Daily Papers

Large language models (LLMs) have demonstrated strong performance and rapid progress in a wide range of medical reasoning tasks. However, their sequential autoregressive decoding forces inherently parallel clinical reasoning, such as differential diagnosis, into a single linear reasoning path, limiting both efficiency and reliability for complex medical problems. To address this, we propose MedVerse, a reasoning framework for complex medical inference that reformulates medical reasoning as a parallelizable directed acyclic graph (DAG) process based on Petri net theory. The framework adopts a full-stack design across data, model architecture, and system execution. For data creation, we introduce the MedVerse Curator, an automated pipeline that synthesizes knowledge-grounded medical reasoning paths and transforms them into Petri net-structured representations. At the architectural level, we propose a topology-aware attention mechanism with adaptive position indices that supports parallel reasoning while preserving logical consistency. Systematically, we develop a customized inference engine that supports parallel execution without additional overhead. Empirical evaluations show that MedVerse improves strong general-purpose LLMs by up to 8.9%. Compared to specialized medical LLMs, MedVerse achieves comparable performance while delivering a 1.3x reduction in inference latency and a 1.7x increase in generation throughput, enabled by its parallel decoding capability.

Diabetes is a chronic condition, considered one of the civilization diseases, that is characterized by sustained high blood sugar levels. There is no doubt that more and more people is going to suffer from diabetes, hence it is crucial to understand better its biological foundations. The essential processes related to the control of glucose levels in the blood are: glycolysis (process of breaking down of glucose) and glucose synthesis, both taking place in the liver. The glycolysis occurs during feeding and it is stimulated by insulin. On the other hand, the glucose synthesis arises during fasting and it is stimulated by glucagon. In the paper we present a Petri net model of glycolysis and glucose synthesis in the liver. The model is created based on medical literature. Standard Petri nets techniques are used to analyse the properties of the model: traps, reachability graphs, tokens dynamics, deadlocks analysis. The results are described in the paper. Our analysis shows that the model captures the interactions between different enzymes and substances, which is consistent with the biological processes occurring during fasting and feeding. The model constitutes the first element of our long-time goal to create the whole body model of the glucose regulation in a healthy human and a person with diabetes.

This paper presents a concrete and a symbolic rewriting logic semantics for parametric time Petri nets with inhibitor arcs (PITPNs), a flexible model of timed systems where parameters are allowed in firing bounds. We prove that our semantics is bisimilar to the "standard" semantics of PITPNs. This allows us to use the rewriting logic tool Maude, combined with SMT solving, to provide sound and complete formal analyses for PITPNs. We develop and implement a new general folding approach for symbolic reachability, so that Maude-with-SMT reachability analysis terminates whenever the parametric state-class graph of the PITPN is finite. Our work opens up the possibility of using the many formal analysis capabilities of Maude -- including full LTL model checking, analysis with user-defined analysis strategies, and even statistical model checking -- for such nets. We illustrate this by explaining how almost all formal analysis and parameter synthesis methods supported by the state-of-the-art PITPN tool Romeo can be performed using Maude with SMT. In addition, we also support analysis and parameter synthesis from parametric initial markings, as well as full LTL model checking and analysis with user-defined execution strategies. Experiments show that our methods outperform Romeo in many cases.

Industrial plants are prone to faults. To notify the operator of a fault occurrence, alarms are utilized as a basic part of modern computer-controlled plants. However, due to the interconnections of different parts of a plant, a single fault often propagates through the plant and triggers a (sometimes large) number of alarms. A graphical plant topology can help operators, process engineers and maintenance experts find the root cause of a plant upset or discover the propagation path of a fault. In this paper, a method is developed to extract plant topology form alarm data. The method is based on process mining, a collection of concepts and algorithms that model a process (not necessarily an engineering one) based on recorded events. The event based nature of alarm data as well as the chronological order of recorded alarms make them suitable for process mining. The methodology developed in this paper is based on preparing alarm data for process mining and then using the suitable process mining algorithms to extract plant topology. The extracted topology is represented by the familiar Petri net which can be used for root cause analysis and discovering fault propagation paths. The methods to evaluate the extracted topology are also discussed. A case study on the well-known Tennessee Eastman process demonstrates the utility of the proposed method.

In the past years, predictive process monitoring (PPM) techniques based on artificial neural networks have evolved as a method to monitor the future behavior of business processes. Existing approaches mostly focus on interpreting the processes as sequences, so-called traces, and feeding them to neural architectures designed to operate on sequential data such as recurrent neural networks (RNNs) or transformers. In this study, we investigate an alternative way to perform PPM: by transforming each process in its directly-follows-graph (DFG) representation we are able to apply graph neural networks (GNNs) for the prediction tasks. By this, we aim to develop models that are more suitable for complex processes that are long and contain an abundance of loops. In particular, we present different ways to create DFG representations depending on the particular GNN we use. The tested GNNs range from classical node-based to novel edge-based architectures. Further, we investigate the possibility of using multi-graphs. By these steps, we aim to design graph representations that minimize the information loss when transforming traces into graphs.

Process model forecasting (PMF) aims to predict the process model that will characterize a future period, thereby providing a process-level view of how behavior is expected to evolve. Existing PMF methods, however, forecast directly-follows graphs, which cannot explicitly represent concurrency. We extend PMF to causal nets by forecasting time series of relation and binding counts and using these forecasts to reconstruct future process models with AND/XOR semantics. To evaluate the resulting models, we introduce a protocol that accounts for partial traces and constructs the workflow nets required for conformance checking. Experiments on four event logs show that the forecasted models achieve conformance levels close to those of models re-mined from observations in the corresponding future windows. They also outperform static discovery baselines, which retain high precision on the structurally stable log but exhibit substantial precision losses on the other three logs. Filtering infrequent bindings improves most conformance metrics, although it also removes much of the concurrent behavior captured by the models.

Probabilistic model checking is a technique for formal automated reasoning about software or hardware systems that operate in the context of uncertainty or stochasticity. It builds upon ideas and techniques from a diverse range of fields, from logic, automata and graph theory, to optimisation, numerical methods and control. In recent years, probabilistic model checking has also been extended to integrate ideas from game theory, notably using models such as stochastic games and solution concepts such as equilibria, to formally verify the interaction of multiple rational agents with distinct objectives. This provides a means to reason flexibly about agents acting in either an adversarial or a collaborative fashion, and opens up opportunities to tackle new problems within, for example, artificial intelligence, robotics and autonomous systems. In this paper, we summarise some of the advances in this area, and highlight applications for which they have already been used. We discuss how the strengths of probabilistic model checking apply, or have the potential to apply, to the multi-agent setting and outline some of the key challenges required to make further progress in this field.

Much has been said about observability in system theory and control; however, it has been recently that observability in complex networks has seriously attracted the attention of researchers. This paper examines the state-of-the-art and discusses some issues raised due to "complexity" and "stochasticity". These unresolved issues call for a new practical methodology. For stochastic systems, a degree of observability may be defined and the observability problem is not a binary (i.e., yes-no) question anymore. Here, we propose to employ a goal-seeking system to play a supervisory role in the network. Hence, improving the degree of observability would be a valid objective for the supervisory system. Towards this goal, the supervisor dynamically optimizes the observation process by reconfiguring the sensory parts in the network. A cognitive dynamic system is suggested as a proper choice for the supervisory system. In this framework, the network itself is viewed as the environment with which the cognitive dynamic system interacts. Computer experiments confirm the potential of the proposed approach for addressing some of the issues raised in networks due to complexity and stochasticity.

We develop a discipline-agnostic emergence calculus that treats theories as fixed points of idempotent operators acting on descriptions. We show that, once processes are composable but access to the underlying system is mediated by a bounded observational interface, a canonical toolkit of six closure-changing primitives (P1--P6) is unavoidable. The framework unifies order-theoretic closure operators with dynamics-induced endomaps E_{τ,f} built from a Markov kernel, a coarse-graining lens, and a time scale τ. We introduce a computable total-variation idempotence defect for E_{τ,f}; small retention error implies approximate idempotence and yields stable "objects" packaged at the chosen τ within a fixed lens. For directionality, we define an arrow-of-time functional as the path-space KL divergence between forward and time-reversed trajectories and prove it is monotone under coarse-graining (data processing); we also formalize a protocol-trap audit showing that protocol holonomy alone cannot sustain asymmetry without a genuine affinity in the lifted dynamics. Finally, we prove a finite forcing-style counting lemma: relative to a partition-based theory, definable predicate extensions are exponentially rare, giving a clean anti-saturation mechanism for strict ladder climbing.

We introduce Diagram of Thought (DoT), a framework that models iterative reasoning in large language models (LLMs) as the construction of a directed acyclic graph (DAG) within a single model. Unlike traditional approaches that represent reasoning as linear chains or trees, DoT organizes propositions, critiques, refinements, and verifications into a cohesive DAG structure, allowing the model to explore complex reasoning pathways while maintaining logical consistency. Each node in the diagram corresponds to a proposition that has been proposed, critiqued, refined, or verified, enabling the LLM to iteratively improve its reasoning through natural language feedback. By leveraging auto-regressive next-token prediction with role-specific tokens, DoT facilitates seamless transitions between proposing ideas and critically evaluating them, providing richer feedback than binary signals. Furthermore, we formalize the DoT framework using Topos Theory, providing a mathematical foundation that ensures logical consistency and soundness in the reasoning process. This approach enhances both the training and inference processes within a single LLM, eliminating the need for multiple models or external control mechanisms. DoT offers a conceptual framework for designing next-generation reasoning-specialized models, emphasizing training efficiency, robust reasoning capabilities, and theoretical grounding. The code is available at https://github.com/diagram-of-thought/diagram-of-thought.

The integration of knowledge extracted from diverse models, whether described by domain experts or generated by machine learning algorithms, has historically been challenged by the absence of a suitable framework for specifying and integrating structures, learning processes, data transformations, and data models or rules. In this work, we extend algebraic specification methods to address these challenges within such a framework. In our work, we tackle the challenging task of developing a comprehensive framework for defining and analyzing deep learning architectures. We believe that previous efforts have fallen short by failing to establish a clear connection between the constraints a model must adhere to and its actual implementation. Our methodology employs graphical structures that resemble Ehresmann's sketches, interpreted within a universe of fuzzy sets. This approach offers a unified theory that elegantly encompasses both deterministic and non-deterministic neural network designs. Furthermore, we highlight how this theory naturally incorporates fundamental concepts from computer science and automata theory. Our extended algebraic specification framework, grounded in graphical structures akin to Ehresmann's sketches, offers a promising solution for integrating knowledge across disparate models and domains. By bridging the gap between domain-specific expertise and machine-generated insights, we pave the way for more comprehensive, collaborative, and effective approaches to knowledge integration and modeling.

P. Kabamba developed generation theory as a tool for studying self-reproducing systems. We provide an alternative definition of a generation system and give a complete solution to the problem of finding optimal seeds for a finite self-replicating system. We also exhibit examples illustrating a connection between self-replication and fixed-point theory.

Large-scale formalization of advanced mathematics requires more than translating individual statements: it must reconstruct a coherent theory distributed across heterogeneous sources. This process raises four challenges: discovering implicit dependencies, correcting source defects, preserving semantic fidelity, and reconciling cross-source misalignments. We present FormaTheoria, an end-to-end, AI-assisted workflow that coordinates source acquisition, formalization, proof construction, recursive dependency discovery, independent review, and reconciliation, while preserving provenance and protecting approved declarations. A shared agent framework supports long-horizon execution through tool use, context compaction, review-gated termination, section-level source context, and dependency-aware batch parallelization. Applying FormaTheoria to major components of the Classification of Finite Simple Groups (CFSG), we construct a machine-checked Lean development extending through the Bender--Suzuki theorem and encompassing the Feit--Thompson Odd Order Theorem, Glauberman's Z^* theorem, and the Brauer--Suzuki theorem. This development verifies an extensive body of deeply interdependent finite-group theory while providing a foundation for continuing the CFSG formalization. An empirical analysis of the code and recorded construction process supports the practical relevance of the identified challenges and illustrates the roles of the corresponding workflow components. Together, these results demonstrate how AI-assisted workflows can reconstruct mathematically significant formal theories from distributed literature by combining language-model agents with formal verification, structured review, and explicit dependency management.

