Exponential Separation Research Articles

Building Strategies into QBF Proofs

Strategy extraction is of great importance for quantified Boolean formulas (QBF), both in solving and proof complexity. So far in the QBF literature, strategy extraction has been algorithmically performed from proofs. Here we devise the first QBF system where (partial) strategies are built into the proof and are piecewise constructed by simple operations along with the derivation. This has several advantages: (1) lines of our calculus have a clear semantic meaning as they are accompanied by semantic objects; (2) partial strategies are represented succinctly (in contrast to some previous approaches); (3) our calculus has strategy extraction by design; and (4) the partial strategies allow new sound inference steps which are disallowed in previous central QBF calculi such as Q-Resolution and long-distance Q-Resolution. The last item (4) allows us to show an exponential separation between our new system and the previously studied reductionless long-distance resolution calculus. Our approach also naturally lifts to dependency QBFs (DQBF), where it yields the first sound and complete CDCL-style calculus for DQBF, thus opening future avenues into CDCL-based DQBF solving.

The Emergence of Chaos in Quantum Mechanics

Nonlinearity in Quantum Mechanics may have extrinsic or intrinsic origins and is a liable route to a chaotic behaviour that can be of difficult observations. In this paper, we propose two forms of nonlinear Hamiltonian, which explicitly depend upon the phase of the wave function and produce chaotic behaviour. To speed up the slow manifestation of chaotic effects, a resonant laser field assisting the time evolution of the systems causes cumulative effects that might be revealed, at least in principle. The nonlinear Schrödinger equation is solved within the two-state approximation; the solution displays features with characteristics similar to those found in chaotic Classical Mechanics: sensitivity on the initial state, dense power spectrum, irregular filling of the Poincaré map and exponential separation of the trajectories of the Bloch vector σ in the Bloch sphere.

Dimension of ergodic measures projected onto self‐similar sets with overlaps

For self-similar sets on $\mathbb{R}$ satisfying the exponential separation condition we show that the natural projections of shift invariant ergodic measures is equal to $\min\{1,\frac{h}{-\chi}\}$, where $h$ and $\chi$ are the entropy and Lyapunov exponent respectively. The proof relies on Shmerkin's recent result on the $L^{q}$ dimension of self-similar measures. We also use the same method to give results on convolutions and orthogonal projections of ergodic measures projected onto self-similar sets.

The Fermi-Pasta-Ulam Quantum Recurrence in The Dynamics of an Elementary Physical Vacuum Cell and The Problem of its Polarization

A model of a Quantum recurrence in the dynamics of an elementary physical vacuum cell within the framework of four coupled Shrodinger equations has been suggested. The model of an elementary vacuum cell shows that a Quantum recurrence which represents the dynamics of virtual transformations in the cell, qualitatively differs from that of Poincare and the Fermi-Pasta-Ulam. Whereas these recurrences develop in time or space, the Quantum recurrence develops in a sequence of Fourier images represented by non exponentially separating functions. The sequence experiences random energy additions but no exponential separation occurs. The Quantum recurrence can be defined as the most frequent array of Fourier images that appear in a certain quantum system during a period of its observation. Different scenarios of the Fourier images sequences interpreted as bosons (electron and positron) and fermions (photons) apearing in the solutions of the model demonstrate that during some periods of its observation they become indistinguishable. The quantum dynamics of every physical vacuum cell depends on the dynamics of many other vacuum cells interacting with it, thus the quasi periodicity (during the period of observation) of the Fourier images recurrence can have infinite periods of time and space and the amplitudes of the Fourier images can vary many orders in their magnitudes. Such recurrence times does not correspond even roughly to the Poincare recurrence time of an isolated macroscopic system. It reminds the behavior of spatially coupled standard mappings with different parameters. The amount of energy in the physical vacuum is infinite but extracting a part of it and converting, it into a time-space form requires a process of periodical transfer of the reversible microscopic system dynamics into that of a macroscopic system. This process can be realized through a resonant interaction between the classical and quantum recurrences developing in these two systems. However, a technical realization of this problem is problematic.

