Unbounded Length Research Articles

Scientific article summarization model with unbounded input length

In recent years, the exponential growth of scientific literature has made it increasingly difficult for researchers and practitioners to keep up with new discoveries and developments in their fields. Thanks to this, text summarization has become one of the primary tasks of natural language processing. Abstractive summarization of long documents, such as scientific articles, requires large neural networks with high memory and computation requirements. Therefore, it is all the more important to find ways to increase the efficiency of long document summarization models. The objects of this research are long document summarization transformer models and the Unlimiformer cross-attention modification. The article reviews the basic principles of transformer attention, which constitutes the primary computational expense in transformer models. More efficient self-attention approaches used for long document summarization models are described, such as the global+sliding window attention used by Longformer. The cross-attention mechanism of Unlimiformer, which allows a model to have unbounded input length, is described in detail. The objective of the study is the development and evaluation of a long document summarization model using the Unlimiformer modification. To achieve this goal, a Longformer Decoder-Encoder model pretrained on the arXiv dataset is modified with Unlimiformer cross-attention. This modification can be applied without additional model fine-tuning, avoiding the cost of further training a large sequence length model. The developed model was evaluated on the arXiv dataset using the ROUGE-1, ROUGE-2 and ROUGE-L metrics. The developed model showed improved results compared to the baseline model, demonstrating the viability of using this approach to improve long document summarization models.

Quantum key distribution with unbounded pulse correlations

Abstract Typical security proofs of quantum key distribution (QKD) require that the emitted signals are independent and identically distributed. In practice, however, this assumption is not met because intrinsic device flaws inevitably introduce correlations between the emitted signals. Although analyses addressing this issue have been recently proposed, they only consider a restrictive scenario in which the correlations have a finite and known maximum length that is much smaller than the total number of emitted signals. While it is expected that the magnitude of the correlations decreases as the pulse separation increases, the assumption that this magnitude is exactly zero after a certain point does not seem to have any physical justification. Concerningly, this means that the available analyses cannot guarantee the security of current QKD implementations. Here, we solve this pressing problem by developing a rigorous framework that, when combined with existing results, can guarantee security against pulse correlations of unbounded length. Our framework is rather general and could be applied to other situations for which the existing analyses consider a scenario that differs slightly from the actual one.

Stability of KdV equation on a network with bounded and unbounded lengths

In this work, we studied the exponential stability of the nonlinear KdV equation posed on a finite star shaped network with finite number of branches. On each branch of the network we define a KdV equation posed on a finite domain [[EQUATION]] or the half-line [[EQUATION]]. We start by proving well-posedness and some regularity results. Then, we state the exponential stability of the linear KdV equation by acting with a damping term on some branches. The main idea is to prove a suitable observability inequality. In the nonlinear case, we obtain two kinds of results: The first result holds for small amplitude solutions, and is proved using a perturbation argument from the linear case but without acting on all edges. The second result is a semiglobal stability result, and it is obtained by proving an observability inequality directly for the nonlinear system, but we need to act with damping terms on all the branches. In this case, we are able to prove the stabilization in weighted spaces.

Ionic strength alters crosslinker-driven self-organization of microtubules.

The microtubule cytoskeleton is a major structural element inside cells that directs self-organization using microtubule-associated proteins and motors. It has been shown that finite-sized, spindle-like microtubule organizations, called "tactoids," can form in vitro spontaneously from mixtures of tubulin and the antiparallel crosslinker, MAP65, from the MAP65/PRC1/Ase family. Here, we probe the ability of MAP65 to form tactoids as a function of the ionic strength of the buffer to attempt to break the electrostatic interactions binding MAP65 to microtubules and inter-MAP65 binding. We observe that, with increasing monovalent salts, the organizations change from finite tactoids to unbounded length bundles, yet the MAP65 binding and crosslinking appear to stay intact. We further explore the effects of ionic strength on the dissociation constant of MAP65 using both microtubule pelleting and single-molecule binding assays. We find that salt can reduce the binding, yet salt never negates it. Instead, we believe that the salt is affecting the ability of the MAP65 to form phase-separated droplets, which cause the nucleation and growth of tactoids, as recently demonstrated.