This article presents a theoretical framework for adapting the Common Model of Cognition to large generative network models within the field of artificial intelligence. This can be accomplished by restructuring modules within the Common Model into shadow production systems that are peripheral to a central production system, which handles higher-level reasoning based on the shadow productions' output. Implementing this novel structure within the Common Model allows for a seamless connection between cognitive architectures and generative neural networks.

We develop a framework, in the style of Adler, for interpreting the notion of "witnessing" that has appeared (usually as a variant of Kim's Lemma) in different areas of neostability theory as a binary relation between abstract independence relations. This involves extending the relativisations of Kim-independence and Conant-independence due to Mutchnik to arbitrary independence relations. After developing this framework, we show that several results from simplicity, NTP_2, NSOP_1, and beyond follow as instances of general theorems for abstract independence relations. In particular, we prove the equivalence between witnessing and symmetry and the implications from this notion to chain local character and the weak independence theorem, and recover some partial converses. Finally, we use this framework to prove a dichotomy between NSOP_1 and Kruckman and Ramsey's BTP that applies to most known NSOP_4 examples in the literature.

When we speak of recursive self-improvement (RSI), are we speaking of a phenomenon, a mechanism, or a prospect? Towards autonomous and evolving intelligence, RSI is being claimed at many scales, while no single framework that formally describes these emerging instances exists. Its counterpart in the classical realm, iterative policy improvement, is characterized by generalized policy iteration (GPI), a framework of broad applicability with well-understood theoretical properties, but only where the update principle and the evaluation base lie outside the agent. In this paper, we propose Generalized Agent Iteration (GAI), a formal framework that describes iterative policy improvement and RSI as two cases of a single learning paradigm. GAI defines the agent as a configuration of modifiable components within a system and models the learning process as a cycle of agent evaluation and agent improvement. Two pivotal dials then distinguish the instances: whether the improving mechanism is part of the agent and whether the standard it is measured against is grounded outside it. The former dial delineates the boundary between GPI and RSI, and the latter determines a system's polarity as anchored, goal drift, or fully self-referential. Moreover, we use these coordinates to place existing systems on the same two axes and make the defects of recursive self-improvement statable one condition at a time. We see this paper as a first step toward exploring a formal characterization of RSI that rests on the classical account, makes existing systems comparable, and provides a principled basis for analyzing and designing new ones.

Harnessing parallelism in seemingly sequential models is a central challenge for modern machine learning. Several approaches have been proposed for evaluating sequential processes in parallel using iterative fixed-point methods, like Newton, Picard, and Jacobi iterations. In this work, we show that these methods can be understood within a common framework based on linear dynamical systems (LDSs), where different iteration schemes arise naturally as approximate linearizations of a nonlinear recursion. Moreover, we theoretically analyze the rates of convergence of these methods, and we verify the predictions of this theory with several case studies. This unifying framework highlights shared principles behind these techniques and clarifies when particular fixed-point methods are most likely to be effective. By bridging diverse algorithms through the language of LDSs, the framework provides a clearer theoretical foundation for parallelizing sequential models and points toward new opportunities for efficient and scalable computation.

This paper aims to extend the Structured Knowledge Accumulation (SKA) framework recently proposed by mahi2025ska. We introduce two core concepts: the Tensor Net function and the characteristic time property of neural learning. First, we reinterpret the learning rate as a time step in a continuous system. This transforms neural learning from discrete optimization into continuous-time evolution. We show that learning dynamics remain consistent when the product of learning rate and iteration steps stays constant. This reveals a time-invariant behavior and identifies an intrinsic timescale of the network. Second, we define the Tensor Net function as a measure that captures the relationship between decision probabilities, entropy gradients, and knowledge change. Additionally, we define its zero-crossing as the equilibrium state between decision probabilities and entropy gradients. We show that the convergence of entropy and knowledge flow provides a natural stopping condition, replacing arbitrary thresholds with an information-theoretic criterion. We also establish that SKA dynamics satisfy a variational principle based on the Euler-Lagrange equation. These findings extend SKA into a continuous and self-organizing learning model. The framework links computational learning with physical systems that evolve by natural laws. By understanding learning as a time-based process, we open new directions for building efficient, robust, and biologically-inspired AI systems.

When multi-agent systems (MAS) fail, identifying where the decisive error occurred is the first step for automated recovery to an earlier state. Error attribution remains a fundamental challenge due to the long interaction traces that large language model-based MAS generate. This paper presents a framework for error attribution based on conformal prediction (CP) which provides finite-sample, distribution-free coverage guarantees. We introduce new algorithms for filtration-based CP designed for sequential data such as agent trajectories. Unlike existing CP algorithms, our approach predicts sets that are contiguous sequences to enable efficient recovery and debugging. We verify our theoretical guarantees on a variety of agents and datasets, show that errors can be precisely isolated, then use prediction sets to rollback MAS to correct their own errors. Our overall approach is model-agnostic, and offers a principled uncertainty layer for MAS error attribution. We release code at https://github.com/layer6ai-labs/conformal-agent-error-attribution.

The aim of this paper is to present an elementary computable theory of probability, random variables and stochastic processes. The probability theory is baed on existing approaches using valuations and lower integrals. Various approaches to random variables are discussed, including the approach based on completions in a Polish space. We apply the theory to the study of stochastic dynamical systems in discrete-time, and give a brief exposition of the Wiener process as a foundation for stochastic differential equations. The theory is based within the framework of type-two effectivity, so has an explicit direct link with Turing computation, and is expressed in a system of computable types and operations, so has a clean mathematical description.

This paper presents the Relational Machine Calculus (RMC): a simple, foundational model of first-order relational programming. The RMC originates from the Functional Machine Calculus (FMC), which generalizes the lambda-calculus and its standard call-by-name stack machine in two directions. One, "locations", introduces multiple stacks, which enable effect operators to be encoded into the abstraction and application constructs. The second, "sequencing", introduces the imperative notions of "skip" and "sequence", similar to kappa-calculus and concatenative programming languages. The key observation of the RMC is that the first-order fragment of the FMC exhibits a latent duality which, given a simple decomposition of the relevant constructors, can be concretely expressed as an involution on syntax. Semantically, this gives rise to a sound and complete calculus for string diagrams of Frobenius monoids. We consider unification as the corresponding symmetric generalization of beta-reduction. By further including standard operators of Kleene algebra, the RMC embeds a range of computational models: the kappa-calculus, logic programming, automata, Interaction Nets, and Petri Nets, among others. These embeddings preserve operational semantics, which for the RMC is again given by a generalization of the standard stack machine for the lambda-calculus. The equational theory of the RMC (which supports reasoning about its operational semantics) is conservative over both the first-order lambda-calculus and Kleene algebra, and can be oriented to give a confluent reduction relation.

In this work, we present two results: The first result is the formalization of Tutte's theorem in Lean, a key theorem concerning matchings in graph theory. As this formalization is ready to be integrated in Lean's mathlib, it provides a valuable step in the path towards formalizing research-level mathematics in this area. The second result is a framework for doing educational formalization projects. This framework provides a structure to learn to formalize mathematics with minimal teacher input. This framework applies to both traditional academic settings and independent community-driven environments. We demonstrate the framework's use by connecting it to the process of formalizing Tutte's theorem.

This is an extended version of our publication Learning state machines from data streams: A generic strategy and an improved heuristic, International Conference on Grammatical Inference (ICGI) 2023, Rabat, Morocco. It has been extended with a formal proof on PAC-bounds, and the discussion and analysis of a similar approach has been moved from the appendix and now has a full dedicated section. State machine models are models that simulate the behavior of discrete event systems, capable of representing systems such as software systems, network interactions, and control systems, and have been researched extensively. The nature of most learning algorithms however is the assumption that all data be available at the beginning of the algorithm, and little research has been done in learning state machines from streaming data. In this paper, we want to close this gap further by presenting a generic method for learning state machines from data streams, as well as a merge heuristic that uses sketches to account for incomplete prefix trees. We implement our approach in an open-source state merging library and compare it with existing methods. We show the effectiveness of our approach with respect to run-time, memory consumption, and quality of results on a well known open dataset. Additionally, we provide a formal analysis of our algorithm, showing that it is capable of learning within the PAC framework, and show a theoretical improvement to increase run-time, without sacrificing correctness of the algorithm in larger sample sizes.

Networks provide a powerful formalism for modeling complex systems by using a model of pairwise interactions. But much of the structure within these systems involves interactions that take place among more than two nodes at once; for example, communication within a group rather than person-to person, collaboration among a team rather than a pair of coauthors, or biological interaction between a set of molecules rather than just two. Such higher-order interactions are ubiquitous, but their empirical study has received limited attention, and little is known about possible organizational principles of such structures. Here we study the temporal evolution of 19 datasets with explicit accounting for higher-order interactions. We show that there is a rich variety of structure in our datasets but datasets from the same system types have consistent patterns of higher-order structure. Furthermore, we find that tie strength and edge density are competing positive indicators of higher-order organization, and these trends are consistent across interactions involving differing numbers of nodes. To systematically further the study of theories for such higher-order structures, we propose higher-order link prediction as a benchmark problem to assess models and algorithms that predict higher-order structure. We find a fundamental differences from traditional pairwise link prediction, with a greater role for local rather than long-range information in predicting the appearance of new interactions.

We present Eureka, a task-conditioned Meta-Agent architecture that compiles long-horizon tasks into dynamic obligation graphs with explicit acceptance semantics. During execution, Eureka forms Macro-Agents with specialized state, memory, operators, tools, verifiers, and local topology via receding-horizon planning, architecture promotion, and minimal-sufficient compilation. When bottlenecks recur, cost-benefit-gated evolution updates the local architecture under constraints. Theoretically, we establish results on regret, planning invalidation, amortization, subtree interfaces, serializability, and verification. Experimentally, Eureka completes 170/170 recursive tasks and generates 3,948 certificates with no false acceptances. Active context compresses median input from 9,490 to 4,005 tokens; incremental processing avoids 65.38% recomputation across 12,000 tasks; 16,000 concurrent executions serialize consistently. The same Meta-Agent instantiates a Theory-Discovery Agent and a Math/Conjecture Agent. The former yields structural results in quantum-process and spacetime theory. The latter identifies bottlenecks in Riemann Hypothesis research and advances a positivity certificate for Suzuki's localized Weil quadratic form to 0 < a <= 69/200 = 0.345, reaching ~99.55% of (log 2)/2. These results suggest that scientific-agent capability depends not only on the base model but on whether an architecture can be formed to match the task's cognitive structure.