Separation Between Read-once Oblivious Algebraic Branching Programs (ROABPs) and Multilinear Depth-three Circuits

We show an exponential separation between two well-studied models of algebraic computation, namely, read-once oblivious algebraic branching programs (ROABPs) and multilinear depth-three circuits. In particular, we show the following: (1) There exists an explicit n -variate polynomial computable by linear sized multilinear depth-three circuits (with only two product gates) such that every ROABP computing it requires 2 Ω (n) size. (2) Any multilinear depth-three circuit computing IMM n,d (the iterated matrix multiplication polynomial formed by multiplying d , n × n symbolic matrices) has n Ω( d ) size. IMM n,d can be easily computed by a poly( n,d ) sized ROABP. (3) Further, the proof of (2) yields an exponential separation between multilinear depth-four and multilinear depth-three circuits: There is an explicit n -variate, degree d polynomial computable by a poly( n ) sized multilinear depth-four circuit such that any multilinear depth-three circuit computing it has size n Ω(d) . This improves upon the quasi-polynomial separation of Reference [36] between these two models. The hard polynomial in (1) is constructed using a novel application of expander graphs in conjunction with the evaluation dimension measure [15, 33, 34, 36], while (2) is proved via a new adaptation of the dimension of the partial derivatives measure of Reference [32]. Our lower bounds hold over any field.

Dynamic QBF Dependencies in Reduction and Expansion

We provide the first proof complexity results for QBF dependency calculi. By showing that the reflexive resolution path dependency scheme admits exponentially shorter Q-resolution proofs on a known family of instances, we answer a question first posed by Slivovsky and Szeider in 2014 [37]. Further, we conceive a method of QBF solving in which dependency recomputation is utilised as a form of inprocessing. Formalising this notion, we introduce a new version of Q-resolution in which a dependency scheme is applied dynamically. We demonstrate the further potential of this approach beyond that of the existing static system with an exponential separation. Last, we show that the same picture emerges in an analogous approach to the universal expansion paradigm.

Exponential Separation of Communication and External Information

We show an exponential gap between communication complexity and external information complexity by analyzing a communication task suggested as a candidate by Braverman [A Hard-to-Compress Interacti...

Exponential Separations in the Energy Complexity of Leader Election

Energy is often the most constrained resource for battery-powered wireless devices, and most of the energy is often spent on transceiver usage (i.e., transmitting and receiving packets) rather than computation. In this article, we study the energy complexity of fundamental problems in several models of wireless radio networks. It turns out that energy complexity is very sensitive to whether the devices can generate random bits and their ability to detect collisions . We consider four collision detection models: Strong-CD (in which transmitters and listeners detect collisions), Sender-CD (in which only transmitters detect collisions), Receiver-CD (in which only listeners detect collisions), and No-CD (in which no one detects collisions). The take-away message of our results is quite surprising. For randomized algorithms, there is an exponential gap between the energy complexity of Sender-CD and Receiver-CD: Randomized: No-CD = Sender-CD > Receiver-CD = Strong-CD and for deterministic algorithms, there is another exponential gap in energy complexity, but in the reverse direction : Deterministic: No-CD = Receiver-CD > Sender-CD = Strong-CD Precisely, the randomized energy complexity of Leader Election is Θ(log * n ) in Sender-CD but Θ(log(log * n )) in Receiver-CD, where n is the number of devices, which is unknown to the devices at the beginning; the deterministic complexity of Leader Election is Θ(log N ) in Receiver-CD but Θ(log log N ) in Sender-CD, where N is the size of the ID space. There is a tradeoff between time and energy. We provide a new upper bound on the time-energy tradeoff curve for randomized algorithms. A critical component of this algorithm is a new deterministic Leader Election algorithm for dense instances, when n = Θ( N ), with inverse Ackermann energy complexity.

