Incremental Recompilation of Knowledge

Approximating a general formula from above and below by Horn formulas (its Horn envelope and Horn core, respectively) was proposed by Selman and Kautz (1991, 1996) as a form of ``knowledge compilation,'' supporting rapid approximate reasoning; on the negative side, this scheme is static in that it supports no updates, and has certain complexity drawbacks pointed out by Kavvadias, Papadimitriou and Sideri (1993). On the other hand, the many frameworks and schemes proposed in the literature for theory update and revision are plagued by serious complexity-theoretic impediments, even in the Horn case, as was pointed out by Eiter and Gottlob (1992), and is further demonstrated in the present paper. More fundamentally, these schemes are not inductive, in that they may lose in a single update any positive properties of the represented sets of formulas (small size, Horn structure, etc.). In this paper we propose a new scheme, incremental recompilation, which combines Horn approximation and model-based updates; this scheme is inductive and very efficient, free of the problems facing its constituents. A set of formulas is represented by an upper and lower Horn approximation. To update, we replace the upper Horn formula by the Horn envelope of its minimum-change update, and similarly the lower one by the Horn core of its update; the key fact which enables this scheme is that Horn envelopes and cores are easy to compute when the underlying formula is the result of a minimum-change update of a Horn formula by a clause. We conjecture that efficient algorithms are possible for more complex updates.Comment: See http://www.jair.org/ for any accompanying file

Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy

We consider quantum computations comprising only commuting gates, known as IQP computations, and provide compelling evidence that the task of sampling their output probability distributions is unlikely to be achievable by any efficient classical means. More specifically we introduce the class post-IQP of languages decided with bounded error by uniform families of IQP circuits with post-selection, and prove first that post-IQP equals the classical class PP. Using this result we show that if the output distributions of uniform IQP circuit families could be classically efficiently sampled, even up to 41% multiplicative error in the probabilities, then the infinite tower of classical complexity classes known as the polynomial hierarchy, would collapse to its third level. We mention some further results on the classical simulation properties of IQP circuit families, in particular showing that if the output distribution results from measurements on only O(log n) lines then it may in fact be classically efficiently sampled.Comment: 13 page

First-order transition in small-world networks

The small-world transition is a first-order transition at zero density $p$ of shortcuts, whereby the normalized shortest-path distance undergoes a discontinuity in the thermodynamic limit. On finite systems the apparent transition is shifted by $\Delta p \sim L^{-d}$. Equivalently a ``persistence size'' $L^* \sim p^{-1/d}$ can be defined in connection with finite-size effects. Assuming $L^* \sim p^{-\tau}$, simple rescaling arguments imply that $\tau=1/d$. We confirm this result by extensive numerical simulation in one to four dimensions, and argue that $\tau=1/d$ implies that this transition is first-order.Comment: 4 pages, 3 figures, To appear in Europhysics Letter

High-Performance Bioinstrumentation for Real-Time Neuroelectrochemical Traumatic Brain Injury Monitoring

Traumatic brain injury (TBI) has been identified as an important cause of death and severe disability in all age groups and particularly in children and young adults. Central to TBIs devastation is a delayed secondary injury that occurs in 30–40% of TBI patients each year, while they are in the hospital Intensive Care Unit (ICU). Secondary injuries reduce survival rate after TBI and usually occur within 7 days post-injury. State-of-art monitoring of secondary brain injuries benefits from the acquisition of high-quality and time-aligned electrical data i.e., ElectroCorticoGraphy (ECoG) recorded by means of strip electrodes placed on the brains surface, and neurochemical data obtained via rapid sampling microdialysis and microfluidics-based biosensors measuring brain tissue levels of glucose, lactate and potassium. This article progresses the field of multi-modal monitoring of the injured human brain by presenting the design and realization of a new, compact, medical-grade amperometry, potentiometry and ECoG recording bioinstrumentation. Our combined TBI instrument enables the high-precision, real-time neuroelectrochemical monitoring of TBI patients, who have undergone craniotomy neurosurgery and are treated sedated in the ICU. Electrical and neurochemical test measurements are presented, confirming the high-performance of the reported TBI bioinstrumentation

