Mogadishu, Somalia – Ending the United States' "forever wars" was a major slogan of Donald Trump's 2024 election campaign, during which he and many of his supporters spoke out against American resources and lives being put to waste in conflicts across the globe. But on February 1, a mere 10 days after being inaugurated for a second time, President Trump announced that the US had carried out air strikes targeting senior leadership of ISIL (ISIS) in Somalia. "These killers, who we found hiding in caves, threatened the United States," his post on X read. This marked Trump's first military action overseas, but it wouldn't be his last. In the time since, the US has provided weapons and support to Israel in its wars in Gaza and across the Middle East; it has launched strikes on Yemen; and even attacked Iran's nuclear facilities.

This draft is a working document, having a summary of nighty-four (94) papers with additional sections on Traceability of Software Requirements (Section 4), Formal Methods and Its Tools (Section 5), Unifying Theories of Programming (UTP) and Theory of Institutions (Section 6). Please refer to abstract of [7,8]. Key difference of this draft from our recently anticipated ones with similar titles, i.e. AACS 2025 [7] and SAIV 2025 [8] is: [7] is a two page submission to ADAPT Annual Conference, Ireland. Submitted on 18th of March, 2025, it went through the light-weight blind review and accepted for poster presentation. Conference was held on 15th of May, 2025; [8] is a nine page paper with additional nine pages of references and summary tables, submitted to Symposium on AI Verification (SAIV 2025) on 24th of April, 2025. It went through rigorous review process. The uploaded version on arXiv.org [8] is the improved one of the submission, after addressing the specific suggestions to improve the paper.

In this paper, we propose DuaShepherd, a novel reward modeling framework that integrates two complementary reward signals, correctness and potential, to enhance the mathematical reasoning capabilities of Large Language Models (LLMs). While correctness-based signals emphasize identification of stepwise errors, potential-based signals focus on the likelihood of reaching the correct final answer. We developed an automated pipeline for constructing large-scale reward modeling dataset with both signals. A unified, multi-head architecture was explored to train the two reward models in a multi-task setup, demonstrating benefits from learning both correctness and potential in parallel. By combining these two signals into a compound probability, our model achieves consistent performance improvements across multiple benchmarks. Empirical evaluations on MATH500 and ProcessBench confirm that this combined reward significantly outperforms models trained on either reward type alone, achieving state-of-the-art performance under comparable resource constraints.

Many researchers are working to address the worldwide mental health crisis by developing therapeutic technologies that increase the accessibility of care, including leveraging large language model (LLM) capabilities in chatbots and socially assistive robots (SARs) used for therapeutic applications. Yet, the effects of these technologies over time remain unexplored. In this study, we use a factorial design to assess the impact of embodiment and time spent engaging in therapeutic exercises on participant disclosures. We assessed transcripts gathered from a two-week study in which 26 university student participants completed daily interactive Cognitive Behavioral Therapy (CBT) exercises in their residences using either an LLM-powered SAR or a disembodied chatbot. We evaluated the levels of active engagement and high intimacy of their disclosures (opinions, judgments, and emotions) during each session and over time. Our findings show significant interactions between time and embodiment for both outcome measures: participant engagement and intimacy increased over time in the physical robot condition, while both measures decreased in the chatbot condition.

The rapid expansion of information from diverse sources has heightened the need for effective automatic text summarization, which condenses documents into shorter, coherent texts. Summarization methods generally fall into two categories: extractive, which selects key segments from the original text, and abstractive, which generates summaries by rephrasing the content coherently. Large language models have advanced the field of abstractive summarization, but they are resource-intensive and face significant challenges in retaining key information across lengthy documents, which we call being "lost in the middle". To address these issues, we propose a hybrid summarization approach that combines extractive and abstractive techniques. Our method splits the document into smaller text chunks, clusters their vector embeddings, generates a summary for each cluster that represents a key idea in the document, and constructs the final summary by relying on a Markov chain graph when selecting the semantic order of ideas.

Lemmatization is crucial for NLP tasks in morphologically rich languages with ambiguous orthography like Arabic, but existing tools face challenges due to inconsistent standards and limited genre coverage. This paper introduces two novel approaches that frame lemmatization as classification into a Lemma-POS-Gloss (LPG) tagset, leveraging machine translation and semantic clustering. We also present a new Arabic lemmatization test set covering diverse genres, standardized alongside existing datasets. We evaluate character level sequence-to-sequence models, which perform competitively and offer complementary value, but are limited to lemma prediction (not LPG) and prone to hallucinating implausible forms. Our results show that classification and clustering yield more robust, interpretable outputs, setting new benchmarks for Arabic lemmatization.

