In this article, We introduce a condition that is both necessary and sufficient for a linear code to achieve minimality when analyzed over the rings $\mathbb{Z}_{n}$.The fundamental inquiry in minimal linear codes is the existence of a $[m,k]$ minimal linear code where $k$ is less than or equal to $m$. W. Lu et al. ( see \cite{nine}) showed that there exists a positive integer $m(k;q)$ such that for $m\geq m(k;q)$ a minimal linear code of length $m$ and dimension $k$ over a finite field $\mathbb{F}_q$ must exist. They give the upper and lower bound of $m(k;q)$. In this manuscript, we establish both an upper and lower bound for $m(k;p^l)$ and $m(k;p_1p_2)$ within the ring $\mathbb{Z}_{p^l}$ and $\mathbb{Z}_{p_1p_2}$ respectively.
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Graph neural networks form a class of deep learning architectures specifically designed to work with graph-structured data. As such, they share the inherent limitations and problems of deep learning, especially regarding the issues of explainability and trustworthiness. We propose $\mu\mathcal{G}$, an original domain-specific language for the specification of graph neural networks that aims to overcome these issues. The language's syntax is introduced, and its meaning is rigorously defined by a denotational semantics. An equivalent characterization in the form of an operational semantics is also provided and, together with a type system, is used to prove the type soundness of $\mu\mathcal{G}$. We show how $\mu\mathcal{G}$ programs can be represented in a more user-friendly graphical visualization, and provide examples of its generality by showing how it can be used to define some of the most popular graph neural network models, or to develop any custom graph processing application.
In the metric distortion problem there is a set of candidates $C$ and voters $V$ in the same metric space. The goal is to select a candidate minimizing the social cost: the sum of distances of the selected candidate from all the voters, and the challenge arises from the algorithm receiving only ordinaL input: each voter's ranking of candidate, while the objective function is cardinal, determined by the underlying metric. The distortion of an algorithm is its worst-case approximation factor of the optimal social cost. A key concept here is the (p,q)-veto core, with $p\in \Delta(V)$ and $q\in \Delta(C)$ being normalized weight vectors representing voters' veto power and candidates' support, respectively. The (p,q)-veto core corresponds to a set of winners from a specific class of deterministic algorithms. Notably, the optimal distortion of $3$ is obtained from this class, by selecting veto core candidates using uniform $p$ and $q$ proportional to candidates' plurality scores. Bounding the distortion of other algorithms from this class is an open problem. Our contribution is twofold. First, we establish upper bounds on the distortion of candidates from the (p,q)-veto core for arbitrary weight vectors $p$ and $q$. Second, we revisit the metric distortion problem through the \emph{learning-augmented} framework, which equips the algorithm with a (machine-learned) prediction regarding the optimal candidate. The quality of this prediction is unknown, and the goal is to optimize the algorithm's performance under accurate predictions (consistency), while simultaneously providing worst-case guarantees under arbitrarily inaccurate predictions (robustness). We propose an algorithm that chooses candidates from the (p,q)-veto core, using a prediction-guided q vector and, leveraging our distortion bounds, we prove that this algorithm achieves the optimal robustness-consistency trade-off.
When verifying liveness properties on a transition system, it is often necessary to discard spurious violating paths by making assumptions on which paths represent realistic executions. Capturing that some property holds under such an assumption in a logical formula is challenging and error-prone, particularly in the modal $\mu$-calculus. In this paper, we present template formulae in the modal $\mu$-calculus that can be instantiated to a broad range of liveness properties. We consider the following assumptions: progress, justness, weak fairness, strong fairness, and hyperfairness, each with respect to actions. The correctness of these formulae has been proven.
We develop domain theory in constructive and predicative univalent foundations (also known as homotopy type theory). That we work predicatively means that we do not assume Voevodsky's propositional resizing axioms. Our work is constructive in the sense that we do not rely on excluded middle or the axiom of (countable) choice. Domain theory studies so-called directed complete posets (dcpos) and Scott continuous maps between them and has applications in a variety of fields, such as programming language semantics, higher-type computability and topology. A common approach to deal with size issues in a predicative foundation is to work with information systems, abstract bases or formal topologies rather than dcpos, and approximable relations rather than Scott continuous functions. In our type-theoretic approach, we instead accept that dcpos may be large and work with type universes to account for this. A priori one might expect that iterative constructions of dcpos may result in a need for ever-increasing universes and are predicatively impossible. We show, through a careful tracking of type universe parameters, that such constructions can be carried out in a predicative setting. In particular, we give a predicative reconstruction of Scott's $D_\infty$ model of the untyped $\lambda$-calculus. Our work is formalised in the Agda proof assistant and its ability to infer universe levels has been invaluable for our purposes.