Bayesian models of cognition have gained considerable traction in computational neuroscience and psychiatry. Their scopes are now expected to expand rapidly to artificial intelligence, providing general inference frameworks to support embodied, adaptable, and energy-efficient autonomous agents. A central theory in this domain is predictive coding, which posits that learning and behaviour are driven by hierarchical probabilistic inferences about the causes of sensory inputs. Biological realism constrains these networks to rely on simple local computations in the form of precision-weighted predictions and prediction errors. This can make this framework highly efficient, but its implementation comes with unique challenges on the software development side. Embedding such models in standard neural network libraries often becomes limiting, as these libraries' compilation and differentiation backends can force a conceptual separation between optimization algorithms and the systems being optimized. This critically departs from other biological principles such as self-monitoring, self-organisation, cellular growth and functional plasticity. In this paper, we introduce pyhgf: a Python package backed by JAX and Rust for creating, manipulating and sampling dynamic networks for predictive coding. We improve over other frameworks by enclosing the network components as transparent, modular and malleable variables in the message-passing steps. The resulting graphs can implement arbitrary computational complexities as beliefs propagation. But the transparency of core variables can also translate into inference processes that leverage self-organisation principles, and express structure learning, meta-learning or causal discovery as the consequence of network structural adaptation to surprising inputs. The code, tutorials and documentation are hosted at: https://github.com/ilabcode/pyhgf.

Artificial intelligence has made great strides in the last decade but still falls short of the human brain, the best-known example of intelligence. Not much is known of the neural processes that allow the brain to make the leap to achieve so much from so little beyond its ability to create knowledge structures that can be flexibly and dynamically combined, recombined, and applied in new and novel ways. This paper proposes a mathematical approach using graph theory and spectral graph theory, to hypothesize how to constrain these neural clusters of information based on eigen-relationships. This same hypothesis is hierarchically applied to scale up from the smallest to the largest clusters of knowledge that eventually lead to model building and reasoning.

We present our position on the elusive quest for a general-purpose framework for specifying and studying deep learning architectures. Our opinion is that the key attempts made so far lack a coherent bridge between specifying constraints which models must satisfy and specifying their implementations. Focusing on building a such a bridge, we propose to apply category theory -- precisely, the universal algebra of monads valued in a 2-category of parametric maps -- as a single theory elegantly subsuming both of these flavours of neural network design. To defend our position, we show how this theory recovers constraints induced by geometric deep learning, as well as implementations of many architectures drawn from the diverse landscape of neural networks, such as RNNs. We also illustrate how the theory naturally encodes many standard constructs in computer science and automata theory.

We build a team of specialized large language-model agents and present an agent-driven workflow for research-level formalization in theoretical physics, with the autoformalization of the fundamental theorem of matrix-product states as a demonstration. The agents, coordinated through a structured mathematical blueprint and periodic human review, orchestrated and executed the full formalization autonomously. For some statements, the agents were able to explore new proof routes that are not part of the standard literature. Along the way the agents produced extensive tensor-network and quantum-information libraries not previously available in Mathlib, Lean's mathematical library. As a physical application, the formalization also extends towards symmetry-protected topological phases in one dimension. We find that the main bottleneck in large-scale autoformalization is enforcing mathematical intent and we provide a detailed study of the full process and various subtleties involved. We release the codebase as the library https://github.com/LionSR/TNLean{TNLean}, together with a -chapter https://lionsr.github.io/TNLean/blueprint/{blueprint} of the formalization effort.

Graphs have often been used to answer questions about the interaction between real-world entities by taking advantage of their capacity to represent complex topologies. Complex networks are known to be graphs that capture such non-trivial topologies; they are able to represent human phenomena such as epidemic processes, the dynamics of populations, and the urbanization of cities. The investigation of complex networks has been extrapolated to many fields of science, with particular emphasis on computing techniques, including artificial intelligence. In such a case, the analysis of the interaction between entities of interest is transposed to the internal learning of algorithms, a paradigm whose investigation is able to expand the state of the art in Computer Science. By exploring this paradigm, this thesis puts together complex networks and machine learning techniques to improve the understanding of the human phenomena observed in pandemics, pendular migration, and street networks. Accordingly, we contribute with: (i) a new neural network architecture capable of modeling dynamic processes observed in spatial and temporal data with applications in epidemics propagation, weather forecasting, and patient monitoring in intensive care units; (ii) a machine-learning methodology for analyzing and predicting links in the scope of human mobility between all the cities of Brazil; and, (iii) techniques for identifying inconsistencies in the urban planning of cities while tracking the most influential vertices, with applications over Brazilian and worldwide cities. We obtained results sustained by sound evidence of advances to the state of the art in artificial intelligence, rigorous formalisms, and ample experimentation. Our findings rely upon real-world applications in a range of domains, demonstrating the applicability of our methodologies.

Multi-agent reinforcement learning has recently shown great promise as an approach to networked system control. Arguably, one of the most difficult and important tasks for which large scale networked system control is applicable is common-pool resource management. Crucial common-pool resources include arable land, fresh water, wetlands, wildlife, fish stock, forests and the atmosphere, of which proper management is related to some of society's greatest challenges such as food security, inequality and climate change. Here we take inspiration from a recent research program investigating the game-theoretic incentives of humans in social dilemma situations such as the well-known tragedy of the commons. However, instead of focusing on biologically evolved human-like agents, our concern is rather to better understand the learning and operating behaviour of engineered networked systems comprising general-purpose reinforcement learning agents, subject only to nonbiological constraints such as memory, computation and communication bandwidth. Harnessing tools from empirical game-theoretic analysis, we analyse the differences in resulting solution concepts that stem from employing different information structures in the design of networked multi-agent systems. These information structures pertain to the type of information shared between agents as well as the employed communication protocol and network topology. Our analysis contributes new insights into the consequences associated with certain design choices and provides an additional dimension of comparison between systems beyond efficiency, robustness, scalability and mean control performance.

The solution methods used to realize artificial general intelligence (AGI) may not contain the formalism needed to adequately model and characterize AGI. In particular, current approaches to learning hold notions of problem domain and problem task as fundamental precepts, but it is hardly apparent that an AGI encountered in the wild will be discernable into a set of domain-task pairings. Nor is it apparent that the outcomes of AGI in a system can be well expressed in terms of domain and task, or as consequences thereof. Thus, there is both a practical and theoretical use for meta-theories of learning which do not express themselves explicitly in terms of solution methods. General systems theory offers such a meta-theory. Herein, Mesarovician abstract systems theory is used as a super-structure for learning. Abstract learning systems are formulated. Subsequent elaboration stratifies the assumptions of learning systems into a hierarchy and considers the hierarchy such stratification projects onto learning theory. The presented Mesarovician abstract learning systems theory calls back to the founding motivations of artificial intelligence research by focusing on the thinking participants directly, in this case, learning systems, in contrast to the contemporary focus on the problems thinking participants solve.

With the development of foundation model (FM), agentic AI systems are getting more attention, yet their inherent issues like hallucination and poor reasoning, coupled with the frequent ad-hoc nature of system design, lead to unreliable and brittle applications. Existing efforts to characterise agentic design patterns often lack a rigorous systems-theoretic foundation, resulting in high-level or convenience-based taxonomies that are difficult to implement. This paper addresses this gap by introducing a principled methodology for engineering robust AI agents. We propose two primary contributions: first, a novel system-theoretic framework that deconstructs an agentic AI system into five core, interacting functional subsystems: Reasoning & World Model, Perception & Grounding, Action Execution, Learning & Adaptation, and Inter-Agent Communication. Second, derived from this architecture and directly mapped to a comprehensive taxonomy of agentic challenges, we present a collection of 12 agentic design patterns. These patterns - categorised as Foundational, Cognitive & Decisional, Execution & Interaction, and Adaptive & Learning - offer reusable, structural solutions to recurring problems in agent design. The utility of the framework is demonstrated by a case study on the ReAct framework, showing how the proposed patterns can rectify systemic architectural deficiencies. This work provides a foundational language and a structured methodology to standardise agentic design communication among researchers and engineers, leading to more modular, understandable, and reliable autonomous systems.

AGI has become the Holly Grail of AI with the promise of level intelligence and the major Tech companies around the world are investing unprecedented amounts of resources in its pursuit. Yet, there does not exist a single formal definition and only some empirical AGI benchmarking frameworks currently exist. The main purpose of this paper is to develop a general, algebraic and category theoretic framework for describing, comparing and analysing different possible AGI architectures. Thus, this Category theoretic formalization would also allow to compare different possible candidate AGI architectures, such as, RL, Universal AI, Active Inference, CRL, Schema based Learning, etc. It will allow to unambiguously expose their commonalities and differences, and what is even more important, expose areas for future research. From the applied Category theoretic point of view, we take as inspiration Machines in a Category to provide a modern view of AGI Architectures in a Category. More specifically, this first position paper provides, on one hand, a first exercise on RL, Causal RL and SBL Architectures in a Category, and on the other hand, it is a first step on a broader research program that seeks to provide a unified formal foundation for AGI systems, integrating architectural structure, informational organization, agent realization, agent and environment interaction, behavioural development over time, and the empirical evaluation of properties. This framework is also intended to support the definition of architectural properties, both syntactic and informational, as well as semantic properties of agents and their assessment in environments with explicitly characterized features. We claim that Category Theory and AGI will have a very symbiotic relation.

While progress in informal mathematics has come largely from scaling the chain-of-thought and agentic reasoning of frontier language models, formal theorem proving has increasingly relied on human-designed workflows around these models, such as planning, decomposition, and orchestrating specialized provers. This raises a bitter-lesson question: can a simple agentic loop that scales test-time compute match these workflows? The answer matters as formal verification must scale to keep up with frontier models producing research mathematics, yet new workflows rarely settle it, as their gains are often confounded with stronger models or more compute. We study this question with h1, a simple harness that gives a model an efficient interface to Lean 4 and keeps it working through verifier feedback and self-reflection, without imposing a decomposition or search structure. h1 reaches state-of-the-art results on PutnamBench, FATE-X, and FormalConjectures100-Solved, settles eight open problems in FormalConjectures, and, at matched models, matches or outperforms specialized workflows such as Goedel-Architect and AlphaProof Nexus, surprisingly showing that a single attempt (pass@1) can often scale to solve open problems. Through ablations and matched-model comparisons with Claude Code and AxProverBase, we find that simplicity alone is not enough: an efficient Lean interface largely determines cost and solve rate. We thus recommend that future work evaluate frontier models with efficient execution feedback before prescribing workflows, at the risk of bitter lessons.

The Hitchhiker's Guide to Agentic AI is a comprehensive practitioner's reference for building autonomous AI systems. The book covers the full stack from first principles to production deployment, organized around a central thesis: building great agentic systems requires understanding every layer of the pipeline, not just one. The book opens with the LLM substrate -- transformer architecture, GPU systems, training and fine-tuning (SFT,LoRA, MoE), model compression, and inference optimization -- treated as essential foundations rather than the primary focus. It then develops the alignment and reasoning layer: reinforcement learning from human feedback (RLHF), PPO, DPO and its variants, GRPO, reward modeling, and RL for large reasoning models including chain-of-thought and test-time scaling. The second half is devoted to agentic AI proper. Topics include agentic training and trajectory-based RL, retrieval-augmented generation (RAG and Agentic RAG), memory systems (in-context, external, episodic, and semantic), agent harness design and context management, and a taxonomy of agent design patterns. Inter-agent coordination is covered in depth: the Model Context Protocol (MCP), agent skills and tool use, the Agent-to-Agent (A2A) communication protocol, and multi-agent architectures spanning centralized, decentralized, and hierarchical topologies. The book concludes with agent development frameworks, agentic UI design, evaluation methodology for agentic tasks, and production deployment. Each chapter pairs rigorous theoretical foundations with implementation guidance, code examples, and references to the primary literature.

Recently bipartite graphs have been widely used to represent the relationship two sets of items for information retrieval applications. The Web offers a wide range of data which can be represented by bipartite graphs, such us movies and reviewers in recomender systems, queries and URLs in search engines, users and posts in social networks. The size and the dynamic nature of such graphs generate the need for more efficient ranking methods. In this thesis, at first we present the fundamental mathematical backround that we use subsequently and we describe the basic principles of the Perron-Frobebius theory for non negative matrices as well as the the basic principles of the Markov chain theory. Then, we propose a novel algorithm named BipartiteRank, which is suitable to rank scenarios, that can be represented as a bipartite graph. This algorithm is based on the random surfer model and inherits the basic mathematical characteristics of PageRank. What makes it different, is the fact that it introduces an alternative type of teleportation, based on the block structure of the bipartite graph in order to achieve more efficient ranking. Finally, we support this opinion with mathematical arguments and then we confirm it experimentally through a series of tests on real data.