Mixed-integer programming models for irregular strip packing based on vertical slices and feasibility cuts

The irregular strip-packing problem, also known as nesting or marker making, is defined as the automatic computation of a non-overlapping placement of a set of non-convex polygons onto a rectangular strip of fixed width and unbounded length, such that the strip length is minimized. Nesting methods based on heuristics are a mature technology, and currently, the only practical solution to this problem. However, recent performance gains of the Mixed-Integer Programming (MIP) solvers, together with the known limitations of the heuristics methods, have encouraged the exploration of exact optimization models for nesting during the last decade. Despite the research effort, there is room to improve the efficiency of the current family of exact MIP models for nesting. In order to bridge this gap, this work introduces a new family of continuous MIP models based on a novel formulation of the NoFit-Polygon Covering Model (NFP-CM), called NFP-CM based on Vertical Slices (NFP-CM-VS). Our new family of MIP models is based on a new convex decomposition of the feasible space of relative placements between pieces into vertical slices, together with a new family of valid inequalities, symmetry breakings, and variable eliminations derived from the former convex decomposition. Our experiments show that our new NFP-CM-VS models outperform the current state-of-the-art MIP models. Ten instances are solved up to optimality within one hour for the first time, including one with 27 pieces. Finally, we provide a detailed reproducibility protocol and dataset as supplementary material to allow the exact replication of our models, experiments, and results.

Higher-order games with dependent types

In previous work on higher-order games, we accounted for finite games of unbounded length by working with continuous outcome functions, which carry implicit game trees. In this work we make such trees explicit. We use concepts from dependent type theory to capture history-dependent games, where the set of available moves at a given position in the game depends on the moves played up to that point. In particular, games are modelled by a W-type, which is essentially the same type used by Aczel to model constructive Zermelo-Frankel set theory (CZF). We have also implemented all our definitions, constructions, results and proofs in the dependently-typed programming language Agda, which, in particular, allows us to run concrete examples of computations of optimal strategies, that is, strategies in subgame perfect equillibrium.

Large cycles in generalized Johnson graphs

AbstractWe count cycles of an unbounded length in generalized Johnson graphs. Asymptotics of the number of such cycles is obtained for certain growth rates of the cycle length.

Unbounded Predicate Inner Product Functional Encryption from Pairings

Predicate inner product functional encryption (P-IPFE) is essentially attribute-based IPFE (AB-IPFE) which additionally hides attributes associated to ciphertexts. In a P-IPFE, a message {textbf {x}} is encrypted under an attribute {textbf {w}} and a secret key is generated for a pair ({textbf {y}}, {textbf {v}}) such that recovery of langle {{textbf {x}}}, {{textbf {y}}}rangle requires the vectors {textbf {w}}, {textbf {v}} to satisfy a linear relation. We call a P-IPFE unbounded if it can encrypt unbounded length attributes and message vectors. bullet zero predicate IPFE. We construct the first unbounded zero predicate IPFE (UZP-IPFE) which recovers langle {{textbf {x}}}, {{textbf {y}}}rangle if langle {{textbf {w}}}, {{textbf {v}}}rangle =0. This construction is inspired by the unbounded IPFE of Tomida and Takashima (ASIACRYPT 2018) and the unbounded zero inner product encryption of Okamoto and Takashima (ASIACRYPT 2012). The UZP-IPFE stands secure against general attackers capable of decrypting the challenge ciphertext. Concretely, it provides full attribute-hiding security in the indistinguishability-based semi-adaptive model under the standard symmetric external Diffie–Hellman assumption. bullet non-zero predicate IPFE. We present the first unbounded non-zero predicate IPFE (UNP-IPFE) that successfully recovers langle {{textbf {x}}}, {{textbf {y}}}rangle if langle {{textbf {w}}}, {{textbf {v}}}rangle ne 0. We generically transform an unbounded quadratic FE (UQFE) scheme to weak attribute-hiding UNP-IPFE in both public and secret key setting. Interestingly, our secret key simulation secure UNP-IPFE has succinct secret keys and is constructed from a novel succinct UQFE that we build in the random oracle model. We leave the problem of constructing a succinct public key UNP-IPFE or UQFE in the standard model as an important open problem.