New Resolution-Based QBF Calculi and Their Proof Complexity

Modern QBF solvers typically use two different paradigms, conflict-driven clause learning (CDCL) solving or expansion solving. Proof systems for quantified Boolean formulas (QBFs) provide a theoretical underpinning for the performance of these solvers, with Q-Resolution and its extensions relating to CDCL solving and ∀Exp+Res relating to expansion solving. This article defines two novel calculi, which are resolution-based and enable unification of some of the principal existing resolution-based QBF calculi, namely Q-resolution, long-distance Q-resolution and the expansion-based calculus ∀Exp+Res. However, the proof complexity of the QBF resolution proof systems is currently not well understood. In this article, we completely determine the relative power of the main QBF resolution systems, settling in particular the relationship between the two different types of resolution-based QBF calculi: proof systems for CDCL-based solvers (Q-resolution, universal, and long-distance Q-resolution) and proof systems for expansion-based solvers (∀Exp+Res and its generalizations IR-calc and IRM-calc defined here). The most challenging part of this comparison is to exhibit hard formulas that underlie the exponential separations of the aforementioned proof systems. To this end, we exhibit a new and elegant proof technique for showing lower bounds in QBF proof systems based on strategy extraction. This technique provides a direct transfer of circuit lower bounds to lengths-of-proofs lower bounds. We use our method to show the hardness of a natural class of parity formulas for Q-resolution and universal Q-resolution. Variants of the formulas are hard for even stronger systems such as long-distance Q-resolution and extensions. With a completely different and novel counting argument, we show the hardness of the prominent formulas of Kleine Büning et al. [51] for the strong expansion-based calculus IR-calc.

Krein–Rutman type property and exponential separation of a noncompact operator

We investigate exponentially separated property for a noncompact linear operator T T on a Banach space. We obtain the relationship between exponentially separated property and the well-known Krein–Rutman type property for a noncompact operator. Under the assumption of an essential spectral gap, we prove that any u u -bounded operator T T with a reproducing cone admits the exponentially separated property and, hence, is of Krein–Rutman type automatically. We also establish an amenable sufficient condition for the exponentially separated property of some degenerate linear parabolic systems.

Characterization of dominated splittings for operator cocycles acting on Banach spaces

Versions of the Oseledets multiplicative ergodic theorem for cocycles acting on infinite-dimensional Banach spaces have been investigated since the pioneering work of Ruelle in 1982 and are a topic of continuing research interest. For a cocycle to induce a continuous splitting in which the growth in one subbundle exponentially dominates the growth in a complementary subbundle requires additional assumptions; a necessary and sufficient condition for the existence of such a dominated splitting (also known as exponential separation) was recently given by J. Bochi and N. Gourmelon for invertible finite-dimensional cocycles in discrete time. We extend this result to cocycles of injective, bounded, and possibly noncompact linear maps acting on Banach spaces (in both discrete and continuous time) using an essentially geometric approach based on a notion of approximate singular value decomposition in Banach spaces. Our method is constructive, and in the finite-dimensional case yields explicit growth estimates on the dominated splitting which may be of independent interest.

Three-stage phase separation kinetics in a model liquid binary mixture: A computational study

We study microscopic aspects of initial phase separation through atomistic molecular dynamics simulation of a structure breaking liquid binary mixture. We find that the phase separation kinetics in a fluid binary mixture model system can indeed be unusual. It can be fast, with a crossover from a pronounced exponential to non-exponential and non-linear dynamics. An important outcome of this work is the quantification of time scales involved in phase separation kinetics at an early stage. The initial exponential phase separation is complete within ∼100 ps. The initial phase separation involves aggregation of small droplets that form rapidly after the quench. This is followed by segregation that gives rise to pattern formation with multiple bands of segregated species. During this initial phase, a particle is found to have moved only about ∼5 molecular diameters. The next stage is slower and characterized by break-up and disappearance of small islands of species trapped inside the domains of other species of the binary mixture. The phase separation in this second stage is highly non-exponential and power-law-like. We identify a new feature in the very late stage of phase separation kinetics that seems to have eluded previous attention, the smoothing of the rugged interface between the two species. This is opposite to the roughening transition one finds on the surface of solids in contact with its vapor phase. The present atomistic simulation provides a molecular picture in terms of molecular motions and displacements.

