We further develop the paraconsistent G\"{o}del modal logic. In this paper, we consider its version endowed with Kripke semantics on $[0,1]$-valued frames with two fuzzy relations $R^+$ and $R^-$ (degrees of trust in assertions and denials) and two valuations $v_1$ and $v_2$ (support of truth and support of falsity) linked with a De Morgan negation $\neg$. We demonstrate that it \emph{does not} extend G\"{o}del modal logic and that $\Box$ and $\lozenge$ are not interdefinable. We also show that several important classes of frames are $\birelKGsquare$ definable (in particular, crisp, mono-relational, and finitely branching). For $\birelKGsquare$ over finitely branching frames, we create a sound and complete constraint tableaux calculus and a decision procedure based upon it. Using the decision procedure we show that $\birelKGsquare$ satisfiability and validity are in PSPACE.
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We present a method for extracting general modules for ontologies formulated in the description logic ALC. A module for an ontology is an ideally substantially smaller ontology that preserves all entailments for a user-specified set of terms. As such, it has applications such as ontology reuse and ontology analysis. Different from classical modules, general modules may use axioms not explicitly present in the input ontology, which allows for additional conciseness. So far, general modules have only been investigated for lightweight description logics. We present the first work that considers the more expressive description logic ALC. In particular, our contribution is a new method based on uniform interpolation supported by some new theoretical results. Our evaluation indicates that our general modules are often smaller than classical modules and uniform interpolants computed by the state-of-the-art, and compared with uniform interpolants, can be computed in a significantly shorter time. Moreover, our method can be used for, and in fact improves, the computation of uniform interpolants and classical modules.
We present quantitative logics with two-step semantics based on the framework of quantitative logics introduced by Arenas et al. (2020) and the two-step semantics defined in the context of weighted logics by Gastin & Monmege (2018). We show that some of the fragments of our logics augmented with a least fixed point operator capture interesting classes of counting problems. Specifically, we answer an open question in the area of descriptive complexity of counting problems by providing logical characterizations of two subclasses of #P, namely SpanL and TotP, that play a significant role in the study of approximable counting problems. Moreover, we define logics that capture FPSPACE and SpanPSPACE, which are counting versions of PSPACE.
Online surgical phase recognition plays a significant role towards building contextual tools that could quantify performance and oversee the execution of surgical workflows. Current approaches are limited since they train spatial feature extractors using frame-level supervision that could lead to incorrect predictions due to similar frames appearing at different phases, and poorly fuse local and global features due to computational constraints which can affect the analysis of long videos commonly encountered in surgical interventions. In this paper, we present a two-stage method, called Long Video Transformer (LoViT) for fusing short- and long-term temporal information that combines a temporally-rich spatial feature extractor and a multi-scale temporal aggregator consisting of two cascaded L-Trans modules based on self-attention, followed by a G-Informer module based on ProbSparse self-attention for processing global temporal information. The multi-scale temporal head then combines local and global features and classifies surgical phases using phase transition-aware supervision. Our approach outperforms state-of-the-art methods on the Cholec80 and AutoLaparo datasets consistently. Compared to Trans-SVNet, LoViT achieves a 2.39 pp (percentage point) improvement in video-level accuracy on Cholec80 and a 3.14 pp improvement on AutoLaparo. Moreover, it achieves a 5.25 pp improvement in phase-level Jaccard on AutoLaparo and a 1.55 pp improvement on Cholec80. Our results demonstrate the effectiveness of our approach in achieving state-of-the-art performance of surgical phase recognition on two datasets of different surgical procedures and temporal sequencing characteristics whilst introducing mechanisms that cope with long videos.
Optimal control problems driven by evolutionary partial differential equations arise in many industrial applications and their numerical solution is known to be a challenging problem. One approach to obtain an optimal feedback control is via the Dynamic Programming principle. Nevertheless, despite many theoretical results, this method has been applied only to very special cases since it suffers from the curse of dimensionality. Our goalis to mitigate this crucial obstruction developing a new version of dynamic programming algorithms based on a tree structure and exploiting the compact representation of the dynamical systems based on tensors notations via a model reduction approach. Here, we want to show how this algorithm can be constructed for general nonlinear control problems and to illustrate its performances on a number of challenging numerical tests. Our numerical results indicate a large decrease in memory requirements, as well as computational time, for the proposed problems. Moreover, we prove the convergence of the algorithm and give some hints on its implementation
The Byzantine consensus problem involves $n$ processes, out of which t < n could be faulty and behave arbitrarily. Three properties characterize consensus: (1) termination, requiring correct (non-faulty) processes to eventually reach a decision, (2) agreement, preventing them from deciding different values, and (3) validity, precluding ``unreasonable'' decisions. But, what is a reasonable decision? Strong validity, a classical property, stipulates that, if all correct processes propose the same value, only that value can be decided. Weak validity, another established property, stipulates that, if all processes are correct and they propose the same value, that value must be decided. The space of possible validity properties is vast. However, their impact on consensus remains unclear. This paper addresses the question of which validity properties allow Byzantine consensus to be solvable with partial synchrony, and at what cost. First, we determine necessary and sufficient conditions for a validity property to make the consensus problem solvable; we say that such validity properties are solvable. Notably, we prove that, if n <= 3t, all solvable validity properties are trivial (there exists an always-admissible decision). Furthermore, we show that, with any non-trivial (and solvable) validity property, consensus requires Omega(t^2) messages. This extends the seminal Dolev-Reischuk bound, originally proven for strong validity, to all non-trivial validity properties. Lastly, we give a general Byzantine consensus algorithm, we call Universal, for any solvable (and non-trivial) validity property. Importantly, Universal incurs O(n^2) message complexity. Thus, together with our lower bound, Universal implies a fundamental result in partial synchrony: with t \in Omega(n), the message complexity of all (non-trivial) consensus variants is Theta(n^2).
