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Autoregressive language models are trained exclusively left-to-right. We explore the complementary factorization, training right-to-left at scale, and ask what reasoning patterns emerge when a model conditions on future context to predict the past. We train LEDOM, an open-source purely reverse autoregressive language model (2B/7B parameters, 435B tokens), and find it develops capabilities distinct from forward models, including abductive inference, question synthesis, and natural resolution of the reversal curse. We then explore one application of the reverse model: combining forward likelihood $P(y \mid x)$ with reverse posterior $P(x \mid y)$ through noisy channel duality. We propose Reverse Reward, which reranks forward outputs using reverse posterior estimates, and prove that bidirectional scoring penalizes hallucinated reasoning chains whose backward reconstruction degrades. Reverse Reward yields gains of up to 6.6\% on AIME 2024 and 15\% on AMC 2023 across multiple strong baselines. We release all models, code, and data here.

The ensemble-averaged dynamics of open quantum systems are typically irreversible. We show that this irreversibility need not hold at the level of individually monitored quantum trajectories. Our main results are analytical stochastic Schrödinger equations for quantum reverse diffusion, along with corresponding stochastic master equations. These equations describe the exact and approximate stochastic reverse processes for continuously monitored Pauli channels, including time-dependent depolarizing noise. We show that the reverse processes generalize the forward dynamics by combining the noise effects of the forward processes with an additional stochastic drift that dynamically steers a quantum state back to its initial configuration. Consequently, the exact reverse stochastic Schrödinger equations admit closed-form solutions that can be implemented in real-time without the need for variational techniques. Our findings establish an analytical framework for quantum state recovery, noise-resilient quantum gates, quantum generative modelling, quantum tomography via forward-reverse cycles, and potential paradigms for quantum error correction based on reverse diffusion.

Reverse engineering can be used to derive a 3D model of an existing physical part when such a model is not readily available. For parts that will be fabricated with subtractive and formative manufacturing processes, existing reverse engineering techniques can be readily applied, but parts produced with additive manufacturing can present new challenges due to the high level of process-induced distortions and unique part attributes. This paper introduces an integrated 3D scanning and process simulation data-driven framework to minimize distortions of reverse-engineered additively manufactured components. This framework employs iterative finite element simulations to predict geometric distortions to minimize errors between the predicted and measured geometrical deviations of the key dimensional characteristics of the part. The effectiveness of this approach is then demonstrated by reverse engineering two Inconel-718 components manufactured using laser powder bed fusion additive manufacturing. This paper presents a remanufacturing process that combines reverse engineering and additive manufacturing, leveraging geometric feature-based part compensation through process simulation. Our ap

Reversible systems exhibit both forward computations and backward computations, where the aim of the latter is to undo the effects of the former. Such systems can be compared via forward-reverse bisimilarity as well as its two components, i.e., forward bisimilarity and reverse bisimilarity. The congruence, equational, and logical properties of these equivalences have already been studied in the setting of sequential processes. In this paper we address concurrent processes and investigate compositionality and axiomatizations of forward bisimilarity, which is interleaving, and reverse and forward-reverse bisimilarities, which are truly concurrent. To uniformly derive expansion laws for the three equivalences, we develop encodings based on the proved trees approach of Degano & Priami. In the case of reverse and forward-reverse bisimilarities, we show that in the encoding every action prefix needs to be extended with the backward ready set of the reached process.

We study the coupling of pairs of reverse plane partitions of the same shape by assigning a certain local interaction between the reverse plane partitions. We show that they are in bijection with a certain Yang-Baxter integrable colored vertex model. By utilizing the Yang-Baxter equation for this colored vertex model, we are able to compute the generating function for the interacting pairs of reverse plane partitions. We also give a bijection between the coupled pairs of reverse plane partitions with the interaction strength set to zero and a single reverse plane partition of the same shape.

