We restate the proposition 1 and its proof here. Given Beta embedding S, S is a fixed point of N N: N (N ( S)) = S. 2. Given Beta embedding S, we have I ({S, S,..., S}) = S. Proof. Then we naturally have S = I ( {S,..., S}) . Here we discuss the computation complexity of representing any given FOL query using the De Morgan's laws (DM) and the disjunctive normal form (DNF). Then for each query structure, we use pre-order traversal starting from the target node/answer to assign an entity/relation to each node/edge iteratively until we instantiate every anchor nodes (the root of the query structure). Table 7: Number of training, validation, and test queries generated for different query structures.

We thank the reviewers for their time and valuable feedback. Below, we clarify a number of important points raised by the reviewers. Reviewers raise concern on multi-modal embeddings. We will highlight this limitation in Sec. R3 suggests that "the authors can adapt the FOL queries to other We argue the differences in tasks and setups below.

We introduce SetBERT, a fine-tuned BERT-based model designed to enhance query embeddings for set operations and Boolean logic queries, such as Intersection (AND), Difference (NOT), and Union (OR). SetBERT significantly improves retrieval performance for logic-structured queries, an area where both traditional and neural retrieval methods typically underperform. We propose an innovative use of inversed-contrastive loss, focusing on identifying the negative sentence, and fine-tuning BERT with a dataset generated via prompt GPT. Furthermore, we demonstrate that, unlike other BERT-based models, fine-tuning with triplet loss actually degrades performance for this specific task. Our experiments reveal that SetBERT-base not only significantly outperforms BERT-base (up to a 63% improvement in Recall) but also achieves performance comparable to the much larger BERT-large model, despite being only one-third the size.

Complex Logical Query Answering (CLQA) over incomplete knowledge graphs is a challenging task. Recently, Query Embedding (QE) methods are proposed to solve CLQA by performing multi-hop logical reasoning. However, most of them only consider historical query context information while ignoring future information, which leads to their failure to capture the complex dependencies behind the elements of a query. In recent years, the transformer architecture has shown a strong ability to model long-range dependencies between words. The bidirectional attention mechanism proposed by the transformer can solve the limitation of these QE methods regarding query context. Still, as a sequence model, it is difficult for the transformer to model complex logical queries with branch structure computation graphs directly. To this end, we propose a neural one-point embedding method called Pathformer based on the tree-like computation graph, i.e., query computation tree. Specifically, Pathformer decomposes the query computation tree into path query sequences by branches and then uses the transformer encoder to recursively encode these path query sequences to obtain the final query embedding. This allows Pathformer to fully utilize future context information to explicitly model the complex interactions between various parts of the path query. Experimental results show that Pathformer outperforms existing competitive neural QE methods, and we found that Pathformer has the potential to be applied to non-one-point embedding space.

Generating accurate Structured Querying Language (SQL) is a long-standing problem, especially in matching users' semantic queries with structured databases and then generating structured SQL. Existing models typically input queries and database schemas into the LLM and rely on the LLM to perform semantic-structure matching and generate structured SQL. However, such solutions overlook the structural information within user queries and databases, which can be utilized to enhance the generation of structured SQL. This oversight can lead to inaccurate or unexecutable SQL generation. To fully exploit the structure, we propose a structure-to-SQL framework, which leverages the inherent structure information to improve the SQL generation of LLMs. Specifically, we introduce our Structure Guided SQL~(SGU-SQL) generation model. SGU-SQL first links user queries and databases in a structure-enhanced manner. It then decomposes complicated linked structures with grammar trees to guide the LLM to generate the SQL step by step. Extensive experiments on two benchmark datasets illustrate that SGU-SQL can outperform sixteen SQL generation baselines.