Experimental Quantum Switching for Exponentially Superior Quantum Communication Complexity.

Finding exponential separation between quantum and classical information tasks is like striking gold in quantum information research. Such an advantage is believed to hold for quantum computing but is proven for quantum communication complexity. Recently, a novel quantum resource called the quantum switch-which creates a coherent superposition of the causal order of events, known as quantum causality-has been harnessed theoretically in a new protocol providing provable exponential separation. We experimentally demonstrate such an advantage by realizing a superposition of communication directions for a two-party distributed computation. Our photonic demonstration employs d-dimensional quantum systems, qudits, up to d=2^{13} dimensions and demonstrates a communication complexity advantage, requiring less than (0.696±0.006) times the communication of any causally ordered protocol. These results elucidate the crucial role of the coherence of communication direction in achieving the exponential separation for the one-way processing task, and open a new path for experimentally exploring the fundamentals and applications of advanced features of indefinite causal structures.

An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model

Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge coloring. For most problems...

Determination and verification of triaxiality dependent cohesive zone parameters of SA333 Grade 6 steel

Cohesive zone models are employed to simulate crack propagation in fracture process zone. It has been demonstrated in the present study that the cohesive zone parameters depend not only on the material, but also on stress triaxiality. To determine cohesive zone parameters, experimental results of 14 three-point bend specimens (TPBB) made of SA333 Grade 6 steel were analyzed using 3D finite element model using WARP3D. Cohesive parameters were determined by carrying out parametric studies of these specimens by varying peak stress ‘T’ to match experimental load-displacement data with the computed data. An exponential traction separation law was used for this purpose. In addition, elastic-plastic finite element analyses (FEA) are conducted to determine the multi-axiality quotient ‘q’ at the crack tip for TPBB specimens and pipe components. This helped to get a relation between peak stress ‘T’ and ‘q’. To extend the range of validity of parameter ‘T’ as a function of ‘q’, experimental results of six piping components with through-wall circumferential crack made of SA333 Grade 6 steel tested earlier were also used similarly. A confidence interval band between q and T is then plotted to study the transferability of cohesive parameters. The variation in ‘T’ for the same value of ‘q’ is attributed to the microstructural changes in the material near crack tip.To test the accuracy of the cohesive parameters determined above, experimental results of six piping components were again used. Using the ‘q’ parameters determined from elastic-plastic FEA on pipes, the upper and lower limits of peak stress ‘T’ from the confidence band determined above were found out. This range of ‘T’ was then used to carry out cohesive zone analyses (CZA) of these piping components to determine load-displacement curves. For each piping component, a set of load-displacement curves were determined by assuming a normal variation of parameter ‘T’ over the maximum and minimum limits. The simulation results were compared with the experimental results and it has been observed that experimental results lie reasonably well within the computed results.

Entropy and Minimal Bit Rates for State Estimation and Model Detection

We study a notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">estimation entropy</i> for continuous-time nonlinear systems, formulated in terms of the number of system trajectories that approximate all other trajectories up to an exponentially decaying error. We also consider an alternative definition of estimation entropy, which uses approximating functions that are not necessarily trajectories of the system, and show that the two entropy notions are equivalent. We establish an upper bound on the estimation entropy in terms of the sum of the desired convergence rate and an upper bound on the matrix measure of the Jacobian, multiplied by the system dimension. A lower bound on the estimation entropy is developed as well. We then turn our attention to state estimation and model detection with quantized and sampled state measurements. We describe an iterative procedure that uses such measurements to generate state estimates that converge to the true state at the desired exponential rate. The average bit rate utilized by this procedure matches the derived upper bound on the estimation entropy, and no other algorithm of this type can perform the same estimation task with bit rates lower than the estimation entropy. Finally, we discuss an application of the estimation procedure in determining, from the quantized state measurements, which of two competing models of a dynamical system is the true model. We show that under a mild assumption of “exponential separation” of the candidate models, detection always happens in finite time.