We consider experimentally and computationally the phenomenon of the reverse Janssen effect, involving the counterintuitive finding that the force on the base of a column containing granular particles may be larger than the weight of the granular material itself. This finding is in contrast to the common Janssen effect, for which the force on the base is smaller than the particle weight, illustrating one of the best-known differences between granular and liquid systems. We find that the reverse Janssen effect is strongly influenced by the pouring protocol: under Earth's gravitational field, we find that the reverse Janssen effect is strongly and consistently influenced by the pouring height, as well as by (to somewhat lesser degree) pouring flux. Pouring grains from the height measured in tens of particle diameters leads to a reverse Janssen effect that is an order of magnitude stronger than the one found for small pouring heights of few particle diameters. This influence of the particles' delivery protocol allows for the development of a better understanding of the general features of the reverse Janssen effect and of the comparison between experiments and simulations reported in

Bayes' rule connects forward and reverse processes in classical probability theory, and its quantum analogue has been discussed in terms of the Petz (transpose) map. For quantum dynamics governed by the Lindblad equation, the corresponding Petz map can also be written in Lindblad form. In classical stochastic systems, the analogue of the Lindblad equation is the Fokker-Planck equation, and applying Bayes' rule to it yields the reverse diffusion equation underlying modern diffusion-based generative models. Here we demonstrate that a semiclassical approximation of the Lindblad equation yields the Fokker-Planck equation for the Wigner function -- a quasiprobability distribution defined on phase space as the Wigner transform of the density operator. Applying the same approximation to the Lindblad equation associated with the Petz map produces an equation that coincides with that obtained from the Fokker-Planck equation via Bayes' rule. This finding establishes a direct correspondence between the Petz map and Bayes' rule, unifying quantum reversibility with classical reverse diffusion.

Humans are accustomed to reading and writing in a forward manner, and this natural bias extends to text understanding in auto-regressive large language models (LLMs). This paper investigates whether LLMs, like humans, struggle with reverse modeling, specifically with reversed text inputs. We found that publicly available pre-trained LLMs cannot understand such inputs. However, LLMs trained from scratch with both forward and reverse texts can understand them equally well during inference across multiple languages. Our case study shows that different-content texts result in different losses if input (to LLMs) in different directions -- some get lower losses for forward while some for reverse. This leads us to a simple and nice solution for data selection based on the loss differences between forward and reverse directions. Using our selected data in continued pretraining can boost LLMs' performance by a large margin across different language understanding benchmarks.

Using data from the Chandra X-Ray Observatory, we revisited the reverse shock in the supernova remnant (SNR) Cassiopeia A.Based on the spectroscopic of a series of annuli in the northwest (NW) and southeast (SE), we get the radial profiles of the S/Si K-alpha line flux ratio and Fe K-alpha line centroid energy. They both show monotonic increase, confirming that the Si- and Fe-rich ejecta are heated by the reverse shock.The abrupt change of the S and Si line flux ratio is clearly observed in Cassiopeia A, leading to the determination of the reverse shock location (~1.71+-0.16 arcmin and ~1.35+-0.18 arcmin in the NW and SE, with respect to the central source). By comparing the radial profiles of S and Si line flux, we find that the reverse shock is moving outward in the frame of the observer, and the velocities are ~3950+-210 km/s and ~2900+-260 km/s in the NW and SE, respectively. In contrast, the velocities become ~1150 km/s (NW) and ~1300 km/s (SE) in the ejecta frame. Our measured reverse shock velocities are quite consistent with those obtained from the X-ray and/or optical images. It therefore supplies a crosscheck of the accuracy for the two available methods to measure the re

We introduce Reverse CAPTCHA, an evaluation framework that tests whether large language models follow invisible Unicode-encoded instructions embedded in otherwise normal-looking text. Unlike traditional CAPTCHAs that distinguish humans from machines, our benchmark exploits a capability gap: models can perceive Unicode control characters that are invisible to human readers. We evaluate five models from two providers across two encoding schemes (zero-width binary and Unicode Tags), four hint levels, two payload framings, and with tool use enabled or disabled. Across 8,308 model outputs, we find that tool use dramatically amplifies compliance (Cohen's h up to 1.37, a large effect), that models exhibit provider-specific encoding preferences (OpenAI models decode zero-width binary; Anthropic models prefer Unicode Tags), and that explicit decoding instructions increase compliance by up to 95 percentage points within a single model and encoding. All pairwise model differences are statistically significant (p < 0.05, Bonferroni-corrected). These results highlight an underexplored attack surface for prompt injection via invisible Unicode payloads.