The innovations in reactive synthesis from {\em Linear Temporal Logics over finite traces} ($\ltlf$) will be amplified by the ability to verify the correctness of the strategies generated by $\ltlf$ synthesis tools. This motivates our work on {\em $\ltlf$ model checking}. $\ltlf$ model checking, however, is not straightforward. The strategies generated by $\ltlf$ synthesis may be represented using {\em terminating} transducers or {\em non-terminating} transducers where executions are of finite-but-unbounded length or infinite length, respectively. For synthesis, there is no evidence that one type of transducer is better than the other since they both demonstrate the same complexity and similar algorithms. In this work, we show that for model checking, the two types of transducers are fundamentally different. Our central result is that $\ltlf$ model checking of non-terminating transducers is \emph{exponentially harder} than that of terminating transducers. We show that the problems are \expspace-complete and $\pspace$-complete, respectively. Hence, considering the feasibility of verification, $\ltlf$ synthesis tools should synthesize terminating transducers. This is, to the best of our knowledge, the \emph{first} evidence to use one transducer over the other in $\ltlf$ synthesis.
When a problem has more than one solution, it is often important, depending on the underlying context, to enumerate (i.e., to list) them all. Even when the enumeration can be done in polynomial delay, that is, spending no more than polynomial time to go from one solution to the next, this can be costly as the number of solutions themselves may be huge, including sometimes exponential. Furthermore, depending on the application, many of these solutions can be considered equivalent. The problem of an efficient enumeration of the equivalence classes or of one representative per class (without generating all the solutions), although identified as a need in many areas, has been addressed only for very few specific cases. In this paper, we provide a general framework that solves this problem in polynomial delay for a wide variety of contexts, including optimization ones that can be addressed by dynamic programming algorithms, and for certain types of equivalence relations between solutions.
In a vertex-colored graph $G = (V, E)$, a subset $S \subseteq V$ is said to be consistent if every vertex has a nearest neighbor in $S$ with the same color. The problem of computing a minimum cardinality consistent subset of a graph is known to be NP-hard. On the positive side, Dey et al. (FCT 2021) show that this problem is solvable in polynomial time when input graphs are restricted to bi-colored trees. In this paper, we give a polynomial-time algorithm for this problem on $k$-colored trees with fixed $k$.
We consider the problem of discovering $K$ related Gaussian directed acyclic graphs (DAGs), where the involved graph structures share a consistent causal order and sparse unions of supports. Under the multi-task learning setting, we propose a $l_1/l_2$-regularized maximum likelihood estimator (MLE) for learning $K$ linear structural equation models. We theoretically show that the joint estimator, by leveraging data across related tasks, can achieve a better sample complexity for recovering the causal order (or topological order) than separate estimations. Moreover, the joint estimator is able to recover non-identifiable DAGs, by estimating them together with some identifiable DAGs. Lastly, our analysis also shows the consistency of union support recovery of the structures. To allow practical implementation, we design a continuous optimization problem whose optimizer is the same as the joint estimator and can be approximated efficiently by an iterative algorithm. We validate the theoretical analysis and the effectiveness of the joint estimator in experiments.
Stock trend forecasting, aiming at predicting the stock future trends, is crucial for investors to seek maximized profits from the stock market. Many event-driven methods utilized the events extracted from news, social media, and discussion board to forecast the stock trend in recent years. However, existing event-driven methods have two main shortcomings: 1) overlooking the influence of event information differentiated by the stock-dependent properties; 2) neglecting the effect of event information from other related stocks. In this paper, we propose a relational event-driven stock trend forecasting (REST) framework, which can address the shortcoming of existing methods. To remedy the first shortcoming, we propose to model the stock context and learn the effect of event information on the stocks under different contexts. To address the second shortcoming, we construct a stock graph and design a new propagation layer to propagate the effect of event information from related stocks. The experimental studies on the real-world data demonstrate the efficiency of our REST framework. The results of investment simulation show that our framework can achieve a higher return of investment than baselines.
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