Short attribute-based signatures for arbitrary Turing machines from standard assumptions

This paper presents the first attribute-based signature (ABS) scheme supporting signing policies representable by Turing machines (TM), based on well-studied computational assumptions. Our work supports arbitraryTMs as signing policies in the sense that the TMs can accept signing attribute strings of unbounded polynomial length and there is no limit on their running time, description size, or space complexity. Moreover, we are able to achieve input-specific running time for the signing algorithm. All other known expressive ABS schemes could at most support signing policies realizable by either arbitrary polynomial-size circuits or TMs having a pre-determined upper bound on the running time. Consequently, those schemes can only deal with signing attribute strings whose lengths are a priori bounded, as well as suffers from the worst-case running time problem. On a more positive note, for the first time in the literature, the signature size of our ABS scheme only depends on the size of the signed message and is completely independent of the size of the signing policy under which the signature is generated. This is a significant achievement from the point of view of communication efficiency. Our ABS construction makes use of indistinguishability obfuscation (IO) for polynomial-size circuits and certain IO-compatible cryptographic tools. Note that, all of these building blocks including IO for polynomial-size circuits are currently known to be realizable under well-studied computational assumptions.

Decomposition of a symbolic element over a countable amenable group into blocks approximating ergodic measures

Consider a subshift over a finite alphabet, X\subset \Lambda^\mathbb{Z} (or X\subset\Lambda^{\mathbb{N}_0} ). With each finite block B\in\Lambda^k appearing in X we associate the empirical measure ascribing to every block C\in\Lambda^l the frequency of occurrences of C in B . By comparing the values ascribed to blocks C we define a metric on the combined space of blocks B and probability measures \mu on X , whose restriction to the space of measures is compatible with the weak- \star topology. Next, in this combined metric space we fix an open set \mathcal{U} containing all ergodic measures, and we say that a block B is “ergodic” if B\in\mathcal{U} . In this paper we prove the following main result: Given \varepsilon>0 , every x\in X decomposes as a concatenation of blocks of bounded lengths in such a way that, after ignoring a set M of coordinates of upper Banach density smaller than \varepsilon , all blocks in the decomposition are ergodic. We also prove a finitistic version of this theorem (about decomposition of long blocks), and a version about decomposition of x\in X into finite blocks of unbounded lengths. The second main result concerns subshifts whose set of ergodic measures is closed. We show that, in this case, no matter how x\in X is partitioned into blocks (as long as their lengths are sufficiently large and bounded), excluding a set M of upper Banach density smaller than \varepsilon , all blocks in the decomposition are ergodic. The first half of the paper is concluded by examples showing, among other things, that the small set M , in both main theorems, cannot be avoided. The second half of the paper is devoted to generalizing the two main results described above to subshifts X\subset\Lambda^G with the action of a countable amenable group G . The role of long blocks is played by blocks whose domains are members of a Følner sequence, while the decomposition of x\in X into blocks (of which majority are ergodic) is obtained with the help of a congruent system of tilings.

