If a lattice L is distributive, then L is also modular . By assuming x y, we have x y = y . Hence, from the distributive property we get: x (y z) ( x y) (x z) y ( x z) 14 Definition A.7. Congruence Lattice Reflexivity: For every element a in A, a is related to itself, denoted as a a; Symmetry: For any elements a and b in A, if a b, then b a; Transitivity: For any elements a, b, and c in A, if a b and b c, then a c . An algebra with no other congruences is called simple . A type F is defined as a set of operation symbols along with their respective arities.

If a lattice L is distributive, then L is also modular . By assuming x y, we have x y = y . Hence, from the distributive property we get: x (y z) ( x y) (x z) y ( x z) 14 Definition A.7. Congruence Lattice Reflexivity: For every element a in A, a is related to itself, denoted as a a; Symmetry: For any elements a and b in A, if a b, then b a; Transitivity: For any elements a, b, and c in A, if a b and b c, then a c . An algebra with no other congruences is called simple . A type F is defined as a set of operation symbols along with their respective arities.

All three contributions have a common motivation e.g. Figure 1 (acknowledged by R2, R3).

By uncovering the contrast between Artificial Intelligence and Natural-born Intelligence as a computational process, we define closed computing and open computing, and implement open computing within chemical reactions. This involves forming a mixture and invalidation of the computational process and the execution environment, which are logically distinct, and coalescing both to create a system that adjusts fluctuations. We model chemical reactions by considering the computation as the chemical reaction and the execution environment as the degree of aggregation of molecules that interact with the reactive environment. This results in a chemical reaction that progresses while repeatedly clustering and de-clustering, where concentration no longer holds significant meaning. Open computing is segmented into Token computing, which focuses on the individual behavior of chemical molecules, and Type computing, which focuses on normative behavior. Ultimately, both are constructed as an interplay between the two. In this system, Token computing demonstrates self-organizing critical phenomena, while Type computing exhibits quantum logic. Through their interplay, the recruitment of fluctuations is realized, giving rise to interactions between quantum logical subspaces corresponding to quantum coherence across different Hilbert spaces. As a result, spike waves are formed, enabling signal transmission. This occurrence may be termed quantum-like coherence, implying the source of enzymes responsible for controlling spike waves and biochemical rhythms.

We advance a general theory of coherent preference that surrenders restrictions embodied in orthodox doctrine. This theory enjoys the property that any preference system admits extension to a complete system of preferences, provided it satisfies a certain coherence requirement analogous to the one de Finetti advanced for his foundations of probability. Unlike de Finetti's theory, the one we set forth requires neither transitivity nor Archimedeanness nor boundedness nor continuity of preference. This theory also enjoys the property that any complete preference system meeting the standard of coherence can be represented by utility in an ordered field extension of the reals. Representability by utility is a corollary of this paper's central result, which at once extends H older's Theorem and strengthens Hahn's Embedding Theorem.

Declarative specifications have a vital role to play in developing safe and dependable software systems. Writing specifications correctly, however, remains particularly challenging. This paper presents a controlled experiment on using large language models (LLMs) to write declarative formulas in the well-known language Alloy. Our use of LLMs is three-fold. One, we employ LLMs to write complete Alloy formulas from given natural language descriptions (in English). Two, we employ LLMs to create alternative but equivalent formulas in Alloy with respect to given Alloy formulas. Three, we employ LLMs to complete sketches of Alloy formulas and populate the holes in the sketches by synthesizing Alloy expressions and operators so that the completed formulas accurately represent the desired properties (that are given in natural language). We conduct the experimental evaluation using 11 well-studied subject specifications and employ two popular LLMs, namely ChatGPT and DeepSeek. The experimental results show that the LLMs generally perform well in synthesizing complete Alloy formulas from input properties given in natural language or in Alloy, and are able to enumerate multiple unique solutions. Moreover, the LLMs are also successful at completing given sketches of Alloy formulas with respect to natural language descriptions of desired properties (without requiring test cases). We believe LLMs offer a very exciting advance in our ability to write specifications, and can help make specifications take a pivotal role in software development and enhance our ability to build robust software.

In order to behave intelligently both humans and machines have to represent their knowledge adequately for how it is used. Humans often use analogies to transfer their knowledge to new domains, or help others with this transfer via explanations. Hence, an important question is: What representation can be used to construct, find, and evaluate analogies? In this paper, we study features of a domain that are important for constructing analogies. We do so by formalizing knowledge domains as categories. We use the well-known example of the analogy between the solar system and the hydrogen atom to demonstrate how to construct domain categories. We also show how functors, pullbacks, and pushouts can be used to define an analogy, describe its core and a corresponding blend of the underlying domains.

A widely used Agile practice for requirements is to produce a set of user stories (also called ``agile product backlog''), which roughly includes a list of pairs (role, feature), where the role handles the feature for a certain purpose. In the context of Software Product Lines, the requirements for a family of similar systems is thus a family of user-story sets, one per system, leading to a 3-dimensional dataset composed of sets of triples (system, role, feature). In this paper, we combine Triadic Concept Analysis (TCA) and Large Language Model (LLM) prompting to suggest the user-story set required to develop a new system relying on the variability logic of an existing system family. This process consists in 1) computing 3-dimensional variability expressed as a set of TCA implications, 2) providing the designer with intelligible design options, 3) capturing the designer's selection of options, 4) proposing a first user-story set corresponding to this selection, 5) consolidating its validity according to the implications identified in step 1, while completing it if necessary, and 6) leveraging LLM to have a more comprehensive website. This process is evaluated with a dataset comprising the user-story sets of 67 similar-purpose websites.

While standard Retrieval-Augmented Generation (RAG) based on chunks, GraphRAG structures knowledge as graphs to leverage the relations among entities. However, previous GraphRAG methods are limited by binary relations: one edge in the graph only connects two entities, which cannot well model the n-ary relations among more than two entities that widely exist in reality. To address this limitation, we propose HyperGraphRAG, a novel hypergraph-based RAG method that represents n-ary relational facts via hyperedges, modeling the complicated n-ary relations in the real world. To retrieve and generate over hypergraphs, we introduce a complete pipeline with a hypergraph construction method, a hypergraph retrieval strategy, and a hypergraph-guided generation mechanism. Experiments across medicine, agriculture, computer science, and law demonstrate that HyperGraphRAG outperforms standard RAG and GraphRAG in accuracy and generation quality.