On the Complexity of the Permanent in Various Computational Models

We answer a question in [Landsberg, Ressayre, 2015], showing the regular determinantal complexity of the determinant det_m is O(m^3). We answer questions in, and generalize results of [Aravind, Joglekar, 2015], showing there is no rank one determinantal expression for perm_m or det_m when m >= 3. Finally we state and prove several "folklore" results relating different models of computation

On the Complexity of the Permanent in Various Computational Models

We answer a question in [Landsberg, Ressayre, 2015], showing the regular determinantal complexity of the determinant det_m is O(m^3). We answer questions in, and generalize results of [Aravind, Joglekar, 2015], showing there is no rank one determinantal expression for perm_m or det_m when m >= 3. Finally we state and prove several "folklore" results relating different models of computation

Quantum Discord and Quantum Computing - An Appraisal

We discuss models of computing that are beyond classical. The primary motivation is to unearth the cause of nonclassical advantages in computation. Completeness results from computational complexity theory lead to the identification of very disparate problems, and offer a kaleidoscopic view into the realm of quantum enhancements in computation. Emphasis is placed on the `power of one qubit' model, and the boundary between quantum and classical correlations as delineated by quantum discord. A recent result by Eastin on the role of this boundary in the efficient classical simulation of quantum computation is discussed. Perceived drawbacks in the interpretation of quantum discord as a relevant certificate of quantum enhancements are addressed.Comment: To be published in the Special Issue of the International Journal of Quantum Information on "Quantum Correlations: entanglement and beyond." 11 pages, 4 figure

Reliability and maintainability assessment factors for reliable fault-tolerant systems

A long term goal of the NASA Langley Research Center is the development of a reliability assessment methodology of sufficient power to enable the credible comparison of the stochastic attributes of one ultrareliable system design against others. This methodology, developed over a 10 year period, is a combined analytic and simulative technique. An analytic component is the Computer Aided Reliability Estimation capability, third generation, or simply CARE III. A simulative component is the Gate Logic Software Simulator capability, or GLOSS. The numerous factors that potentially have a degrading effect on system reliability and the ways in which these factors that are peculiar to highly reliable fault tolerant systems are accounted for in credible reliability assessments. Also presented are the modeling difficulties that result from their inclusion and the ways in which CARE III and GLOSS mitigate the intractability of the heretofore unworkable mathematics

Neurons and symbols: a manifesto

We discuss the purpose of neural-symbolic integration including its principles, mechanisms and applications. We outline a cognitive computational model for neural-symbolic integration, position the model in the broader context of multi-agent systems, machine learning and automated reasoning, and list some of the challenges for the area of neural-symbolic computation to achieve the promise of effective integration of robust learning and expressive reasoning under uncertainty

Exponential Time Complexity of the Permanent and the Tutte Polynomial

We show conditional lower bounds for well-studied #P-hard problems: (a) The number of satisfying assignments of a 2-CNF formula with n variables cannot be counted in time exp(o(n)), and the same is true for computing the number of all independent sets in an n-vertex graph. (b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed in time exp(o(n)). (c) The Tutte polynomial of an n-vertex multigraph cannot be computed in time exp(o(n)) at most evaluation points (x,y) in the case of multigraphs, and it cannot be computed in time exp(o(n/polylog n)) in the case of simple graphs. Our lower bounds are relative to (variants of) the Exponential Time Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF formulas cannot be decided in time exp(o(n)). We relax this hypothesis by introducing its counting version #ETH, namely that the satisfying assignments cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds, we transfer the sparsification lemma for d-CNF formulas to the counting setting