Classification Computational Complexity Research Articles

FV 12. Non-invasive Suppression of Essential Tremor via Phase-Locked Disruption of its Temporal Coherence

Aberrant neural oscillations hallmark the pathophysiology of numerous neurological and psychiatric disorders. Here, we first report a method to accurately track the phase of neural oscillations in real-time by a Hilbert transform that avoids the characteristic Gibbs distortion at the end of the signal, aka endpoint-corrected Hilbert transform (ecHT). The ecHT method maintains the same computational complexity class of the original Hilbert transform allowing implementation in simple digital hardware. We then used the ecHT method to show that the aberrant neural oscillation that hallmarks treatment-resistant essential tremor (ET), the most common adult movement disorder, can be noninvasively supressed via transcranial electrical stimulation at a fixed phase lag over the cerebellar hemisphere ipsilateral to the tremor movement. This was tested in a quadruple-replication randomized way including stimulation at 6 fixed phases, sham and without phase-locking. In a total of 11 subjects, the suppression of ET activity was sustained after the end of the stimulation and was phenomenologically predicted, post-hoc, from the features of the tremor movement before the start of the stimulation. To test for reproducibility, 6 of the original participants werer stimulated exactly the same way three years after the original experiments: the observed, significant stimulation response remained, i.e. responders continued to respond and non-responders did not. Finally, we used a highly-comparative feature extraction (> 8000 features) with statistical learning and neurophysiological computational modelling to show that the suppression of ET activity can be mechanistically attributed to a disruption of the temporal coherence in the tremor movement that can be originated in a higher bursting entropy at the cortico-olivo-cerebello-thalamic circuitry. The suppression of aberrant neural oscillation via phase-locked driven disruption of temporal coherence may represent a powerful neuromodulatory strategy to treat neurological and psychiatric disorders.

English

In this paper, an original reduction algorithm for solving simultaneous multivariate polynomial equations is presented. The algorithm is exponential in complexity, but the well-known algorithms, such as the extended Euclidean algorithm and Buchberger's algorithm, are superexponential. The superexponential complexity of the well-known algorithms is due to their not being "minimal" in a certain sense. Buchberger's algorithm produces a Groebner basis. The proposed original reduction algorithm achieves the required task, via computation of determinants of parametric Sylvester matrices, and produces a Rabin basis, which is shown to be minimal, when two multivariate polynomials are reduced at a time. The minimality of Rabin basis allows us to prove exponential lower bounds for the space complexity of an algebraic proof of certification, for a specific computational problem in the computational complexity class PSPACE, showing that the complexity classes PSPACE and P cannot be the same. By the same reasoning, it follows that Co-NP is not the same as either of the complexity classes NP and P, and that the polynomial time hierarchy does not collapse. It is also shown that the class BPP of languages decidable by bounded error probabilistic algorithms with (probabilistic) polynomial time proofs for the membership of input words is not the same as any one of the complexity classes P, NP and Co-NP. It follows again, from the discussions, that the complexity classes NP and P are not the same, by relativization of BPP, with respect to P and NP.

Quantum Computational Advantage with String Order Parameters of One-Dimensional Symmetry-Protected Topological Order.

Nonlocal games with advantageous quantum strategies give arguably the most fundamental demonstration of the power of quantum resources over their classical counterparts. Recently, certain multiplayer generalizations of nonlocal games have been used to prove unconditional separations between limited computational complexity classes of shallow-depth circuits. Here, we show advantageous strategies for these nonlocal games for generic ground states of one-dimensional symmetry-protected topological orders (SPTOs), when a discrete invariant of a SPTO known as a twist phase is nontrivial and -1. Our construction demonstrates that sufficiently large string order parameters of such SPTOs are indicative of globally constrained correlations useful for the unconditional computational separation.

On the Complexity of Finding the Maximum Entropy Compatible Quantum State

Herein we study the problem of recovering a density operator from a set of compatible marginals, motivated by limitations of physical observations. Given that the set of compatible density operators is not singular, we adopt Jaynes’ principle and wish to characterize a compatible density operator with maximum entropy. We first show that comparing the entropy of compatible density operators is complete for the quantum computational complexity class QSZK, even for the simplest case of 3-chains. Then, we focus on the particular case of quantum Markov chains and trees and establish that for these cases, there exists a procedure polynomial in the number of subsystems that constructs the maximum entropy compatible density operator. Moreover, we extend the Chow–Liu algorithm to the same subclass of quantum states.