Tetris is Hard, Even to Approximate

In the popular computer game of Tetris, the player is given a sequence of tetromino pieces and must pack them into a rectangular gameboard initially occupied by a given configuration of filled squares; any completely filled row of the gameboard is cleared and all pieces above it drop by one row. We prove that in the offline version of Tetris, it is NP-complete to maximize the number of cleared rows, maximize the number of tetrises (quadruples of rows simultaneously filled and cleared), minimize the maximum height of an occupied square, or maximize the number of pieces placed before the game ends. We furthermore show the extreme inapproximability of the first and last of these objectives to within a factor of p^(1-epsilon), when given a sequence of p pieces, and the inapproximability of the third objective to within a factor of (2 - epsilon), for any epsilon>0. Our results hold under several variations on the rules of Tetris, including different models of rotation, limitations on player agility, and restricted piece sets.Comment: 56 pages, 11 figure

Matching Kasteleyn Cities for Spin Glass Ground States

As spin glass materials have extremely slow dynamics, devious numerical methods are needed to study low-temperature states. A simple and fast optimization version of the classical Kasteleyn treatment of the Ising model is described and applied to two-dimensional Ising spin glasses. The algorithm combines the Pfaffian and matching approaches to directly strip droplet excitations from an excited state. Extended ground states in Ising spin glasses on a torus, which are optimized over all boundary conditions, are used to compute precise values for ground state energy densities.Comment: 4 pages, 2 figures; minor clarification

The computational difficulty of finding MPS ground states

We determine the computational difficulty of finding ground states of one-dimensional (1D) Hamiltonians which are known to be Matrix Product States (MPS). To this end, we construct a class of 1D frustration free Hamiltonians with unique MPS ground states and a polynomial gap above, for which finding the ground state is at least as hard as factoring. By lifting the requirement of a unique ground state, we obtain a class for which finding the ground state solves an NP-complete problem. Therefore, for these Hamiltonians it is not even possible to certify that the ground state has been found. Our results thus imply that in order to prove convergence of variational methods over MPS, as the Density Matrix Renormalization Group, one has to put more requirements than just MPS ground states and a polynomial spectral gap.Comment: 5 pages. v2: accepted version, Journal-Ref adde

Positivity of energy for asymptotically locally AdS spacetimes

We derive necessary conditions for the spinorial Witten-Nester energy to be well-defined for asymptotically locally AdS spacetimes. We find that the conformal boundary should admit a spinor satisfying certain differential conditions and in odd dimensions the boundary metric should be conformally Einstein. We show that these conditions are satisfied by asymptotically AdS spacetimes. The gravitational energy (obtained using the holographic stress energy tensor) and the spinorial energy are equal in even dimensions and differ by a bounded quantity related to the conformal anomaly in odd dimensions.Comment: 36 pages, 1 figure; minor corrections, JHEP versio

Exactly solvable models of adaptive networks

A satisfiability (SAT-UNSAT) transition takes place for many optimization problems when the number of constraints, graphically represented by links between variables nodes, is brought above some threshold. If the network of constraints is allowed to adapt by redistributing its links, the SAT-UNSAT transition may be delayed and preceded by an intermediate phase where the structure self-organizes to satisfy the constraints. We present an analytic approach, based on the recently introduced cavity method for large deviations, which exactly describes the two phase transitions delimiting this adaptive intermediate phase. We give explicit results for random bond models subject to the connectivity or rigidity percolation transitions, and compare them with numerical simulations.Comment: 4 pages, 4 figure

Phase transition for cutting-plane approach to vertex-cover problem

We study the vertex-cover problem which is an NP-hard optimization problem and a prototypical model exhibiting phase transitions on random graphs, e.g., Erdoes-Renyi (ER) random graphs. These phase transitions coincide with changes of the solution space structure, e.g, for the ER ensemble at connectivity c=e=2.7183 from replica symmetric to replica-symmetry broken. For the vertex-cover problem, also the typical complexity of exact branch-and-bound algorithms, which proceed by exploring the landscape of feasible configurations, change close to this phase transition from "easy" to "hard". In this work, we consider an algorithm which has a completely different strategy: The problem is mapped onto a linear programming problem augmented by a cutting-plane approach, hence the algorithm operates in a space OUTSIDE the space of feasible configurations until the final step, where a solution is found. Here we show that this type of algorithm also exhibits an "easy-hard" transition around c=e, which strongly indicates that the typical hardness of a problem is fundamental to the problem and not due to a specific representation of the problem.Comment: 4 pages, 3 figure