Temporal reasoning with conditionals is more complex than both classical temporal reasoning and reasoning with timeless conditionals, and can lead to some rather counter-intuitive conclusions. For instance, Aristotle's famous "Sea Battle Tomorrow" puzzle leads to a fatalistic conclusion: whether there will be a sea battle tomorrow or not, but that is necessarily the case now. We propose a branching-time logic LTC to formalise reasoning about temporal conditionals and provide that logic with adequate formal semantics. The logic LTC extends the Nexttime fragment of CTL*, with operators for model updates, restricting the domain to only future moments where antecedent is still possible to satisfy. We provide formal semantics for these operators that implements the restrictor interpretation of antecedents of temporalized conditionals, by suitably restricting the domain of discourse. As a motivating example, we demonstrate that a naturally formalised in our logic version of the `Sea Battle' argument renders it unsound, thereby providing a solution to the problem with fatalist conclusion that it entails, because its underlying reasoning per cases argument no longer applies when these cases are treated not as material implications but as temporal conditionals. On the technical side, we analyze the semantics of LTC and provide a series of reductions of LTC-formulae, first recursively eliminating the dynamic update operators and then the path quantifiers in such formulae. Using these reductions we obtain a sound and complete axiomatization for LTC, and reduce its decision problem to that of the modal logic KD.
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We consider the statistical linear inverse problem of making inference on an unknown source function in an elliptic partial differential equation from noisy observations of its solution. We employ nonparametric Bayesian procedures based on Gaussian priors, leading to convenient conjugate formulae for posterior inference. We review recent results providing theoretical guarantees on the quality of the resulting posterior-based estimation and uncertainty quantification, and we discuss the application of the theory to the important classes of Gaussian series priors defined on the Dirichlet-Laplacian eigenbasis and Mat\'ern process priors. We provide an implementation of posterior inference for both classes of priors, and investigate its performance in a numerical simulation study.
Although recently several foundation models for satellite remote sensing imagery have been proposed, they fail to address major challenges of real/operational applications. Indeed, embeddings that don't take into account the spectral, spatial and temporal dimensions of the data as well as the irregular or unaligned temporal sampling are of little use for most real world uses.As a consequence, we propose an ALIgned Sits Encoder (ALISE), a novel approach that leverages the spatial, spectral, and temporal dimensions of irregular and unaligned SITS while producing aligned latent representations. Unlike SSL models currently available for SITS, ALISE incorporates a flexible query mechanism to project the SITS into a common and learned temporal projection space. Additionally, thanks to a multi-view framework, we explore integration of instance discrimination along a masked autoencoding task to SITS. The quality of the produced representation is assessed through three downstream tasks: crop segmentation (PASTIS), land cover segmentation (MultiSenGE), and a novel crop change detection dataset. Furthermore, the change detection task is performed without supervision. The results suggest that the use of aligned representations is more effective than previous SSL methods for linear probing segmentation tasks.
A common method for estimating the Hessian operator from random samples on a low-dimensional manifold involves locally fitting a quadratic polynomial. Although widely used, it is unclear if this estimator introduces bias, especially in complex manifolds with boundaries and nonuniform sampling. Rigorous theoretical guarantees of its asymptotic behavior have been lacking. We show that, under mild conditions, this estimator asymptotically converges to the Hessian operator, with nonuniform sampling and curvature effects proving negligible, even near boundaries. Our analysis framework simplifies the intensive computations required for direct analysis.
A new decoder for the SIF test problems of the CUTEst collection is described, which produces problem files allowing the computation of values and derivatives of the objective function and constraints of most \cutest\ problems directly within ``native'' Matlab, Python or Julia, without any additional installation or interfacing with MEX files or Fortran programs. When used with Matlab, the new problem files optionally support reduced-precision computations.
We prove the convergence of a damped Newton's method for the nonlinear system resulting from a discretization of the second boundary value problem for the Monge-Ampere equation. The boundary condition is enforced through the use of the notion of asymptotic cone. The differential operator is discretized based on a discrete analogue of the subdifferential.
In this work we propose a discretization of the second boundary condition for the Monge-Ampere equation arising in geometric optics and optimal transport. The discretization we propose is the natural generalization of the popular Oliker-Prussner method proposed in 1988. For the discretization of the differential operator, we use a discrete analogue of the subdifferential. Existence, unicity and stability of the solutions to the discrete problem are established. Convergence results to the continuous problem are given.
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.
In the finite difference approximation of the fractional Laplacian the stiffness matrix is typically dense and needs to be approximated numerically. The effect of the accuracy in approximating the stiffness matrix on the accuracy in the whole computation is analyzed and shown to be significant. Four such approximations are discussed. While they are shown to work well with the recently developed grid-over finite difference method (GoFD) for the numerical solution of boundary value problems of the fractional Laplacian, they differ in accuracy, economics to compute, performance of preconditioning, and asymptotic decay away from the diagonal line. In addition, two preconditioners based on sparse and circulant matrices are discussed for the iterative solution of linear systems associated with the stiffness matrix. Numerical results in two and three dimensions are presented.
Combining data from various sources empowers researchers to explore innovative questions, for example those raised by conducting healthcare monitoring studies. However, the lack of a unique identifier often poses challenges. Record linkage procedures determine whether pairs of observations collected on different occasions belong to the same individual using partially identifying variables (e.g. birth year, postal code). Existing methodologies typically involve a compromise between computational efficiency and accuracy. Traditional approaches simplify this task by condensing information, yet they neglect dependencies among linkage decisions and disregard the one-to-one relationship required to establish coherent links. Modern approaches offer a comprehensive representation of the data generation process, at the expense of computational overhead and reduced flexibility. We propose a flexible method, that adapts to varying data complexities, addressing registration errors and accommodating changes of the identifying information over time. Our approach balances accuracy and scalability, estimating the linkage using a Stochastic Expectation Maximisation algorithm on a latent variable model. We illustrate the ability of our methodology to connect observations using large real data applications and demonstrate the robustness of our model to the linking variables quality in a simulation study. The proposed algorithm FlexRL is implemented and available in an open source R package.
We consider a non-linear Bayesian data assimilation model for the periodic two-dimensional Navier-Stokes equations with initial condition modelled by a Gaussian process prior. We show that if the system is updated with sufficiently many discrete noisy measurements of the velocity field, then the posterior distribution eventually concentrates near the ground truth solution of the time evolution equation, and in particular that the initial condition is recovered consistently by the posterior mean vector field. We further show that the convergence rate can in general not be faster than inverse logarithmic in sample size, but describe specific conditions on the initial conditions when faster rates are possible. In the proofs we provide an explicit quantitative estimate for backward uniqueness of solutions of the two-dimensional Navier-Stokes equations.
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