Descriptive Complexity, Computational Tractability, and the Logical and Cognitive Foundations of Mathematics

In computational complexity theory, decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them. In theoretical computer science, it is commonly accepted that only functions for solving problems in the complexity class P, solvable by a deterministic Turing machine in polynomial time, are considered to be tractable. In cognitive science and philosophy, this tractability result has been used to argue that only functions in P can feasibly work as computational models of human cognitive capacities. One interesting area of computational complexity theory is descriptive complexity, which connects the expressive strength of systems of logic with the computational complexity classes. In descriptive complexity theory, it is established that only first-order (classical) systems are connected to P, or one of its subclasses. Consequently, second-order systems of logic are considered to be computationally intractable, and may therefore seem to be unfit to model human cognitive capacities. This would be problematic when we think of the role of logic as the foundations of mathematics. In order to express many important mathematical concepts and systematically prove theorems involving them, we need to have a system of logic stronger than classical first-order logic. But if such a system is considered to be intractable, it means that the logical foundation of mathematics can be prohibitively complex for human cognition. In this paper I will argue, however, that this problem is the result of an unjustified direct use of computational complexity classes in cognitive modelling. Placing my account in the recent literature on the topic, I argue that the problem can be solved by considering computational complexity for humanly relevant problem solving algorithms and input sizes.

On NP-hard graph properties characterized by the spectrum

Properties of graphs that can be characterized by the spectrum of the adjacency matrix of the graph have been studied systematically recently. Motivated by the complexity of these properties, we show that there are such properties for which testing whether a graph has that property can be NP-hard (or belong to other computational complexity classes consisting of even harder problems). In addition, we discuss a possible spectral characterization of some well-known NP-hard properties. In particular, for every integer k≥6 we construct a pair of k-regular cospectral graphs, where one graph is Hamiltonian and the other one not.

Transportation infrastructure network design in the presence of modal competition: computational complexity classification and a genetic algorithm

In this paper we analyze the computational complexity of transportation infrastructure network design problems, in the presence of a competing transportation mode. Some of these problems have previously been introduced in the literature. All problems studied have a common objective: the maximization of the number of travelers using the new network to be built. The differences between them are due to two factors. The first one is the constraints that the new network should satisfy: (1) budget constraint, (2) no-cycle constraint, (3) both constraints. The second factor is the topology of the network formed by the feasible links and stations: (1) a general network, (2) a forest. By combining these two factors, in total we analyze six problems, five of them are shown to be NP-hard, the sixth being trivial. Due to the NP-hardness of these problems, a genetic algorithm is proposed. Computational experiments show the applicability of this algorithm.

Quantum complexity of time evolution with chaotic Hamiltonians

We study the quantum complexity of time evolution in large-N chaotic systems, with the SYK model as our main example. This complexity is expected to increase linearly for exponential time prior to saturating at its maximum value, and is related to the length of minimal geodesics on the manifold of unitary operators that act on Hilbert space. Using the Euler-Arnold formalism, we demonstrate that there is always a geodesic between the identity and the time evolution operator e−iHt whose length grows linearly with time. This geodesic is minimal until there is an obstruction to its minimality, after which it can fail to be a minimum either locally or globally. We identify a criterion — the Eigenstate Complexity Hypothesis (ECH) — which bounds the overlap between off- diagonal energy eigenstate projectors and the k-local operators of the theory, and use it to argue that the linear geodesic will at least be a local minimum for exponential time. We show numerically that the large-N SYK model (which is chaotic) satisfies ECH and thus has no local obstructions to linear growth of complexity for exponential time, as expected from holographic duality. In contrast, we also study the case with N = 2 fermions (which is integrable) and find short-time linear complexity growth followed by oscillations. Our analysis relates complexity to familiar properties of physical theories like their spectra and the structure of energy eigenstates and has implications for the hypothesized computational complexity class separations PSPACE BQP/poly and PSPACE BQSUBEXP/subexp, and the “fast-forwarding” of quantum Hamiltonians.

Arithmetics based on computability logic

This paper is a brief survey of number theories based on em computability logic (CoL) a game-semantically conceived logic of computational tasks of resources. Such theories, termed em clarithmetics, are conservative extensions of first-order Peano arithmetic. 
 
 The first section of the paper lays out the conceptual basis of CoL and describes the relevant fragment of its formal language, with so called parallel connectives, choice connectives and quantifiers, and blind quantifiers. Both syntactically and semantically, this is a conservative generalization of the language of classical logic. 
 
 Clarithmetics, based on the corresponding fragment of CoL in the same sense as Peano 
 arithmetic is based on classical logic, are discussed in the second section. The axioms and inference rules of the system of clarithmetic named ${\bf CLA11}$ are presented, and the main results on this system are stated: constructive soundness, extensional completeness, and intensional completeness. 
 