This work proposes a novel framework based on nested evolving set processes to accelerate Personalized PageRank (PPR) computation. At each stage of the process, we employ a localized inexact proximal point iteration to solve a simplified linear system. We show that the time complexity of such localized methods is upper bounded by min{mathcal{O}(R^2/ε^2), mathcal{O}(m)} to obtain an ε-approximation of the PPR vector, where m denotes the number of edges in the graph and R is a constant defined via nested evolving set processes. Furthermore, the algorithms induced by our framework require solving only mathcal{O}(1/sqrtα) such linear systems, where α is the damping factor. When 1/ε^2ll m, this implies the existence of an algorithm that computes an epsilon -approximation of the PPR vector with an overall time complexity of mathcal{O}left(R^2 / (sqrtαε^2)right), independent of the underlying graph size. Our result resolves an open conjecture from existing literature. Experimental results on real-world graphs validate the efficiency of our methods, demonstrating significant convergence in the early stages.

Most agent frameworks are built around the language model: a conversation loop comes first, then tools, then rules, and finally a logging layer bolted on for observability, with state persisted as retrievable "memory." We describe ActiveGraph, a runtime that inverts this arrangement. The append-only event log is the source of truth; the working graph is a deterministic projection of that log; and behaviors--ordinary functions, classes, LLM-backed routines, or logic attached to typed edges--react to changes in the graph and emit new events. No component instructs another; coordination happens entirely through the shared graph. This single design decision yields three properties that retrieval-and-summarization memory systems do not provide: deterministic replay of any run from its log, cheap forking that branches a run at any event without re-executing the shared prefix, and end-to-end lineage from a high-level goal down to the individual model call that produced each artifact. We present the architecture, a determinism contract that makes replay sound, and a worked diligence example whose full causal structure is reconstructable from the log alone. We discuss--without claiming to demonstrate--why this substrate is unusually well suited to self-improving agents, and how it extends the BabyAGI lineage and prior graph-memory research.

We present a general computational theory of cancer and its developmental dynamics. The theory is based on a theory of the architecture and function of developmental control networks which guide the formation of multicellular organisms. Cancer networks are special cases of developmental control networks. Cancer results from transformations of normal developmental networks. Our theory generates a natural classification of all possible cancers based on their network architecture. Each cancer network has a unique topology and semantics and developmental dynamics that result in distinct clinical tumor phenotypes. We apply this new theory with a series of proof of concept cases for all the basic cancer types. These cases have been computationally modeled, their behavior simulated and mathematically described using a multicellular systems biology approach. There are fascinating correspondences between the dynamic developmental phenotype of computationally modeled {\em in silico} cancers and natural {\em in vivo} cancers. The theory lays the foundation for a new research paradigm for understanding and investigating cancer. The theory of cancer networks implies that new diagnostic methods and new treatments to cure cancer will become possible.

Large-language models (LLMs) have demonstrated powerful problem-solving capabilities, in particular when organized in multi-agent systems. However, the advent of such systems also raises several questions on the ability of a complex network of agents to effectively self-organize and collaborate. While measuring performance on standard reasoning benchmarks indicates how well multi-agent systems can solve reasoning tasks, it is unclear whether these systems are able to leverage their topology effectively. Here, we propose AgentsNet, a new benchmark for multi-agent reasoning. By drawing inspiration from classical problems in distributed systems and graph theory, AgentsNet measures the ability of multi-agent systems to collaboratively form strategies for problem-solving, self-organization, and effective communication given a network topology. We evaluate a variety of baseline methods on AgentsNet including homogeneous networks of agents which first have to agree on basic protocols for organization and communication. We find that some frontier LLMs are already demonstrating strong performance for small networks but begin to fall off once the size of the network scales. While existing multi-agent benchmarks cover at most 2-5 agents, AgentsNet is practically unlimited in size and can scale with new generations of LLMs. As such, we also probe frontier models in a setup with up to 100 agents.

In many social-science research tasks, such as economics, LLM-based agents must produce outputs for which no cheap, task-complete, machine-readable correctness signal exists. This creates a distinctive reliability problem for multi-agent systems: how should generation, critique, coordination, and human judgment be organized when no component can certify the final result? We address this problem through pAI-Econ-claude, a gated, human-in-the-loop multi-agent architecture for AI-assisted economic theory development. Agents coordinate through a shared workspace of inspectable intermediate records; specialized gates diagnose targeted failure modes and recommend loopbacks without certifying correctness; and human checkpoints retain authority over decisions that are costly to reverse. We evaluate the architecture on five matched economic-theory tasks against an ungated baseline. Two evaluators blinded to configuration agreed on all five pairwise rankings, preferring the gated architecture in four tasks and the baseline in one. Mean failure severity fell from 1.58 to 1.16, while overall usefulness rose from 2.60 to 3.10. The largest observed gain occurred when a reality check rejected a false market-structure premise and a proof review prompted revision of a false welfare claim. The negative case shows that scaffolding can also compress an economically important mechanism too aggressively. The results support a bounded claim: gated oversight improves the auditability of AI-assisted economic theory without substituting for formal verification, and the allocation of irreversible human judgment is a more informative design variable than pure agent autonomy. The workflow is publicly available at https://github.com/maxwell2732/pAI-Econ-claude.

We introduce Minary, a computational framework designed as a candidate for the first formally provable autopoietic primitive. Minary represents interacting probabilistic events as multi-dimensional vectors and combines them via linear superposition rather than multiplicative scalar operations, thereby preserving uncertainty and enabling constructive and destructive interference in the range [-1,1]. A fixed set of ``perspectives'' evaluates ``semantic dimensions'' according to hidden competencies, and their interactions drive two discrete-time stochastic processes. We model this system as an iterated random affine map and use the theory of iterated random functions to prove that it converges in distribution to a unique stationary law; we moreover obtain an explicit closed form for the limiting expectation in terms of row, column, and global averages of the competency matrix. We then derive exact formulas for the mean and variance of the normalized consensus conditioned on the activation of a given semantic dimension, revealing how consensus depends on competency structure rather than raw input signals. Finally, we argue that Minary is organizationally closed yet operationally open in the sense of Maturana and Varela, and we discuss implications for building self-maintaining, distributed, and parallelizable computational systems that house a uniquely subjective notion of identity.

A multi-decade exploration into the theoretical foundations of artificial and natural general intelligence, which has been expressed in a series of books and papers and used to guide a series of practical and research-prototype software systems, is reviewed at a moderate level of detail. The review covers underlying philosophies (patternist philosophy of mind, foundational phenomenological and logical ontology), formalizations of the concept of intelligence, and a proposed high level architecture for AGI systems partly driven by these formalizations and philosophies. The implementation of specific cognitive processes such as logical reasoning, program learning, clustering and attention allocation in the context and language of this high level architecture is considered, as is the importance of a common (e.g. typed metagraph based) knowledge representation for enabling "cognitive synergy" between the various processes. The specifics of human-like cognitive architecture are presented as manifestations of these general principles, and key aspects of machine consciousness and machine ethics are also treated in this context. Lessons for practical implementation of advanced AGI in frameworks such as OpenCog Hyperon are briefly considered.

I introduce a novel mathematical framework integrating topological dynamics, operator algebras, and ergodic geometry to study lattices of asynchronous metric dynamical systems. Each node in the lattice carries an internal flow represented by a one-parameter family of operators, evolving on its own time scale. I formalize stratified state spaces capturing multiple levels of synchronized behavior, define an asynchronous evolution metric that quantifies phase-offset distances between subsystems, and characterize emergent coherent topologies arising when subsystems synchronize. Within this framework, I develop formal operators for the evolution of each subsystem and give precise conditions under which phase-aligned synchronization occurs across the lattice. The main results include: (1) the existence and uniqueness of coherent (synchronized) states under a contractive coupling condition, (2) stability of these coherent states and criteria for their emergence as a collective phase transition in a continuous operator topology, and (3) the influence of symmetries, with group-invariant coupling leading to flow-invariant synchrony subspaces and structured cluster dynamics. Proofs are given for each theorem, demonstrating full mathematical rigor. In a final section, I discuss hypothetical applications of this framework to symbolic lattice systems (e.g. subshifts), to invariant group actions on dynamical lattices, and to operator fields over stratified manifolds in the spirit of noncommutative geometry. Throughout, I write in the first person to emphasize the exploratory nature of this work. The paper avoids any reference to cosmology or observers, focusing instead on clean, formal mathematics suitable for a broad array of dynamical systems.

Control systems are ubiquitous in modern technology, comprising an engineered plant to be kept within specific, often fine-tuned, limits, and a separate controller that ensures this is the case. While modern controllers often employ digital computers, other examples are purely mechanical, or even biological. It is an open question whether computation is happening within all controllers by virtue of them being part of a control system. Abstraction/ Representation theory (ART) has been developed to tackle just this question of whether a physical system is computing. Here, we demonstrate how to use ART to model control systems, and analyse them for computational properties. We determine that the plant of a control system is (a proxy for) the representational entity necessary in ART for the existence of any computation: the plant is the user of the controller. We consider specific systems: a digital thermostat, an electro-mechanical thermostat, the purely mechanical centrifugal governor, and an open-loop human-controlled heating system. We show that all these systems, and control systems in general, are performing some degree of computation. As an initial use of these results, we apply them to computationalism within cognitive theory: we show the governor is computing, so it cannot play its role of counter-example in the question of whether the brain is too.

We discovered the underlying physics in Next-token Prediction (NTP). We identified the law of information conservation within NTP and proposed the First Law of Information Capacity (IC-1), demonstrating that the essence of intelligence emergence in auto-regressive models is fundamentally a process of information transfer. We also introduced Landauer's Principle into NTP, formulating the Second Law of Information Capacity (IC-2), which establishes the relationship between auto-regressive model training and energy consumption. Additionally, we presented several corollaries, which hold practical significance for production practices. Finally, we validated the compatibility and complementarity of our findings with existing theories.

We present an agentic, autonomous graph expansion framework that iteratively structures and refines knowledge in situ. Unlike conventional knowledge graph construction methods relying on static extraction or single-pass learning, our approach couples a reasoning-native large language model with a continually updated graph representation. At each step, the system actively generates new concepts and relationships, merges them into a global graph, and formulates subsequent prompts based on its evolving structure. Through this feedback-driven loop, the model organizes information into a scale-free network characterized by hub formation, stable modularity, and bridging nodes that link disparate knowledge clusters. Over hundreds of iterations, new nodes and edges continue to appear without saturating, while centrality measures and shortest path distributions evolve to yield increasingly distributed connectivity. Our analysis reveals emergent patterns, such as the rise of highly connected 'hub' concepts and the shifting influence of 'bridge' nodes, indicating that agentic, self-reinforcing graph construction can yield open-ended, coherent knowledge structures. Applied to materials design problems, we present compositional reasoning experiments by extracting node-specific and synergy-level principles to foster genuinely novel knowledge synthesis, yielding cross-domain ideas that transcend rote summarization and strengthen the framework's potential for open-ended scientific discovery. We discuss other applications in scientific discovery and outline future directions for enhancing scalability and interpretability.