Distributed graph pattern matching via bounded dual simulation

Graph Pattern Matching (GPM) finds subgraphs of a large data graph that are similar to an input query graph. It has many applications, such as pattern recognition, detecting plagiarism, and finding communities in social networks. Current real-world applications generate massive amounts of data that cannot be stored on the memory of a single machine, which raises the need for distributed storage and processing. Recent relaxed GPM models, although of polynomial time complexity, are nevertheless not distributed by nature. Moreover, the existing relaxed GPM algorithms are limited in terms of scalability. In this paper, we propose Bounded Dual Simulation (BDSim) as a new relaxed model for a scalable evaluation of GPM in massive graphs. BDSim captures more semantic similarities compared to graph simulation, dual simulation, and even strong simulation. It preserves the vertices’ proximity by eliminating cycles of unbounded length from the resulting match graph. Furthermore, we propose distributed vertex-centric algorithms to evaluate BDSim. We prove their effectiveness and efficiency through detailed theoretical validation and extensive experiments conducted on real-world and synthetic datasets. To the best of our knowledge, BDSim is the first relaxed GPM model that captures the cyclic structure of the query graph while being feasible in cubic time.

Optimal cyclic locally repairable codes with unbounded length from their zeros

In distributed storage systems, locally repairable codes (LRCs) can effectively reduce the complexity of repairing failed nodes. In this paper, we constructed six classes of optimal cyclic (r,δ)-LRCs with unbounded length and minimum distance δ+2≤d≤2δ from their zeros. Two classes of optimal cyclic (r,δ)-LRCs with large minimum distance are also presented.

Graphs containing finite induced paths of unbounded length

The age $\mathcal{A}(G)$ of a graph $G$ (undirected and without loops) is the collection of finite induced subgraphs of $G$, considered up to isomorphy and ordered by embeddability. It is well-quasi-ordered (wqo) for this order if it contains no infinite antichain. A graph is \emph{path-minimal} if it contains finite induced paths of unbounded length and every induced subgraph $G'$ with this property embeds $G$. We construct $2^{\aleph_0}$ path-minimal graphs whose ages are pairwise incomparable with set inclusion and which are wqo. Our construction is based on uniformly recurrent sequences and lexicographical sums of labelled graphs.

Online Coloring and a New Type of Adversary for Online Graph Problems

We introduce a new type of adversary for online graph problems thus allowing parametrized analysis of online algorithms with respect to a natural parameter. The new adversary is parameterized by a single integer \(\kappa \), which upper bounds the number of connected components that the adversary can use at any time during the presentation of the online graph G. We call this adversary “\(\kappa \)-components-bounded”, or \(\kappa \)-CB for short. On one hand, this adversary is restricted compared to the classical adversary because of the \(\kappa \)-CB constraint. On the other hand, we seek competitive ratios parameterized only by \(\kappa \) with no dependence on the input length n, thereby giving the new adversary power to use arbitrarily large inputs. We study online coloring under the \(\kappa \)-CB adversary. We obtain a finer analysis of the existing algorithms FirstFit and CBIP by computing their competitive ratios on trees and bipartite graphs under the new adversary: (1) Perhaps surprisingly, FirstFit outperforms CBIP on trees; (2) The competitive ratio of CBIP on bipartite graphs is simply \(\kappa \). We also study several well known classes of graphs, such as 3-colorable, \(C_k\)-free, d-inductive, planar, and bounded treewidth, with respect to online coloring under the \(\kappa \)-CB adversary. We demonstrate that the extra adversarial power of unbounded input length outweighs the restriction on the number of connected components leading to non-existence of competitive algorithms for these classes.

Limit theorems for chains with unbounded variable length memory which satisfy Cramer condition

We consider a class of variable length Markov chains with a binary alphabet in which context tree is defined by adding finite trees with uniformly bounded height to the vertices of an infinite comb tree. Such type of Markov chain models the spike neuron patterns and also extends the class of persistent random walks. The main interest is the limiting properties of the empirical distribution of symbols from the alphabet. We obtain the strong law of large numbers, central limit theorem, and exact asymptotics for large and moderate deviations. The presence of an intrinsic renewal structure is the subject of discussion in the literature. Proofs are based on the construction of a renewals of the chain and the applying corresponding properties of the compound (or generalized) renewal processes.