 
 In the final section two potential applications of clarithmetics are addressed: clarithmetics as declarative programming languages in an extreme sense, and as tools for separating computational complexity classes. When clarithmetics or similar CoL-based theories are viewed as programming languages, programming reduces to proof-search, as programs can be mechanically extracted from proofs; such programs also serves as their own formal verifications, thus fully neutralizing the notorious (and generally undecidable) program verification problem. The second application reduces the problem of separating various computational complexity classes to separating the corresponding versions of clarithmetic, the potential benefits of which stem from the belief that separating theories should generally be easier than separating complexity classes directly. 

Two-agent scheduling with deteriorating jobs on a single parallel-batching machine: refining computational complexity

Tang et al. (Eur J Oper Res 263:401–411, 2017) have recently introduced a parallel-batching machine scheduling problem with linearly deteriorating jobs of two agents and presented a computational complexity classification of various special cases of this problem, including a number of NP-hardness proofs. We refine these results by demonstrating strong NP-hardness of several special cases, which are proved NP-hard in the ordinary sense in Tang et al. (Eur J Oper Res 263:401–411, 2017). Our reduction employs the problem studied in the first issue of Journal of Scheduling.

Superregular grammars do not provide additional explanatory power but allow for a compact analysis of animal song.

A pervasive belief with regard to the differences between human language and animal vocal sequences (song) is that they belong to different classes of computational complexity, with animal song belonging to regular languages, whereas human language is superregular. This argument, however, lacks empirical evidence since superregular analyses of animal song are understudied. The goal of this paper is to perform a superregular analysis of animal song, using data from gibbons as a case study, and demonstrate that a superregular analysis can be effectively used with non-human data. A key finding is that a superregular analysis does not increase explanatory power but rather provides for compact analysis: fewer grammatical rules are necessary once superregularity is allowed. This pattern is analogous to a previous computational analysis of human language, and accordingly, the null hypothesis, that human language and animal song are governed by the same type of grammatical systems, cannot be rejected.

Principled network reliability approximation: A counting-based approach

As engineered systems expand, become more interdependent, and operate in real-time, reliability assessment is key to inform investment and decision making. However, network reliability problems are known to be #P-complete, a computational complexity class believed to be intractable, and thus motivate the quest for approximations. Based on their theoretical foundations, reliability evaluation methods can be grouped as: (i) exact or bounds, (ii) guarantee-less sampling, and (iii) probably approximately correct (PAC). Group (i) is well regarded due to its useful byproducts, but it does not scale in practice. Group (ii) scales well and verifies desirable properties, such as the bounded relative error, but it lacks error guarantees. Group (iii) is of great interest when precision and scalability are required. We introduce K-RelNet, an extended counting-based method that delivers PAC guarantees for the K-terminal reliability problem. We also put our developments in context relative to classical and emerging techniques to facilitate dissemination. Then, we test in a fair way the performance of competitive methods using various benchmark systems. We note the range of application of algorithms and suggest a foundation for future computational reliability and resilience engineering, given the need for principled uncertainty quantification across complex networked systems.

Non-Line-of-Sight Identification Based on Unsupervised Machine Learning in Ultra Wideband Systems

Identification of line-of-sight (LOS) and non-line-of-sight (NLOS) propagation conditions is very useful in ultra wideband localization systems. In the identification, supervised machine learning is often used, but it requires exorbitant efforts to maintain and label the LOS and NLOS database. In this paper, we apply unsupervised machine learning approach called “expectation maximization for Gaussian mixture models” to classify LOS and NLOS components. The key advantage of applying unsupervised machine learning is that it does not require any rigorous and explicit labeling of the database at a certain location. The simulation results demonstrate that by using the proposed algorithm, LOS and NLOS signals can be classified with 86.50% correct rate, 12.70% false negative, and 0.8% false positive rate. We also compare the proposed algorithm with the existing cutting-edge supervised machine learning algorithms in terms of computational complexity and signals' classification performance.

Computational Complexity Relationship between Compaction, Vertex-Compaction, and Retraction

In this paper, we show a very close relationship between the compaction, vertex-compaction, and retraction problems for reflexive and bipartite graphs. Similar to a long-standing open problem concerning whether the compaction and retraction problems are polynomially equivalent, the relationships that we present relate to our problems concerning whether the compaction and vertex-compaction problems are polynomially equivalent, and whether the vertex-compaction and retraction problems are polynomially equivalent. The relationships that we present also relate to the constraint satisfaction problem, providing evidence that similar to the compaction and retraction problems, it is also likely to be difficult to give a complete computational complexity classification of the vertex-compaction problem for every fixed reflexive or bipartite graph. In this paper, we however give a complete computational complexity classification of the vertex-compaction problem for all graphs, including even partially reflexive graphs, with four or fewer vertices, by giving proofs based on mostly just knowing the computational complexity classification results of the compaction problem for such graphs determined earlier by the author. Our results show that the compaction, vertex-compaction, and retraction problems are polynomially equivalent for every graph with four or fewer vertices.