Companion paper [118] developed a powerful \emph{Random duality theory} (RDT) based analytical program to statistically characterize performance of \emph{descending} phase retrieval algorithms (dPR) (these include all variants of gradient descents and among them widely popular Wirtinger flows). We here generalize the program and show how it can be utilized to handle rank $d$ positive definite phase retrieval (PR) measurements (with special cases $d=1$ and $d=2$ serving as emulations of the real and complex phase retrievals, respectively). In particular, we observe that the minimal sample complexity ratio (number of measurements scaled by the dimension of the unknown signal) which ensures dPR's success exhibits a phase transition (PT) phenomenon. For both plain and lifted RDT we determine phase transitions locations. To complement theoretical results we implement a log barrier gradient descent variant and observe that, even in small dimensional scenarios (with problem sizes on the order of 100), the simulated phase transitions are in an excellent agreement with the theoretical predictions.

We analyze the relation between spectral initializers and theoretical limits of \emph{descending} phase retrieval algorithms (dPR). In companion paper [104], for any sample complexity ratio, $α$, \emph{parametric manifold}, ${\mathcal {PM}}(α)$, is recognized as a critically important structure that generically determines dPRs abilities to solve phase retrieval (PR). Moreover, overlap between the algorithmic solution and the true signal is positioned as a key ${\mathcal {PM}}$'s component. We here consider the so-called \emph{overlap optimal} spectral initializers (OptSpins) as dPR's starting points and develop a generic \emph{Random duality theory} (RDT) based program to statistically characterize them. In particular, we determine the functional structure of OptSpins and evaluate the starting overlaps that they provide for the dPRs. Since ${\mathcal {PM}}$'s so-called \emph{flat regions} are highly susceptible to \emph{local jitteriness} and as such are key obstacles on dPR's path towards PR's global optimum, a precise characterization of the starting overlap allows to determine if such regions can be successfully circumvented. Through the presented theoretical analysis we observe two key points in that regard: \textbf{\emph{(i)}} dPR's theoretical phase transition (critical $α$ above which they solve PR) might be difficult to practically achieve as the ${\mathcal {PM}}$'s flat regions are large causing the associated OptSpins to fall exactly within them; and \textbf{\emph{(ii)}} Opting for so-called ``\emph{safer compression}'' and slightly increasing $α$ (by say $15\%$) shrinks flat regions and allows OptSpins to fall outside them and dPRs to ultimately solve PR. Numerical simulations are conducted as well and shown to be in an excellent agreement with theoretical predictions.

Group structures are common in regression analysis. They can appear in the form of categorical predictors represented by groups of dummy variables or in the context of additive modeling, where each predictor can be expressed as a set of basis functions forming a group; in applications such as gene expression analysis and financial market modeling, groupings exist naturally in the data. For instance, genes that influence similar traits form groups in gene expression data, while stocks from the same sector form groups in financial data. In these scenarios, group shrinkage plays an important role: when there is insufficient evidence to suggest the significance of predictors within a group, the entire group of predictors is shrunk towards zero. This reduces the noise from individual "spurious predictors", which tend to appear more frequently in high-dimensional settings, and decreases model complexity, thereby reducing the risk of overfitting. 1 Within the Bayesian framework, there has been extensive research focusing on the application of continuous shrinkage priors for linear regression problems involving group predictor variables. Traditional approaches, such as the group lasso[31, 24], the group bridge [16], and the group horseshoe [29] primarily apply shrinkage at the group level and do not consider within-group shrinkage.

We study theoretical limits of \emph{descending} phase retrieval algorithms. Utilizing \emph{Random duality theory} (RDT) we develop a generic program that allows statistical characterization of various algorithmic performance metrics. Through these we identify the concepts of \emph{parametric manifold} and its \emph{funneling points} as key mathematical objects that govern the underlying algorithms' behavior. An isomorphism between single funneling point manifolds and global convergence of descending algorithms is established. The structure and shape of the parametric manifold as well as its dependence on the sample complexity are studied through both plain and lifted RDT. Emergence of a phase transition is observed. Namely, as sample complexity increases, parametric manifold transitions from a multi to a single funneling point structure. This in return corresponds to a transition from the scenarios where descending algorithms generically fail to the scenarios where they succeed in solving phase retrieval. We also develop and implement a practical algorithmic variant that in a hybrid alternating fashion combines a barrier and a plain gradient descent. Even though the theoretical results are obtained for infinite dimensional scenarios (and consequently non-jittery parametric manifolds), we observe a strong agrement between theoretical and simulated phase transitions predictions for fairly small dimensions on the order of a few hundreds.