Hopf algebra structures for the backward error analysis of ergodic stochastic differential equations
While backward error analysis does not generalise straightforwardly to the strong and weak approximation of stochastic differential equations, it extends for the sampling of ergodic dynamics. The calculation of the modified equation relies on tedious calculations and there is no expression of the modified vector field, in opposition to the deterministic setting. We uncover in this paper the Hopf algebra structures associated to the laws of composition and substitution of exotic aromatic S-series, relying on the new idea of clumping. We use these algebraic structures to provide the algebraic foundations of stochastic numerical analysis with S-series, as well as an explicit expression of the modified vector field as an exotic aromatic B-series.
The novelty of the current work is precisely to propose a statistical procedure to combine estimates of the modal parameters provided by any set of Operational Modal Analysis (OMA) algorithms so as to avoid preference for a particular one and also to derive an approximate joint probability distribution of the modal parameters, from which engineering statistics of interest such as mean value and variance are readily provided. The effectiveness of the proposed strategy is assessed considering measured data from an actual centrifugal compressor. The statistics obtained for both forward and backward modal parameters are finally compared against modal parameters identified during standard stability verification testing (SVT) of centrifugal compressors prior to shipment, using classical Experimental Modal Analysis (EMA) algorithms. The current work demonstrates that combination of OMA algorithms can provide quite accurate estimates for both the modal parameters and the associated uncertainties with low computational costs.
In relational verification, judicious alignment of computational steps facilitates proof of relations between programs using simple relational assertions. Relational Hoare logics (RHL) provide compositional rules that embody various alignments of executions. Seemingly more flexible alignments can be expressed in terms of product automata based on program transition relations. A single degenerate alignment rule (self-composition), atop a complete Hoare logic, comprises a RHL for $\forall\forall$ properties that is complete in the ordinary logical sense (Cook'78). The notion of alignment completeness was previously proposed as a more satisfactory measure, and some rules were shown to be alignment complete with respect to a few ad hoc forms of alignment automata. This paper proves alignment completeness with respect to a general class of $\forall\forall$ alignment automata, for a RHL comprised of standard rules together with a rule of semantics-preserving rewrites based on Kleene algebra with tests. A new logic for $\forall\exists$ properties is introduced and shown to be alignment complete. The $\forall\forall$ and $\forall\exists$ automata are shown to be semantically complete. Thus the logics are both complete in the ordinary sense. Recent work by D'Osualdo et al highlights the importance of completeness relative to assumptions (which we term entailment completeness), and presents $\forall\forall$ examples seemingly beyond the scope of RHLs. Additional rules enable these examples to be proved in our RHL, shedding light on the open problem of entailment completeness.
We present a novel data-driven strategy to choose the hyperparameter $k$ in the $k$-NN regression estimator without using any hold-out data. We treat the problem of choosing the hyperparameter as an iterative procedure (over $k$) and propose using an easily implemented in practice strategy based on the idea of early stopping and the minimum discrepancy principle. This model selection strategy is proven to be minimax-optimal over some smoothness function classes, for instance, the Lipschitz functions class on a bounded domain. The novel method often improves statistical performance on artificial and real-world data sets in comparison to other model selection strategies, such as the Hold-out method, 5-fold cross-validation, and AIC criterion. The novelty of the strategy comes from reducing the computational time of the model selection procedure while preserving the statistical (minimax) optimality of the resulting estimator. More precisely, given a sample of size $n$, if one should choose $k$ among $\left\{ 1, \ldots, n \right\}$, and $\left\{ f^1, \ldots, f^n \right\}$ are the estimators of the regression function, the minimum discrepancy principle requires the calculation of a fraction of the estimators, while this is not the case for the generalized cross-validation, Akaike's AIC criteria, or Lepskii principle.
Let $q$ be an odd prime power and let $\mathbb{F}_{q^2}$ be the finite field with $q^2$ elements. In this paper, we determine the differential spectrum of the power function $F(x)=x^{2q+1}$ over $\mathbb{F}_{q^2}$. When the characteristic of $\mathbb{F}_{q^2}$ is $3$, we also determine the value distribution of the Walsh spectrum of $F$, showing that it is $4$-valued, and use the obtained result to determine the weight distribution of a $4$-weight cyclic code.
When and why can a neural network be successfully trained? This article provides an overview of optimization algorithms and theory for training neural networks. First, we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum, and then discuss practical solutions including careful initialization and normalization methods. Second, we review generic optimization methods used in training neural networks, such as SGD, adaptive gradient methods and distributed methods, and theoretical results for these algorithms. Third, we review existing research on the global issues of neural network training, including results on bad local minima, mode connectivity, lottery ticket hypothesis and infinite-width analysis.
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