results showed significant improvements in the IR task and provided a new perspective on IR for RC: it is effective to teach which part of the passage answers the question rather than to give only a relevance score to the whole passage.Comment: 10 pages, 6 figure. Accepted as a full paper at CIKM 201