Multi-hop logical reasoning over knowledge graph (KG) plays a fundamental role in many artificial intelligence tasks. Recent complex query embedding (CQE) methods for reasoning focus on static KGs, while temporal knowledge graphs (TKGs) have not been fully explored. Reasoning over TKGs has two challenges: 1. The query should answer entities or timestamps; 2. The operators should consider both set logic on entity set and temporal logic on timestamp set. To bridge this gap, we define the multi-hop logical reasoning problem on TKGs. With generated three datasets, we propose the first temporal CQE named Temporal Feature-Logic Embedding framework (TFLEX) to answer the temporal complex queries. We utilize vector logic to compute the logic part of Temporal Feature-Logic embeddings, thus naturally modeling all First-Order Logic (FOL) operations on entity set. In addition, our framework extends vector logic on timestamp set to cope with three extra temporal operators (After, Before and Between). Experiments on numerous query patterns demonstrate the effectiveness of our method.

Logical reasoning over incomplete knowledge graphs to answer complex logical queries is a challenging task. With the emergence of new entities and relations in constantly evolving KGs, inductive logical reasoning over KGs has become a crucial problem. However, previous PLMs-based methods struggle to model the logical structures of complex queries, which limits their ability to generalize within the same structure. In this paper, we propose a structure-modeled textual encoding framework for inductive logical reasoning over KGs. It encodes linearized query structures and entities using pre-trained language models to find answers. For structure modeling of complex queries, we design stepwise instructions that implicitly prompt PLMs on the execution order of geometric operations in each query. We further separately model different geometric operations (i.e., projection, intersection, and union) on the representation space using a pre-trained encoder with additional attention and maxout layers to enhance structured modeling. We conduct experiments on two inductive logical reasoning datasets and three transductive datasets. The results demonstrate the effectiveness of our method on logical reasoning over KGs in both inductive and transductive settings.

Knowledge graphs (KGs) capture knowledge in the form of head--relation--tail triples and are a crucial component in many AI systems. There are two important reasoning tasks on KGs: (1) single-hop knowledge graph completion, which involves predicting individual links in the KG; and (2), multi-hop reasoning, where the goal is to predict which KG entities satisfy a given logical query. Embedding-based methods solve both tasks by first computing an embedding for each entity and relation, then using them to form predictions. However, existing scalable KG embedding frameworks only support single-hop knowledge graph completion and cannot be applied to the more challenging multi-hop reasoning task. Here we present Scalable Multi-hOp REasoning (SMORE), the first general framework for both single-hop and multi-hop reasoning in KGs. Using a single machine SMORE can perform multi-hop reasoning in Freebase KG (86M entities, 338M edges), which is 1,500x larger than previously considered KGs. The key to SMORE's runtime performance is a novel bidirectional rejection sampling that achieves a square root reduction of the complexity of online training data generation. Furthermore, SMORE exploits asynchronous scheduling, overlapping CPU-based data sampling, GPU-based embedding computation, and frequent CPU--GPU IO. SMORE increases throughput (i.e., training speed) over prior multi-hop KG frameworks by 2.2x with minimal GPU memory requirements (2GB for training 400-dim embeddings on 86M-node Freebase) and achieves near linear speed-up with the number of GPUs. Moreover, on the simpler single-hop knowledge graph completion task SMORE achieves comparable or even better runtime performance to state-of-the-art frameworks on both single GPU and multi-GPU settings.

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. Here, we present BetaE, a probabilistic embedding framework for answering arbitrary FOL queries over KGs. BetaE is the first method that can handle a complete set of first-order logical operations: conjunction ($\wedge$), disjunction ($\vee$), and negation ($\neg$). A key insight of BetaE is to use probabilistic distributions with bounded support, specifically the Beta distribution, and embed queries/entities as distributions, which as a consequence allows us to also faithfully model uncertainty. Logical operations are performed in the embedding space by neural operators over the probabilistic embeddings. We demonstrate the performance of BetaE on answering arbitrary FOL queries on three large, incomplete KGs. While being more general, BetaE also increases relative performance by up to 25.4% over the current state-of-the-art KG reasoning methods that can only handle conjunctive queries without negation.