New Constructions of Optimal Locally Repairable Codes With Super-Linear Length

As an important coding scheme in modern distributed storage systems, locally repairable codes (LRCs) have attracted a lot of attentions from perspectives of both practical applications and theoretical research. As a major topic in the research of LRCs, bounds and constructions of the corresponding optimal codes are of particular concerns. In this work, codes with (r,δ)-locality which have optimal minimal distance w.r.t. the bound given by Prakash et al. are considered. Through parity-check matrix approach, constructions of both optimal (r,δ)-LRCs with all symbol locality ( (r,δ) <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sub> -LRCs) and optimal (r,δ)-LRCs with information locality ( (r,δ) <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> -LRCs) are provided. As a generalization of a work of Xing and Yuan, these constructions are built on a connection between sparse hypergraphs and optimal (r,δ)-LRCs. With the help of constructions of large sparse hypergraphs, the lengths of codes obtained from our construction can be super-linear in the alphabet size. This improves upon previous constructions when the minimal distance of the code is at least 3δ+1. As two applications, optimal H-LRCs with super-linear lengths and GSD codes with unbounded lengths are also constructed.

Causal Path Identification for Timed and Sequential Circuits

Self-timed and pulse mode circuit modules are often implemented using combinational logic with feedback, or require timing constraints for the circuit to function correctly. Identifying causal paths through these circuit modules is a fundamental requirement to enable automatic system level timing closure, performance analysis, and timing-driven optimizations when using static timing analysis algorithms. A methodology and algorithm for identifying causal paths in cyclic sequential circuits under arbitrary timing models are presented. The algorithm identifies causal paths in a sequential circuit implemented with standard cell gates. The algorithm is demonstrated by identifying causal paths for over 100 asynchronous finite-state machines and timed circuits designed using a variety of timing models. Cyclic paths, pulse generating paths, and unbounded length self-enabling loops are identified inside these circuit blocks. Some of the causal paths in these designs have never been previously identified. No preliminary versions of this work have been previously published.

Cyclic and Convolutional Codes With Locality

Locally recoverable (LRC) codes and their variants have been extensively studied in recent years. In this paper we focus on cyclic constructions of LRC codes and derive conditions on the zeros of the code that support the property of hierarchical locality. As a result, we obtain a general family of hierarchical LRC codes for a new range of code parameters. We also observe that our approach enables one to represent an LRC code in quasicyclic form, and use this representation to construct tail-biting convolutional LRC codes with locality. Among other results, we extend the general approach to cyclic codes with locality to multidimensional cyclic codes, yielding new families of LRC codes with availability, and construct a family of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -ary cyclic hierarchical LRC codes of unbounded length.

Inverse single facility location problem on a tree with balancing on the distance of server to clients

&lt;p style='text-indent:20px;'&gt;We introduce a case of inverse single facility location problem on a tree where by minimum modifying in the length of edges, the difference of distances between the farthest and nearest clients to a given facility is minimized. Two cases are considered: bounded and unbounded nonnegative edge lengths. In the unbounded case, we show the problem can be reduced to solve the problem on a star graph. Then an &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ O(nlogn) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; algorithm is developed to find the optimal solution. For the bounded edge lengths case an algorithm with time complexity &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ O(n^2) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is presented.&lt;/p&gt;

Repeated-Root Constacyclic Codes With Pair-Metric

Symbol-pair codes are block codes when considering the pair-metric, and are designed to protect against pair-errors occur in the pair-read vector which consisting of overlapping pairs of symbols. A code that achieves the analog of Singleton bound for symbol-pair codes is called a maximum distance separable (MDS) symbol-pair code. The contribution of this letter is twofold. First, we give a lower bound of minimum pair-distance of repeated-root constacyclic codes in terms of minimum Hamming distance, which extends the lower bound given by Chen, Lin and Liu. Second, we give three new classes of MDS symbol-pair codes. As we know, for an MDS symbol-pair code with a fixed minimum pair-distance, the larger the code length of the code has, the higher the code rate is. The first class of MDS symbol-pair codes we show has unbounded code length and the other two classes of MDS symbol-pair codes have larger code lengths than the code lengths of existing MDS symbol-pair codes under the same minimum pair-distance 6.