Reverse engineering is a complex process essential to software-security tasks such as vulnerability discovery and malware analysis. Significant research and engineering effort has gone into developing tools to support reverse engineers. However, little work has been done to understand the way reverse engineers think when analyzing programs, leaving tool developers to make interface design decisions based only on intuition. This paper takes a first step toward a better understanding of reverse engineers' processes, with the goal of producing insights for improving interaction design for reverse engineering tools. We present the results of a semi-structured, observational interview study of reverse engineers (N=16). Each observation investigated the questions reverse engineers ask as they probe a program, how they answer these questions, and the decisions they make throughout the reverse engineering process. From the interview responses, we distill a model of the reverse engineering process, divided into three phases: overview, sub-component scanning, and focused experimentation. Each analysis phase's results feed the next as reverse engineers' mental representations become more conc

Large language models (LLMs) have a surprising failure: when trained on "A has a feature B", they do not generalize to "B is a feature of A", which is termed the Reversal Curse. Even when training with trillions of tokens this issue still appears due to Zipf's law - hence even if we train on the entire internet. This work proposes an alternative training scheme, called reverse training, whereby all words are used twice, doubling the amount of available tokens. The LLM is trained in both forward and reverse directions by reversing the training strings while preserving (i.e., not reversing) chosen substrings, such as entities. We show that data-matched reverse-trained models provide superior performance to standard models on standard tasks, and compute-matched reverse-trained models provide far superior performance on reversal tasks, helping resolve the reversal curse issue.

This manuscript establishes several sufficient conditions for the validity of both the reverse order law and forward order law for NDMPI. Additionally, some characterization of the reverse order law of the NDMPI is obtained. We also explore the applications of the reverse order law within this framework. Finally, we demonstrate the additivity of the NDMPI, supported by illustrative examples.

Reversible systems feature both forward computations and backward computations, where the latter undo the effects of the former in a causally consistent manner. The compositionality properties and equational characterizations of strong and weak variants of forward-reverse bisimilarity as well as of its two components, i.e., forward bisimilarity and reverse bisimilarity, have been investigated on a minimal process calculus for nondeterministic reversible systems that are sequential, so as to be neutral with respect to interleaving vs. truly concurrent semantics of parallel composition. In this paper we provide logical characterizations for the considered bisimilarities based on forward and backward modalities, which reveals that strong and weak reverse bisimilarities respectively correspond to strong and weak reverse trace equivalences. Moreover, we establish a clear connection between weak forward-reverse bisimilarity and branching bisimilarity, so that the former inherits two further logical characterizations from the latter over a specific class of processes.

Quantum annealers conventionally use forward annealing to generate heuristic solutions. Reverse annealing can potentially generate better solutions but necessitates an appropriate initial state. Ways to find such states are generally unknown or highly problem dependent, offer limited success, and severely restrict the scope of reverse annealing. We use a general method that improves the overall solution quality and quantity by feeding reverse annealing with low-quality solutions obtained from forward annealing. An experimental demonstration of solving the graph coloring problem using the D-Wave quantum annealers shows that our method is able to convert invalid solutions obtained from forward annealing to at least one valid solution obtained after assisted reverse annealing for $57\%$ of $459$ random Erdős-Rényi graphs. Our method significantly outperforms random initial states, obtains more unique solutions on average, and widens the applicability of reverse annealing. Although the average number of valid solutions obtained drops exponentially with the problem size, a scaling analysis for the graph coloring problem shows that our method effectively extends the computational reach o

In this paper we prove a reverse Hölder inequality for the variable exponent Muckenhoupt weights $\mathcal{A}_{p(\cdot)}$, introduced by the first author, Fiorenza, and Neugeabauer. All of our estimates are quantitative, showing the dependence of the exponent function on the $\mathcal{A}_{p(\cdot)}$ characteristic. As an application, we use the reverse Hölder inequality to prove that the matrix $\mathcal{A}_{p(\cdot)}$ weights, introduced in our previous paper, have both a right and left-openness property. This result is new even in the scalar case.

Previous work has shown that reverse differential categories give an abstract setting for gradient-based learning of functions between Euclidean spaces. However, reverse differential categories are not suited to handle gradient-based learning for functions between more general spaces such as smooth manifolds. In this paper, we propose a setting to handle this, which we call reverse tangent categories: tangent categories with an involution operation for their differential bundles.

Reverse engineering has been a standard practice in the hardware community for some time. It has only been within the last ten years that reverse engineering, or "program comprehension", has grown into the current sub-discipline of software engineering. Traditional software engineering is primarily focused on the development and design of new software. However, most programmers work on software that other people have designed and developed. Up to 50% of a software maintainers time can be spent determining the intent of source code. The growing demand to reevaluate and reimplement legacy software systems, brought on by the proliferation of clientserver and World Wide Web technologies, has underscored the need for reverse engineering tools and techniques. This paper introduces the terminology of reverse engineering and gives some of the obstacles that make reverse engineering difficult. Although reverse engineering remains heavily dependent on the human component, a number of automated tools are presented that aid the reverse engineer.