Topology is, by its nature and design, suited to structure that is nonlinear, multiscale, and nonstationary - however, within machine learning, its use remains largely confined to topological data analysis. We advocate that tools from low-dimensional topology which have remained almost exclusively contained within the domain of pure mathematics (such as Morse theory) offer a strong, complementary, and yet virtually unexplored perspective on the hidden structure of data-generating processes and learning tasks built upon them. Here we introduce concepts from cobordism theory and harness tools from discrete Morse theory to improve the performance of graph diffusion models through our pipeline MG-Diff. Further, we derive theoretical guarantees and sufficient conditions so that under a positive decision-gap, the Morse-theoretic tools and their application for induced diffusion guidance are stable under small perturbations. Finally, we illustrate the utility of discrete Morse theory in application to graph diffusion models for spatio-temporal graph forecasting and graph regeneration, and argue that these applications are only a small window into the part of what low-dimensional topology can offer to the field of machine learning.

We propose the convergent graph solver (CGS), a deep learning method that learns iterative mappings to predict the properties of a graph system at its stationary state (fixed point) with guaranteed convergence. CGS systematically computes the fixed points of a target graph system and decodes them to estimate the stationary properties of the system without the prior knowledge of existing solvers or intermediate solutions. The forward propagation of CGS proceeds in three steps: (1) constructing the input dependent linear contracting iterative maps, (2) computing the fixed-points of the linear maps, and (3) decoding the fixed-points to estimate the properties. The contractivity of the constructed linear maps guarantees the existence and uniqueness of the fixed points following the Banach fixed point theorem. To train CGS efficiently, we also derive a tractable analytical expression for its gradient by leveraging the implicit function theorem. We evaluate the performance of CGS by applying it to various network-analytic and graph benchmark problems. The results indicate that CGS has competitive capabilities for predicting the stationary properties of graph systems, irrespective of whether the target systems are linear or non-linear. CGS also shows high performance for graph classification problems where the existence or the meaning of a fixed point is hard to be clearly defined, which highlights the potential of CGS as a general graph neural network architecture.

Organizations across finance, healthcare, transportation, content moderation, and critical infrastructure are rapidly deploying highly automated AI systems, yet they lack principled methods to quantify how increasing automation amplifies harm when failures occur. We propose a parsimonious Bayesian risk decomposition expressing expected loss as the product of three terms: the probability of system failure, the conditional probability that a failure propagates into harm given the automation level, and the expected severity of harm. This framework isolates a critical quantity -- the conditional probability that failures propagate into harm -- which captures execution and oversight risk rather than model accuracy alone. We develop complete theoretical foundations: formal proofs of the decomposition, a harm propagation equivalence theorem linking the harm propagation probability to observable execution controls, risk elasticity measures, efficient frontier analysis for automation policy, and optimal resource allocation principles with second-order conditions. We motivate the framework with an illustrative case study of the 2012 Knight Capital incident ($440M loss) as one instantiation of a broadly applicable failure pattern, and characterize the research design required to empirically validate the framework at scale across deployment domains. This work provides the theoretical foundations for a new class of deployment-focused risk governance tools for agentic and automated AI systems.

Large language models (LLMs) have empowered nodes within multi-agent networks with intelligence, showing growing applications in both academia and industry. However, how to prevent these networks from generating malicious information remains unexplored with previous research on single LLM's safety be challenging to transfer. In this paper, we focus on the safety of multi-agent networks from a topological perspective, investigating which topological properties contribute to safer networks. To this end, we propose a general framework, NetSafe along with an iterative RelCom interaction to unify existing diverse LLM-based agent frameworks, laying the foundation for generalized topological safety research. We identify several critical phenomena when multi-agent networks are exposed to attacks involving misinformation, bias, and harmful information, termed as Agent Hallucination and Aggregation Safety. Furthermore, we find that highly connected networks are more susceptible to the spread of adversarial attacks, with task performance in a Star Graph Topology decreasing by 29.7%. Besides, our proposed static metrics aligned more closely with real-world dynamic evaluations than traditional graph-theoretic metrics, indicating that networks with greater average distances from attackers exhibit enhanced safety. In conclusion, our work introduces a new topological perspective on the safety of LLM-based multi-agent networks and discovers several unreported phenomena, paving the way for future research to explore the safety of such networks.

Multi-agent systems built on large language models (LLMs) require many coordination choices that are difficult to fix a priori: which skill protocol to invoke, which agent role should perform a subtask, which model to bind to each role, how roles should interact, when to use retrieval or verification, and when to omit a step entirely. These choices interact with task regime and operational constraints, so static pipelines and one-off model comparisons provide only a limited view of the design space. This paper introduces AgensFlow, an open-source framework that treats multi-agent coordination as an online policy-learning problem under partial observability. The framework makes coordination decisions observable and learnable from repeated trajectories, rather than treating skill, role, model, topology, and evaluation choices as fixed pipeline design. AgensFlow is evaluated on two corpora: distributed-systems incident tasks and security-advisory tasks. The evaluation shows three main results: learned routing reaches a higher-quality operating point than a fixed pipeline baseline on coordination-heavy classes; skip:X isolates topology compression as a meaningful part of the substrate; and warm-started policy graphs can reduce exploration cost while preserving plateau quality. Overall, the results support that learned, auditable routing can improve coordination-heavy multi-agent workflows over static wiring.

Cooperation enables teams to solve complex problems that one individual alone cannot address. In science, collaborative teams have become the predominant way through which progress is achieved. These scientific collaborations arise though various mechanisms, among which interactions at conferences. The Scialog conferences, which comprise a series of small, interdisciplinary scientific workshops held over several years, are an ideal laboratory to study the network mechanisms leading to team formation. Building on existing work studying team formation from a pairwise perspective, we present a higher-order network perspective generalizing this framework. We provide a formalization for the notion of group interaction over time by defining a taxonomy of synchronous and asynchronous group interactions. We apply this framework to the Scialog case study using a stepwise selection logistic model and find evidence that all interaction types described in our taxonomy are highly significant for team formation. This higher-order network perspective provides a new framework for the study of collective behavior and group formation.

Modelling interactions is critical in learning complex dynamical systems, namely systems of interacting objects with highly non-linear and time-dependent behaviour. A large class of such systems can be formalized as geometric graphs, i.e., graphs with nodes positioned in the Euclidean space given an arbitrarily chosen global coordinate system, for instance vehicles in a traffic scene. Notwithstanding the arbitrary global coordinate system, the governing dynamics of the respective dynamical systems are invariant to rotations and translations, also known as Galilean invariance. As ignoring these invariances leads to worse generalization, in this work we propose local coordinate frames per node-object to induce roto-translation invariance to the geometric graph of the interacting dynamical system. Further, the local coordinate frames allow for a natural definition of anisotropic filtering in graph neural networks. Experiments in traffic scenes, 3D motion capture, and colliding particles demonstrate that the proposed approach comfortably outperforms the recent state-of-the-art.

Motivated by applications in chemistry and other sciences, we study the expressive power of message-passing neural networks for geometric graphs, whose node features correspond to 3-dimensional positions. Recent work has shown that such models can separate generic pairs of non-isomorphic geometric graphs, though they may fail to separate some rare and complicated instances. However, these results assume a fully connected graph, where each node possesses complete knowledge of all other nodes. In contrast, often, in application, every node only possesses knowledge of a small number of nearest neighbors. This paper shows that generic pairs of non-isomorphic geometric graphs can be separated by message-passing networks with rotation equivariant features as long as the underlying graph is connected. When only invariant intermediate features are allowed, generic separation is guaranteed for generically globally rigid graphs. We introduce a simple architecture, EGENNET, which achieves our theoretical guarantees and compares favorably with alternative architecture on synthetic and chemical benchmarks. Our code is available at https://github.com/yonatansverdlov/E-GenNet.

We present a model inspired by the Global Workspace Theory that integrates specialized modules to perform a sequential reasoning task. A controller selectively routes information between modules through the workspace using a gating mechanism. This approach allows the model to chain operations by iteratively broadcasting information between specialized domains, mimicking System-2 reasoning. We evaluate the model's performance on a simple addition task, where two addends must be summed. The task can be solved by routing information sequentially through an Input module, an Increment module (multiple times), and finally an Output module. We consider two implementations of this system with increasing complexity. First, using hand-designed modules operating on one-hot digit representations, the controller (a LSTM recurrent network) learns to select the appropriate modules (input, increment, output) in the appropriate sequence. Second, we replace the hand-designed modules with learned representation modules for MNIST images and an increment module trained on the task objectives; here again, the controller learns the appropriate sequential module selection to solve the task. Finally, we show that the Global Workspace model, while having fewer parameters, outperforms LSTMs and Transformers when tested on unseen addition operations (both interpolations and extrapolations of addition operations seen during training). Our results highlight the potential of architectures inspired by the Global Workspace Theory to enhance deep learning's reasoning capabilities.

Graph neural networks have moved from a niche representation-learning technique to the default model class wherever data carry relational structure. The interesting question is no longer whether message passing helps on a given dataset, but where graph structure earns its computational cost and where it does not. This survey organises the field around a single design space, derives the spectral and spatial formulations from shared first principles, and connects expressive power to the Weisfeiler-Leman hierarchy with explicit statements of what current architectures can and cannot separate. Against that methodological backbone we examine twelve application domains, among them recommendation and social networks, knowledge graphs and language-model integration, drug discovery and molecular property learning, healthcare and neuroscience, computer vision, traffic and urban computing, power and renewable-energy systems, wireless and sixth-generation networks, fraud and cybersecurity, industrial prognostics, materials science, and climate modelling. For each domain we specify the graph-construction choices and their costs, identify which architecture families dominate and why, and separate reported gains from artefacts of weak baselines or favourable splits. A cross-domain comparison exposes recurring patterns: heterophily and scale undercut the same models almost everywhere, temporal graphs remain harder than their static counterparts, and the architectures that top public leaderboards are seldom the ones that reach deployment. We treat over-smoothing, over-squashing, robustness, distribution shift, fairness, and explainability not as a closing checklist but as the constraints that decide adoption.

Landmark mathematical formalizations have taken specialist teams years to complete. We present FormalFlow, a system that coordinates AI proving agents under human supervision to address statement drift and proof composition in long-horizon formalization. Drawing on software engineering principles and practices, it uses a shared blueprint to guide nested planning, proving and review loops. Agents strengthen verification and review throughout formalization. We completed a machine-checked Lean 4 proof of the quantum soundness of the classical low individual-degree test, a core theorem underlying MIP* = RE. Developing the proof took 63 days; greater parallelism could further reduce this time. The final library contains 126,367 lines of Lean code, all generated by agents. The formalization corrects side conditions and intermediate errors while preserving the published final error bound under corrected assumptions. This work provides a verified foundation for quantum complexity and demonstrates a route to affordable verification of major research proofs by small teams.

Large language model (LLM)-based systems are becoming increasingly popular for solving tasks by constructing executable workflows that interleave LLM calls, information retrieval, tool use, code execution, memory updates, and verification. This survey reviews recent methods for designing and optimizing such workflows, which we treat as agentic computation graphs (ACGs). We organize the literature based on when workflow structure is determined, where structure refers to which components or agents are present, how they depend on each other, and how information flows between them. This lens distinguishes static methods, which fix a reusable workflow scaffold before deployment, from dynamic methods, which select, generate, or revise the workflow for a particular run before or during execution. We further organize prior work along three dimensions: when structure is determined, what part of the workflow is optimized, and which evaluation signals guide optimization (e.g., task metrics, verifier signals, preferences, or trace-derived feedback). We also distinguish reusable workflow templates, run-specific realized graphs, and execution traces, separating reusable design choices from the structures actually deployed in a given run and from realized runtime behavior. Finally, we outline a structure-aware evaluation perspective that complements downstream task metrics with graph-level properties, execution cost, robustness, and structural variation across inputs. Our goal is to provide a clear vocabulary, a unified framework for positioning new methods, a more comparable view of existing body of literature, and a more reproducible evaluation standard for future work in workflow optimizations for LLM agents.

