One of the fundamental problems in Artificial Intelligence is to perform complex multi-hop logical reasoning over the facts captured by a knowledge graph (KG). This problem is challenging, because KGs can be massive and incomplete. Recent approaches embed KG entities in a low dimensional space and then use these embeddings to find the answer entities. However, it has been an outstanding challenge of how to handle arbitrary first-order logic (FOL) queries as present methods are limited to only a subset of FOL operators. In particular, the negation operator is not supported. An additional limitation of present methods is also that they cannot naturally model uncertainty.

Query embedding (QE)---which aims to embed entities and first-order logical (FOL) queries in low-dimensional spaces---has shown great power in multi-hop reasoning over knowledge graphs. Recently, embedding entities and queries with geometric shapes becomes a promising direction, as geometric shapes can naturally represent answer sets of queries and logical relationships among them. However, existing geometry-based models have difficulty in modeling queries with negation, which significantly limits their applicability. To address this challenge, we propose a novel query embedding model, namely \textbf{Con}e \textbf{E}mbeddings (ConE), which is the first geometry-based QE model that can handle all the FOL operations, including conjunction, disjunction, and negation. Specifically, ConE represents entities and queries as Cartesian products of two-dimensional cones, where the intersection and union of cones naturally model the conjunction and disjunction operations. By further noticing that the closure of complement of cones remains cones, we design geometric complement operators in the embedding space for the negation operations. Experiments demonstrate that ConE significantly outperforms existing state-of-the-art methods on benchmark datasets.

We present a mechanistic interpretability study of GPT-2 that causally examines how sentiment information is processed across its transformer layers. Using systematic activation patching across all 12 layers, we test the hypothesized two-stage sentiment architecture comprising early lexical detection and mid-layer contextual integration. Our experiments confirm that early layers (0-3) act as lexical sentiment detectors, encoding stable, position specific polarity signals that are largely independent of context. However, all three contextual integration hypotheses: Middle Layer Concentration, Phenomenon Specificity, and Distributed Processing are falsified. Instead of mid-layer specialization, we find that contextual phenomena such as negation, sarcasm, domain shifts etc. are integrated primarily in late layers (8-11) through a unified, non-modular mechanism. These experimental findings provide causal evidence that GPT-2's sentiment computation differs from the predicted hierarchical pattern, highlighting the need for further empirical characterization of contextual integration in large language models.

Large Language Models (LLMs) have transformed natural language processing and hold growing promise for advancing science, healthcare, and decision-making. Yet their training paradigms remain dominated by affirmation-based inference, akin to \textit{modus ponens}, where accepted premises yield predicted consequents. While effective for generative fluency, this one-directional approach leaves models vulnerable to logical fallacies, adversarial manipulation, and failures in causal reasoning. This paper makes two contributions. First, it demonstrates how existing LLMs from major platforms exhibit systematic weaknesses when reasoning in scientific domains with negation, counterexamples, or faulty premises \footnote{Code to recreate these experiments are at https://github.com/hannahdavidsoncollege-maker/ScientificReasoningForEnvironment-MedicineWithLLMs. Second, it introduces a dual-reasoning training framework that integrates affirmative generation with structured counterfactual denial. Grounded in formal logic, cognitive science, and adversarial training, this training paradigm formalizes a computational analogue of ``denying the antecedent'' as a mechanism for disconfirmation and robustness. By coupling generative synthesis with explicit negation-aware objectives, the framework enables models that not only affirm valid inferences but also reject invalid ones, yielding systems that are more resilient, interpretable, and aligned with human reasoning.

Contrastive vision-language models continue to be the dominant approach for image and text retrieval. Contrastive Language-Image Pre-training (CLIP) trains two neural networks in contrastive manner to align their image and text embeddings in a shared latent space. Recent results evaluating CLIP on negated or paraphrased text have shown mixed performance because negation changes meaning radically with minimal lexical changes, while paraphrasing can create very different textual expressions with the same intended meaning. This poses a significant challenge for improving the evaluation results and alignment of vision-language models. To address this challenge, this paper evaluates the combination of paraphrasing and negation, proposes a new CLIP contrastive loss function accounting for both paraphrasing and negation, and applies LLM-generated training triples consisting of original, paraphrased and negated textual captions to CLIP-like training models. The approach, called SemCLIP, is shown to move paraphrased captions towards the original image embeddings while pushing negated captions further away in embedding space. Empirically, SemCLIP is shown to be capable of preserving CLIP's performance while increasing considerably the distances to negated captions. On the CC-Neg benchmark using an original over negation image-retrieval accuracy metric, SemCLIP improves accuracy from 68.1% to 78.1%. Although results are mixed when compared with CLIP on the Sugarcrepe++ benchmark, SemCLIP's performance is generally better than the models trained with negated captions. This robustness to negation extends to downstream zero-shot classification tasks where SemCLIP pre-trained on Sugarcrepe++ performs better than CLIP on all tested downstream tasks. These results indicate that SemCLIP can achieve significant robustness to semantic transformations.

LLMs still struggle with hallucination, especially when confronted with symbolic triggers like modifiers, negation, numbers, exceptions, and named entities. Yet, we lack a clear understanding of where these symbolic hallucinations originate, making it crucial to systematically handle such triggers and localize the emergence of hallucination inside the model. While prior work explored localization using statistical techniques like LSC and activation variance analysis, these methods treat all tokens equally and overlook the role symbolic linguistic knowledge plays in triggering hallucinations. So far, no approach has investigated how symbolic elements specifically drive hallucination failures across model layers, nor has symbolic linguistic knowledge been used as the foundation for a localization framework. We propose the first symbolic localization framework that leverages symbolic linguistic and semantic knowledge to meaningfully trace the development of hallucinations across all model layers. By focusing on how models process symbolic triggers, we analyze five models using HaluEval and TruthfulQA. Our symbolic knowledge approach reveals that attention variance for these linguistic elements explodes to critical instability in early layers (2-4), with negation triggering catastrophic variance levels, demonstrating that symbolic semantic processing breaks down from the very beginning. Through the lens of symbolic linguistic knowledge, despite larger model sizes, hallucination rates remain consistently high (78.3%-83.7% across Gemma variants), with steep attention drops for symbolic semantic triggers throughout deeper layers. Our findings demonstrate that hallucination is fundamentally a symbolic linguistic processing failure, not a general generation problem, revealing that symbolic semantic knowledge provides the key to understanding and localizing hallucination mechanisms in LLMs.

This problem is challenging, because KGs can be massive and incomplete.

To show Proposition 1, we need the following defition and lemma. Suppose that K is a proper cone. A sector-cone is always axially symmetric. Therefore, when C is convex, it is axially symmetric. Therefore, a sector-cone is always axially symmetric.