The more the merrier? Increasing group size may be detrimental to decision-making performance in nominal groups.

Demonstrability—the extent to which group members can recognize a correct solution to a problem—has a significant effect on group performance. However, the interplay between group size, demonstrability and performance is not well understood. This paper addresses these gaps by studying the joint effect of two factors—the difficulty of solving a problem and the difficulty of verifying the correctness of a solution—on the ability of groups of varying sizes to converge to correct solutions. Our empirical investigations use problem instances from different computational complexity classes, NP-Complete (NPC) and PSPACE-complete (PSC), that exhibit similar solution difficulty but differ in verification difficulty. Our study focuses on nominal groups to isolate the effect of problem complexity on performance. We show that NPC problems have higher demonstrability than PSC problems: participants were significantly more likely to recognize correct and incorrect solutions for NPC problems than for PSC problems. We further show that increasing the group size can actually decrease group performance for some problems of low demonstrability. We analytically derive the boundary that distinguishes these problems from others for which group performance monotonically improves with group size. These findings increase our understanding of the mechanisms that underlie group problem-solving processes, and can inform the design of systems and processes that would better facilitate collective decision-making.

Simple paths with exact and forbidden lengths

AbstractWe study new decision and optimization problems of finding a simple path between two given vertices in an arc weighted directed multigraph such that the path length is equal to a given number or it does not fall into the given forbidden intervals (gaps). A fairly complete computational complexity classification is provided and exact and approximation algorithms are suggested.

A PROPOSAL OF KC-SVM CLASSIFIER FOR EEG FEATURES OBTAINED THROUGH SF FILTERING WITH APPLICATION ON BCI DATA

The objective of this paper is to classify Brain Computer Interfacing (BCI) competition III, dataset IIIA with low computational complexity and high classification accuracy rate. This study presents the investigation of self-filtering (SF) as a pre-processing technique and Wigner trispectrum as feature extraction technique followed by the proposed [Formula: see text]-means clustering SVM (KC-SVM) classifier. KC-SVM is an improved form of classifier, which comprises KC-SVM with 10-fold cross validation technique. To analyze each stage of classification, different evaluation indexes of respective methods are considered. Substantial improvement on the performance of proposed classification method is scrutinized. Performance measurement accuracy 98.1%, Kappa coefficient 0.94, mutual information of feature vectors 0.14 and Fisher score 6.5 are achieved for considered dataset. In summary, this paper demonstrates that the classification accuracy of proposed classifier together with SF and Wigner–Ville Trispectrum (WT) satisfies the required criteria of classification.

Efficiently Controllable Graphs.

We investigate graphs that can be disconnected into small components by removing a vanishingly small fraction of their vertices. We show that, when a controllable quantum network is described by such a graph and the gaps in eigenfrequencies and in transition frequencies are bounded exponentially in the number of vertices, the network is efficiently controllable, in the sense that universal quantum computation can be performed using a control sequence polynomial in the size of the network while controlling a vanishingly small fraction of subsystems. We show that networks corresponding to finite-dimensional lattices are efficiently controllable and explore generalizations to percolation clusters and random graphs. We show that the classical computational complexity of estimating the ground state of Hamiltonians described by controllable graphs is polynomial in the number of subsystems or qubits.

Evolutionary dynamics on any population structure.

Evolution occurs in populations of reproducing individuals. The structure of a population can affect which traits evolve. Understanding evolutionary game dynamics in structured populations remains difficult. Mathematical results are known for special structures in which all individuals have the same number of neighbours. The general case, in which the number of neighbours can vary, has remained open. For arbitrary selection intensity, the problem is in a computational complexity class that suggests there is no efficient algorithm. Whether a simple solution for weak selection exists has remained unanswered. Here we provide a solution for weak selection that applies to any graph or network. Our method relies on calculating the coalescence times of random walks. We evaluate large numbers of diverse population structures for their propensity to favour cooperation. We study how small changes in population structure-graph surgery-affect evolutionary outcomes. We find that cooperation flourishes most in societies that are based on strong pairwise ties.

Quantum invariants of 3-manifolds and NP vs #P

The computational complexity class #P captures the di_culty of counting the satisfying assignments to a boolean formula. In this work, we use basic tools from quantum computation to give a proof th...