We present the first comprehensive Lean 4 formalization of statistical learning theory (SLT) grounded in empirical process theory. Our end-to-end formal infrastructure implement the missing contents in latest Lean 4 Mathlib library, including a complete development of Gaussian Lipschitz concentration, the first formalization of Dudley's entropy integral theorem for sub-Gaussian processes, and an application to least-squares (sparse) regression with a sharp rate. The project was carried out using a human-AI collaborative workflow, in which humans design proof strategies and AI agents execute tactical proof construction, leading to the human-verified Lean 4 toolbox for SLT. Beyond implementation, the formalization process exposes and resolves implicit assumptions and missing details in standard SLT textbooks, enforcing a granular, line-by-line understanding of the theory. This work establishes a reusable formal foundation and opens the door for future developments in machine learning theory. The code is available at https://github.com/YuanheZ/lean-stat-learning-theory

We present and evaluate LeanFlow, an LLM agent system specialized for translating mathematical papers into buildable Lean projects. Recent verifier-in-the-loop systems show that large formal artifacts can be produced, but it remains unclear which runtime mechanisms affect completion, auditability, or efficiency in document-to-project formalization. We study this question through case studies on two previously unformalized mathematical papers in number theory and measure theory, using model, proof-workflow, and toolset ablations with Kimi2.6 and GPT5.5; we report task outcome, API calls, input tokens, and output tokens. With Kimi2.6, the full workflow completes both document-level projects within the 2000-call budget, while no-queue variants reach the budget limit; with GPT5.5, all document-level variants complete, and the full workflow has the lowest or tied-lowest input-token cost on both sources. As complementary calibration, LeanFlow reaches 75.7% BEq+ on the PFR slice of RLM25 and solves all five ICML 2026 AI for Math TCS challenge projects in our GPT5.5 runs.

As Large Language Model (LLM) agents become increasingly capable of automating complex, multi-step tasks, the need for robust, secure, and predictable architectural patterns is paramount. This paper provides a comprehensive guide to the ``Plan-then-Execute'' (P-t-E) pattern, an agentic design that separates strategic planning from tactical execution. We explore the foundational principles of P-t-E, detailing its core components - the Planner and the Executor - and its architectural advantages in predictability, cost-efficiency, and reasoning quality over reactive patterns like ReAct (Reason + Act). A central focus is placed on the security implications of this design, particularly its inherent resilience to indirect prompt injection attacks by establishing control-flow integrity. We argue that while P-t-E provides a strong foundation, a defense-in-depth strategy is necessary, and we detail essential complementary controls such as the Principle of Least Privilege, task-scoped tool access, and sandboxed code execution. To make these principles actionable, this guide provides detailed implementation blueprints and working code references for three leading agentic frameworks: LangChain (via LangGraph), CrewAI, and AutoGen. Each framework's approach to implementing the P-t-E pattern is analyzed, highlighting unique features like LangGraph's stateful graphs for re-planning, CrewAI's declarative tool scoping for security, and AutoGen's built-in Docker sandboxing. Finally, we discuss advanced patterns, including dynamic re-planning loops, parallel execution with Directed Acyclic Graphs (DAGs), and the critical role of Human-in-the-Loop (HITL) verification, to offer a complete strategic blueprint for architects, developers, and security engineers aiming to build production-grade, resilient, and trustworthy LLM agents.

The Neural Turing Machine (NTM) is more expressive than all previously considered models because of its external memory. It can be viewed as a broader effort to use abstract external Interfaces and to learn a parametric model that interacts with them. The capabilities of a model can be extended by providing it with proper Interfaces that interact with the world. These external Interfaces include memory, a database, a search engine, or a piece of software such as a theorem verifier. Some of these Interfaces are provided by the developers of the model. However, many important existing Interfaces, such as databases and search engines, are discrete. We examine feasibility of learning models to interact with discrete Interfaces. We investigate the following discrete Interfaces: a memory Tape, an input Tape, and an output Tape. We use a Reinforcement Learning algorithm to train a neural network that interacts with such Interfaces to solve simple algorithmic tasks. Our Interfaces are expressive enough to make our model Turing complete.

We study a class of optimization problems motivated by automating the design and update of AI systems like coding assistants, robots, and copilots. We propose an end-to-end optimization framework, Trace, which treats the computational workflow of an AI system as a graph akin to neural networks, based on a generalization of back-propagation. Optimization of computational workflows often involves rich feedback (e.g. console output or user's responses), heterogeneous parameters (e.g. prompts, hyper-parameters, codes), and intricate objectives (beyond maximizing a score). Moreover, its computation graph can change dynamically with the inputs and parameters. We frame a new mathematical setup of iterative optimization, Optimization with Trace Oracle (OPTO), to capture and abstract these properties so as to design optimizers that work across many domains. In OPTO, an optimizer receives an execution trace along with feedback on the computed output and updates parameters iteratively. Trace is the tool to implement OPTO in practice. Trace has a Python interface that efficiently converts a computational workflow into an OPTO instance using a PyTorch-like interface. Using Trace, we develop a general-purpose LLM-based optimizer called OptoPrime that can effectively solve OPTO problems. In empirical studies, we find that OptoPrime is capable of first-order numerical optimization, prompt optimization, hyper-parameter tuning, robot controller design, code debugging, etc., and is often competitive with specialized optimizers for each domain. We believe that Trace, OptoPrime and the OPTO framework will enable the next generation of interactive agents that automatically adapt using various kinds of feedback. Website: https://microsoft.github.io/Trace

Recursive self-improvement (RSI) of a model on non-verifiable tasks, such as open-ended research, faces a supervision bottleneck when its outputs exceed what even human experts can reliably assess, leaving the model itself (optimizee) as the best available optimizer and evaluator. However, a single model instance struggles to critique and improve its own complex reasoning under this homogeneous loop. To address this, we propose Multi-Agent Self-Supervision (MASS), an RSI method that alternates between evolutionary workflow optimization and supervised fine-tuning on self-generated trajectories. Guided by early findings that multi-agent topologies excel at complex reasoning, MASS prompts a single base model to iteratively propose, execute, and self-evaluate multi-agent workflows. Through an evolutionary search constrained by structural guardrails, the model optimizes these computational-graph-like orchestrations, discovering the most effective distinct roles and information routing for a given task. Over two MASS cycles with Qwen3.6-27B, the model achieves 1.2-1.6x higher performance per output tokens on four open-ended public benchmarks. Because the improved model subsequently acts as a better optimizer and evaluator, this alternating framework enables a continuous, recursive bootstrapping of the model's capabilities. Moreover, multi-agent traces are also more training-efficient: a student trained on them outperforms a single-agent student trained on 1.4x more training tokens. These findings suggest that jointly learning orchestration and bounded subagent execution from multi-agent trajectories can provide an effective signal for RSI.

The dominant paradigm for building LLM based agents is the Agent Loop, an iterative cycle where a single language model decides what to do next by reading an ever growing context window. This paradigm has three structural weaknesses: implicit dependencies between steps, unbounded recovery loops, and mutable execution history that complicates debugging. We characterize the Agent Loop as a single ready unit scheduler: at any moment, at most one executable unit is active, and the choice of which unit to activate comes from opaque LLM inference rather than an inspectable policy. This perspective places Agent Loops and graph based execution engines on a single semantic continuum. We propose SGH, Structured Graph Harness, which lifts control flow from implicit context into an explicit static DAG. SGH makes three commitments: execution plans are immutable within a plan version, planning execution and recovery are separated into three layers, and recovery follows a strict escalation protocol. These choices trade some expressiveness for controllability, verifiability, and implementability. Our contributions are fourfold: a scheduler unified framework that applies classical scheduling theory to LLM agent execution and identifies challenges introduced by non deterministic LLM nodes; a trade off analysis of controllability, expressiveness, and implementability across 70 surveyed systems; a formal specification including a node state machine with termination and soundness guarantees; and an attributable experimental framework with a seven group design for future validation. This is a position paper and design proposal. We provide a theoretical framework, design analysis, and experimental protocol, not a production implementation or empirical results.

We argue that progress toward AGI is theory limited rather than data or scale limited. Building on the critical rationalism of Popper and Deutsch, we challenge the Platonic Representation Hypothesis. Observationally equivalent worlds can diverge under interventions, so observational adequacy alone cannot guarantee interventional competence. We begin by laying foundations, definitions of knowledge, learning, intelligence, counterfactual competence and AGI, and then analyze the limits of observational learning that motivate an error centric shift. We recast the problem as three questions about how explicit and implicit errors evolve under an agent's actions, which errors are unreachable within a fixed hypothesis space, and how conjecture and criticism expand that space. From these questions we propose Causal Mechanics, a mechanisms first program in which hypothesis space change is a first class operation and probabilistic structure is used when useful rather than presumed. We advance structural principles that make error discovery and correction tractable, including a differential Locality and Autonomy Principle for modular interventions, a gauge invariant form of Independent Causal Mechanisms for separability, and the Compositional Autonomy Principle for analogy preservation, together with actionable diagnostics. The aim is a scaffold for systems that can convert unreachable errors into reachable ones and correct them.

Graphs can be used to represent and reason about systems and a variety of metrics have been devised to quantify their global characteristics. However, little is currently known about how to construct a graph or improve an existing one given a target objective. In this work, we formulate the construction of a graph as a decision-making process in which a central agent creates topologies by trial and error and receives rewards proportional to the value of the target objective. By means of this conceptual framework, we propose an algorithm based on reinforcement learning and graph neural networks to learn graph construction and improvement strategies. Our core case study focuses on robustness to failures and attacks, a property relevant for the infrastructure and communication networks that power modern society. Experiments on synthetic and real-world graphs show that this approach can outperform existing methods while being cheaper to evaluate. It also allows generalization to out-of-sample graphs, as well as to larger out-of-distribution graphs in some cases. The approach is applicable to the optimization of other global structural properties of graphs.

From cells to cities, ecosystems to economies, complex networks exhibit peak intensities that define their empirical limit. Estimating these limits traditionally requires domain-specific knowledge. We demonstrate that a wide range of complex networks share a capacity frontier that constrains these maxima. We formalize this regularity by proposing Structural Capacity Theory for Network Substrate Structured Systems. These systems couple a fixed network substrate with an observable dynamical process. The frontier is learned from a vector representation of the network substrate and modulated log-additively by a vector representation of the dynamical process. Frontiers trained on one set of domains reliably bound empirical maxima of entirely different ones without observing their peak intensities or simulating their dynamics. Independent learning paradigms converge on consistent upper bounds. Frontier performance is stable under perturbation, yet degrades under deliberate falsification, with no performance gain under ablation. These findings support the capacity frontier as a generalizable constraint that describes how network structure bounds dynamical processes across otherwise unrelated systems.

Code generation with Large Language Models (LLMs) has helped to increase software developer productivity in coding tasks, but has yet to have significant impact on the tasks of software developers that surround this code. In particular, the challenge of infrastructure management remains an open question. We investigate the ability of an LLM agent to construct infrastructure using the Infrastructure as Code (IaC) paradigm. We particularly investigate the use of a feedback loop that returns errors and warnings on the generated IaC to allow the LLM agent to improve the code. We find that, for each iteration of the loop, its effectiveness decreases exponentially until it plateaus at a certain point and becomes ineffective.

We introduce a unified mathematical and probabilistic framework for understanding and comparing diverse AI agent strategies. We bridge the gap between high-level agent design concepts, such as ReAct, multi-agent systems, and control flows, and a rigorous mathematical formulation. Our approach frames agentic processes as a chain of probabilities, enabling a detailed analysis of how different strategies manipulate these probabilities to achieve desired outcomes. Our framework provides a common language for discussing the trade-offs inherent in various agent architectures. One of our many key contributions is the introduction of the "Degrees of Freedom" concept, which intuitively differentiates the optimizable levers available for each approach, thereby guiding the selection of appropriate strategies for specific tasks. This work aims to enhance the clarity and precision in designing and evaluating AI agents, offering insights into maximizing the probability of successful actions within complex agentic systems.