Strongly Exponentially Separated Linear Systems

In the study of linear differential systems, an important concept is that of exponential separation. It is closely related to the concept of exponential dichotomy and has played a key role in the theory of Lyapunov exponents. Usually in the study of exponential separation, it is assumed that the coefficient matrix A(t) is bounded in norm. Our first aim here is to develop a theory of exponential separation which applies to unbounded systems. It turns that in order to have a reasonable theory it is necessary to add the assumption that the angle between the two separated subspaces is bounded below (note this follows automatically for bounded systems). Our second aim is to show that if a bounded linear Hamiltonian system is exponentially separated into two subspaces of the same dimension, then it must have an exponential dichotomy.

Rumor spreading with bounded in-degree

In the gossip-based model of communication for disseminating information in a network, in each time unit, every node u can contact a single random neighbor v but can possibly be contacted by many nodes.In the present paper, we consider a restricted model where at each node only one incoming call can be answered in one time unit. We study the implied weaker version of the well-studied pull protocol, which we call restricted pull.We prove an exponential separation of the rumor spreading time between two variants of the protocol (the answered call among a set of calls is chosen adversarially or uniformly at random). Further, we show that if the answered call is chosen randomly, the slowdown of restricted pull versus the classic pull protocol can w.h.p. be upper bounded by O(ψ(G)⋅log⁡n), where ψ(G)=max{u,v}∈E⁡d(u)/d(v)≤Δδ with Δ and δ being the largest and smallest degree of the network G=(V,E) and d(u) being the degree of u.

Classical versus quantum communication in XOR games

We introduce an intermediate setting between quantum nonlocality and communication complexity problems. More precisely, we study the value of XOR games when Alice and Bob are allowed to use a limited amount (c bits) of one-way classical communication compared to their value when they are allowed to use the same amount of one-way quantum communication (c qubits). The key quantity here is the ratio between the quantum and classical value. We provide a universal way to obtain Bell inequality violations of general Bell functionals from XOR games for which the previous quotient is larger than 1. This allows, in particular, to find (unbounded) Bell inequality violations from communication complexity problems in the same spirit as the recent work by Buhrman et al. (PNAS 113(12):3191–3196, 2016). We also provide an example of a XOR game for which the previous quotient is optimal (up to a logarithmic factor) in terms of the amount of information c. Interestingly, this game has only polynomially many inputs per player. For the related problem of separating the classical versus quantum communication complexity of a function, the known examples attaining exponential separation require exponentially many inputs per party.

Fluid particles only separate exponentially in the dissipation range of turbulence after extremely long times

In this paper we consider how the statistical moments of the separation between two fluid particles grow with time when their separation lies in the dissipation range of turbulence. In this range the fluid velocity field varies smoothly, and the relative velocity of two fluid particles depends linearly upon their separation. While this may suggest that the rate at which fluid particles separate is exponential in time, this is not guaranteed because the strain-rate governing their separation is a strongly fluctuating quantity in turbulence. Indeed, the recent paper by Afik \& Steinberg (Nat. Commun. \textbf{8}: 468, 2017) argues that there is no convincing evidence that the moments of the separation between fluid particles grow exponentially with time in the dissipation range of turbulence. Motivated by this, we use Direct Numerical Simulations (DNS) to compute the moments of the particle separation over very long periods of time to see if we ever see evidence for exponential separation. Our results show that if the initial separation between the particles is infinitesimal, the moments of the particle separation first grow as power laws in time, but we then observe, for the first time, convincing evidence that at sufficiently long times the moments do grow exponentially. However, this exponential growth is only observed after extremely long times $\gtrsim 200\tau_\eta$, where $\tau_\eta$ is the Kolmogorov timescale. This is due to fluctuations in the strain-rate about its mean value measured along the particle trajectories, the effect of which on the moments of the particle separation persists for very long times. We also consider the Backward-in-Time (BIT) moments of the particle separation, and observe that they too grow exponentially in the long-time regime. However, a dramatic consequence of the exponential separation is that at long-times the difference between..