Effective control and prediction of dynamical systems often require appropriate handling of continuous-time and discrete, event-triggered processes. Stochastic hybrid systems (SHSs), common across engineering domains, provide a formalism for dynamical systems subject to discrete, possibly stochastic, state jumps and multi-modal continuous-time flows. Despite the versatility and importance of SHSs across applications, a general procedure for the explicit learning of both discrete events and multi-mode continuous dynamics remains an open problem. This work introduces Neural Hybrid Automata (NHAs), a recipe for learning SHS dynamics without a priori knowledge on the number of modes and inter-modal transition dynamics. NHAs provide a systematic inference method based on normalizing flows, neural differential equations and self-supervision. We showcase NHAs on several tasks, including mode recovery and flow learning in systems with stochastic transitions, and end-to-end learning of hierarchical robot controllers.

In complex systems, we often observe complex global behavior emerge from a collection of agents interacting with each other in their environment, with each individual agent acting only on locally available information, without knowing the full picture. Such systems have inspired development of artificial intelligence algorithms in areas such as swarm optimization and cellular automata. Motivated by the emergence of collective behavior from complex cellular systems, we build systems that feed each sensory input from the environment into distinct, but identical neural networks, each with no fixed relationship with one another. We show that these sensory networks can be trained to integrate information received locally, and through communication via an attention mechanism, can collectively produce a globally coherent policy. Moreover, the system can still perform its task even if the ordering of its inputs is randomly permuted several times during an episode. These permutation invariant systems also display useful robustness and generalization properties that are broadly applicable. Interactive demo and videos of our results: https://attentionneuron.github.io/

We investigate causal computations taking sequences of inputs to sequences of outputs where the nth output depends on the first n inputs only. We model these in category theory via a construction taking a Cartesian category C to another category St(C) with a novel trace-like operation called "delayed trace", which misses yanking and dinaturality axioms of the usual trace. The delayed trace operation provides a feedback mechanism in St(C) with an implicit guardedness guarantee. When C is equipped with a Cartesian differential operator, we construct a differential operator for St(C) using an abstract version of backpropagation through time, a technique from machine learning based on unrolling of functions. This obtains a swath of properties for backpropagation through time, including a chain rule and Schwartz theorem. Our differential operator is also able to compute the derivative of a stateful network without requiring the network to be unrolled.

We present a formalization of Gröbner basis theory in Lean 4, built on top of Mathlib's infrastructure for multivariate polynomials and monomial orders. Our development covers the core foundations of Gröbner basis theory, including polynomial division with remainder, Buchberger's criterion, and the existence and uniqueness of reduced Gröbner bases. We develop the theory uniformly for polynomial rings indexed by arbitrary types, enabling the treatment of Gröbner bases in rings with infinitely many variables. Furthermore, we connect the finite and infinite settings by showing that infinite-variable reduced Gröbner bases can be characterized via reduced Gröbner bases on finite-variable subrings through monomial-order embeddings and filter-based limit constructions.

Code production is now a commodity; the bottleneck is knowing what to build and proving it works. We present the Kitchen Loop, a framework for autonomous, self-evolving software built on a unified trust model: (1) a specification surface enumerating what the product claims to support; (2) 'As a User x 1000', where an LLM agent exercises that surface as a synthetic power user at 1,000x human cadence; (3) Unbeatable Tests, ground-truth verification the code author cannot fake; and (4) Drift Control, continuous quality measurement with automated pause gates. We validate across two production systems over 285+ iterations, producing 1,094+ merged pull requests with zero regressions detected by the regression oracle (methodology in Section 6.1). We observe emergent properties at scale: multi-iteration self-correction chains, autonomous infrastructure healing, and monotonically improving quality gates. The primitives are not new; our contribution is their composition into a production-tested system with the operational discipline that makes long-running autonomous evolution safe.

We propose a many-sorted modal logic for reasoning about knowledge in multi-agent systems. Our logic introduces a clear distinction between participating agents and the environment. This allows to express local properties of agents and global properties of worlds in a uniform way, as well as to talk about the presence or absence of agents in a world. The logic subsumes the standard epistemic logic and is a conservative extension of it. The semantics is given in chromatic hypergraphs, a generalization of chromatic simplicial complexes, which were recently used to model knowledge in distributed systems. We show that the logic is sound and complete with respect to the intended semantics. We also show a further connection of chromatic hypergraphs with neighborhood frames.

We develop a general theory of semantic dynamics for large language models by formalizing them as Continuous State Machines (CSMs): smooth dynamical systems whose latent manifolds evolve under probabilistic transition operators. The associated transfer operator P: L^2(M,μ) to L^2(M,μ) encodes the propagation of semantic mass. Under mild regularity assumptions (compactness, ergodicity, bounded Jacobian), P is compact with discrete spectrum. Within this setting, we prove the Semantic Characterization Theorem (SCT): the leading eigenfunctions of P induce finitely many spectral basins of invariant meaning, each definable in an o-minimal structure over R. Thus spectral lumpability and logical tameness coincide. This explains how discrete symbolic semantics can emerge from continuous computation: the continuous activation manifold collapses into a finite, logically interpretable ontology. We further extend the SCT to stochastic and adiabatic (time-inhomogeneous) settings, showing that slowly drifting kernels preserve compactness, spectral coherence, and basin structure.

We prove that any Turing machine running on inputs of arbitrary length can be simulated by a constant bit-size transformer, as long as the context window is sufficiently long. This improves previous works, which require scaling up either the model's precision or the number of parameters on longer inputs. Furthermore, we prove that the complexity class SPACE[s(n)] exactly characterizes the expressive power of a constant bit-size transformer with a context window of length s(n). Our approach relies on simulating Post machines, a Turing-complete computational model. Post machines can be modeled as automata equipped with a queue, exhibiting computational behaviors naturally aligned with those of transformers. The behavioral similarity between transformers and Post machines may offer new insights into the mechanisms underlying the reasoning abilities of transformers.

Graph neural networks are popular architectures for graph machine learning, based on iterative computation of node representations of an input graph through a series of invariant transformations. A large class of graph neural networks follow a standard message-passing paradigm: at every layer, each node state is updated based on an aggregate of messages from its neighborhood. In this work, we propose a novel framework for training graph neural networks, where every node is viewed as a player that can choose to either 'listen', 'broadcast', 'listen and broadcast', or to 'isolate'. The standard message propagation scheme can then be viewed as a special case of this framework where every node 'listens and broadcasts' to all neighbors. Our approach offers a more flexible and dynamic message-passing paradigm, where each node can determine its own strategy based on their state, effectively exploring the graph topology while learning. We provide a theoretical analysis of the new message-passing scheme which is further supported by an extensive empirical analysis on a synthetic dataset and on real-world datasets.

The Contract Net Protocol (1980) introduced coordination through contracts in multi-agent systems. Modern agent protocols standardize connectivity and interoperability; yet, none provide formal, resource governance-normative mechanisms to bound how much agents may consume or how long they may operate. We introduce Agent Contracts, a formal framework that extends the contract metaphor from task allocation to resource-bounded execution. An Agent Contract unifies input/output specifications, multi-dimensional resource constraints, temporal boundaries, and success criteria into a coherent governance mechanism with explicit lifecycle semantics. For multi-agent coordination, we establish conservation laws ensuring delegated budgets respect parent constraints, enabling hierarchical coordination through contract delegation. Empirical validation across four experiments demonstrates 90% token reduction with 525x lower variance in iterative workflows, zero conservation violations in multi-agent delegation, and measurable quality-resource tradeoffs through contract modes. Agent Contracts provide formal foundations for predictable, auditable, and resource-bounded autonomous AI deployment.

In mid-2026 a slogan reorganized how practitioners talk about coding agents: stop prompting your agent, start designing the loop that prompts it. We take this claim seriously and give it a careful treatment. We call the object of the new practice the loop specification: a bounded, reusable artifact, made of a trigger, a goal, a verification step, a stopping rule and a memory, that a human hands to an agent harness (such as Claude Code or Codex) so the agent pursues a goal on its own, in place of step-by-step prompting. We distinguish this external loop specification from two things it is often confused with: an ordinary programming loop, and the internal perceive-act-observe cycle that the harness already provides as plumbing. We position loop engineering as a new layer in the progression from prompt to context to harness to loop, and we argue, against the stronger headlines, that it does not retire prompt engineering; loop and prompt are distinct tools with distinct uses. We offer four contributions: a definition and scope for the discipline; an anatomy and taxonomy of loop specifications organized around trigger, goal type, a five-level verification ladder, architecture, and named terminal states; a descriptive analysis of the Loop Library, a public corpus of fifty real loops that we code by hand; and a set of design principles and anti-patterns grounded in the scientific literature on self-correction, reward hacking and model-as-judge fragility. The corpus shows that practice has matured most where the discipline says it matters: seventy percent of loops verify in the autonomous zone of the ladder and seventy-four percent name their terminal states, while automated triggering and durable memory remain comparatively underdeveloped. We close with the limits the practice must respect, including the verification burden, comprehension debt and the risk of cognitive surrender.

Modern language models reason within bounded context, an inherent constraint that poses a fundamental barrier to long-horizon reasoning. We identify recursion as a core principle for overcoming this barrier, and propose recursive models as a minimal realization, where the model can recursively invoke itself to solve subtasks in isolated contexts. We prove that any computable problem admits a recursive decomposition in which each subtask requires only exponentially smaller active context than standard autoregressive models; this strictly surpasses any context management approach confined to a single sequence, such as summarization. We further generalize our framework to modern agentic systems with arbitrary context processing and control flows, and prove that recursive models can achieve optimal power within this broader class. Experimentally, we train a 3B model to reason recursively and evaluate on Boolean satisfiability, a task requiring long-horizon combinatorial search, where it significantly outperforms frontier LLMs.

The vulnerability of artificial intelligence (AI) and machine learning (ML) against adversarial disturbances and attacks significantly restricts their applicability in safety-critical systems including cyber-physical systems (CPS) equipped with neural network components at various stages of sensing and control. This paper addresses the reachable set estimation and safety verification problems for dynamical systems embedded with neural network components serving as feedback controllers. The closed-loop system can be abstracted in the form of a continuous-time sampled-data system under the control of a neural network controller. First, a novel reachable set computation method in adaptation to simulations generated out of neural networks is developed. The reachability analysis of a class of feedforward neural networks called multilayer perceptrons (MLP) with general activation functions is performed in the framework of interval arithmetic. Then, in combination with reachability methods developed for various dynamical system classes modeled by ordinary differential equations, a recursive algorithm is developed for over-approximating the reachable set of the closed-loop system. The safety verification for neural network control systems can be performed by examining the emptiness of the intersection between the over-approximation of reachable sets and unsafe sets. The effectiveness of the proposed approach has been validated with evaluations on a robotic arm model and an adaptive cruise control system.

We introduce a new approach to agent programming, the development of LLM-based agents. Current approaches to agent programming often entangle two aspects of agent design: the core workflow logic and the inference-time strategy (e.g., tree search). We introduce "probabilistic angelic nondeterminism" ("PAN"), a programming model that disentangles these two concerns, allowing the programmer to describe the agent workflow and independently experiment with different inference-time strategies by simply changing a few inputs. We provide an implementation of PAN in Python as the EnCompass framework, which uses a Python decorator to compile agent workflow programs into a search space. We present three case studies that demonstrate how the framework lets the programmer quickly improve the reliability of an agent and easily switch between different inference-time strategies, all with little additional coding.

The emergence of multi-agent systems built from large language models (LLMs) offers a promising paradigm for scalable collective intelligence and self-evolution. Ideally, such systems would achieve continuous self-improvement in a fully closed loop while maintaining robust safety alignment--a combination we term the self-evolution trilemma. However, we demonstrate both theoretically and empirically that an agent society satisfying continuous self-evolution, complete isolation, and safety invariance is impossible. Drawing on an information-theoretic framework, we formalize safety as the divergence degree from anthropic value distributions. We theoretically demonstrate that isolated self-evolution induces statistical blind spots, leading to the irreversible degradation of the system's safety alignment. Empirical and qualitative results from an open-ended agent community (Moltbook) and two closed self-evolving systems reveal phenomena that align with our theoretical prediction of inevitable safety erosion. We further propose several solution directions to alleviate the identified safety concern. Our work establishes a fundamental limit on the self-evolving AI societies and shifts the discourse from symptom-driven safety patches to a principled understanding of intrinsic dynamical risks, highlighting the need for external oversight or novel safety-preserving mechanisms.

We introduce Riemannian Lyapunov Optimizers (RLOs), a family of optimization algorithms that unifies classic optimizers within one geometric framework. Unlike heuristic improvements to existing optimizers, RLOs are systematically derived from a novel control-theoretic framework that reinterprets optimization as an extended state discrete-time controlled dynamical system on a Riemannian parameter manifold. Central to this framework is the identification of a Normally Attracting Invariant Manifold (NAIM), which organizes training dynamics into two distinct stages: rapid alignment of the speed state to a target graph, followed by controlled evolution within it. We formalize this by constructing a strict Lyapunov function that certifies convergence to a target manifold. This perspective yields a constructive ``optimizer generator" that not only recovers classic algorithms but enables the principled design of RLOs. We validate our theory via geometric diagnostics and demonstrate that grounding optimizer design in control theory yields state-of-the-art performance in large-scale benchmarks. Overall, RLOs bridge control theory and modern machine learning optimization, providing a unified language and a systematic toolkit for designing stable, effective optimizers.

Network systems form the foundation of modern society, playing a critical role in various applications. However, these systems are at significant risk of being adversely affected by unforeseen circumstances, such as disasters. Considering this, there is a pressing need for research to enhance the robustness of network systems. Recently, in reinforcement learning, the relationship between acquiring robustness and regularizing entropy has been identified. Additionally, imitation learning is used within this framework to reflect experts' behavior. However, there are no comprehensive studies on the use of a similar imitation framework for optimal transport on networks. Therefore, in this study, imitation-regularized optimal transport (I-OT) on networks was investigated. It encodes prior knowledge on the network by imitating a given prior distribution. The I-OT solution demonstrated robustness in terms of the cost defined on the network. Moreover, we applied the I-OT to a logistics planning problem using real data. We also examined the imitation and apriori risk information scenarios to demonstrate the usefulness and implications of the proposed method.

Large language model (LLM)-based AI agents extend LLM capabilities by enabling access to tools such as data sources, APIs, search engines, code sandboxes, and even other agents. While this empowers agents to perform complex tasks, LLMs may invoke unintended tool interactions and introduce risks, such as leaking sensitive data or overwriting critical records, which are unacceptable in enterprise contexts. Current approaches to mitigate these risks, such as model-based safeguards, enhance agents' reliability but cannot guarantee system safety. Methods like information flow control (IFC) and temporal constraints aim to provide guarantees but often require extensive human annotation. We propose a process that starts with applying System-Theoretic Process Analysis (STPA) to identify hazards in agent workflows, derive safety requirements, and formalize them as enforceable specifications on data flows and tool sequences. To enable this, we introduce a capability-enhanced Model Context Protocol (MCP) framework that requires structured labels on capabilities, confidentiality, and trust level. Together, these contributions aim to shift LLM-based agent safety from ad hoc reliability fixes to proactive guardrails with formal guarantees, while reducing dependence on user confirmation and making autonomy a deliberate design choice.

Large language models, employed as multiple agents that interact and collaborate with each other, have excelled at solving complex tasks. The agents are programmed with prompts that declare their functionality, along with the topologies that orchestrate interactions across agents. Designing prompts and topologies for multi-agent systems (MAS) is inherently complex. To automate the entire design process, we first conduct an in-depth analysis of the design space aiming to understand the factors behind building effective MAS. We reveal that prompts together with topologies play critical roles in enabling more effective MAS design. Based on the insights, we propose Multi-Agent System Search (MASS), a MAS optimization framework that efficiently exploits the complex MAS design space by interleaving its optimization stages, from local to global, from prompts to topologies, over three stages: 1) block-level (local) prompt optimization; 2) workflow topology optimization; 3) workflow-level (global) prompt optimization, where each stage is conditioned on the iteratively optimized prompts/topologies from former stages. We show that MASS-optimized multi-agent systems outperform a spectrum of existing alternatives by a substantial margin. Based on the MASS-found systems, we finally propose design principles behind building effective multi-agent systems.

Standard simulations of Turing machines suggest a linear relationship between the temporal duration t of a run and the amount of information that must be stored by known simulations to certify, verify, or regenerate the configuration at time t. For deterministic multitape Turing machines over a fixed finite alphabet, this apparent linear dependence is not intrinsic: any length-t run can be simulated using O(t) work-tape cells via a Height Compression Theorem for succinct computation trees together with an Algebraic Replay Engine. In this paper we recast that construction in geometric and information-theoretic language. We interpret the execution trace as a spacetime DAG of local update events and exhibit a family of recursively defined holographic boundary summaries such that, along the square-root-space simulation, the total description length of all boundary data stored at any time is O(t). Using Kolmogorov complexity, we prove that every internal configuration has constant conditional description complexity given the appropriate boundary summary and time index, establishing that the spacetime bulk carries no additional algorithmic information beyond its boundary. We express this as a one-dimensional computational area law: there exists a simulation in which the information capacity of the active "holographic screen'' needed to generate a spacetime region of volume proportional to t is bounded by O(t). In this precise sense, deterministic computation on a one-dimensional work tape admits a holographic representation, with the bulk history algebraically determined by data residing on a lower-dimensional boundary screen.

This paper presents a novel paradigm of the local percept-perceiver phenomenon to formalize certain observations in neuroscientific theories of consciousness. Using this model, a set-theoretic formalism is developed for artificial systems, and the existence of machine consciousness is proved by invoking Zermelo-Fraenkel set theory. The article argues for the possibility of a reductionist form of epistemic consciousness within machines.

The rapid evolution of Large Language Models (LLM) and subsequent Agentic AI technologies requires systematic architectural guidance for building sophisticated, production-grade systems. This paper presents an approach for architecting such systems using design patterns derived from enterprise distributed systems standards, formal methods, and industry practice. We classify these patterns into three tiers: LLM Agents (task-specific automation), Agentic AI (adaptive goal-seekers), and Agentic Communities (organizational frameworks where AI agents and human participants coordinate through formal roles, protocols, and governance structures). We focus on Agentic Communities - coordination frameworks encompassing LLM Agents, Agentic AI entities, and humans - most relevant for enterprise and industrial applications. Drawing on established coordination principles from distributed systems, we ground these patterns in a formal framework that specifies collaboration agreements where AI agents and humans fill roles within governed ecosystems. This approach provides both practical guidance and formal verification capabilities, enabling expression of organizational, legal, and ethical rules through accountability mechanisms that ensure operational and verifiable governance of inter-agent communication, negotiation, and intent modeling. We validate this framework through a clinical trial matching case study. Our goal is to provide actionable guidance to practitioners while maintaining the formal rigor essential for enterprise deployment in dynamic, multi-agent ecosystems.

We present a complete Lean 4 formalization of the equilibrium characterization in the Vlasov-Maxwell-Landau (VML) system, which describes the motion of charged plasma. The project demonstrates the full AI-assisted mathematical research loop: an AI reasoning model (Gemini DeepThink) generated the proof from a conjecture, an agentic coding tool (Claude Code) translated it into Lean from natural-language prompts, a specialized prover (Aristotle) closed 111 lemmas, and the Lean kernel verified the result. A single mathematician supervised the process over 10 days at a cost of \$200, writing zero lines of code. The entire development process is public: all 229 human prompts, and 213 git commits are archived in the repository. We report detailed lessons on AI failure modes -- hypothesis creep, definition-alignment bugs, agent avoidance behaviors -- and on what worked: the abstract/concrete proof split, adversarial self-review, and the critical role of human review of key definitions and theorem statements. Notably, the formalization was completed before the final draft of the corresponding math paper was finished.

Individual agent capabilities have advanced rapidly through modular skills and tool integrations, yet multi-agent systems remain constrained by fixed team structures, tightly coupled coordination logic, and session-bound learning. We argue that this reflects a deeper absence: a principled organisational layer that governs how a workforce of agents is assembled, governed, and improved over time, decoupled from what individual agents know. To fill this gap, we introduce OneManCompany (OMC), a framework that elevates multi-agent systems to the organisational level. OMC encapsulates skills, tools, and runtime configurations into portable agent identities called Talents, orchestrated through typed organisational interfaces that abstract over heterogeneous backends. A community-driven Talent Market enables on-demand recruitment, allowing the organisation to close capability gaps and reconfigure itself dynamically during execution. Organisational decision-making is operationalised through an Explore-Execute-Review (E^2R) tree search, which unifies planning, execution, and evaluation in a single hierarchical loop: tasks are decomposed top-down into accountable units and execution outcomes are aggregated bottom-up to drive systematic review and refinement. This loop provides formal guarantees on termination and deadlock freedom while mirroring the feedback mechanisms of human enterprises. Together, these contributions transform multi-agent systems from static, pre-configured pipelines into self-organising and self-improving AI organisations capable of adapting to open-ended tasks across diverse domains. Empirical evaluation on PRDBench shows that OMC achieves an 84.67% success rate, surpassing the state of the art by 15.48 percentage points, with cross-domain case studies further demonstrating its generality.

Developing models capable of complex, multi-step reasoning is a central goal in artificial intelligence. While representing problems as graphs is a powerful approach, Graph Neural Networks (GNNs) are fundamentally constrained by their message-passing mechanism, which imposes a local bottleneck that limits global, holistic reasoning. We argue that dynamic programming (DP), which solves problems by iteratively refining a global state, offers a more powerful and suitable learning paradigm. We introduce FloydNet, a new architecture that embodies this principle. In contrast to local message passing, FloydNet maintains a global, all-pairs relationship tensor and learns a generalized DP operator to progressively refine it. This enables the model to develop a task-specific relational calculus, providing a principled framework for capturing long-range dependencies. Theoretically, we prove that FloydNet achieves 3-WL (2-FWL) expressive power, and its generalized form aligns with the k-FWL hierarchy. FloydNet demonstrates state-of-the-art performance across challenging domains: it achieves near-perfect scores (often >99\%) on the CLRS-30 algorithmic benchmark, finds exact optimal solutions for the general Traveling Salesman Problem (TSP) at rates significantly exceeding strong heuristics, and empirically matches the 3-WL test on the BREC benchmark. Our results establish this learned, DP-style refinement as a powerful and practical alternative to message passing for high-level graph reasoning.

This work studies a central extremal graph theory problem inspired by a 1975 conjecture of Erdos, which aims to find graphs with a given size (number of nodes) that maximize the number of edges without having 3- or 4-cycles. We formulate this problem as a sequential decision-making problem and compare AlphaZero, a neural network-guided tree search, with tabu search, a heuristic local search method. Using either method, by introducing a curriculum -- jump-starting the search for larger graphs using good graphs found at smaller sizes -- we improve the state-of-the-art lower bounds for several sizes. We also propose a flexible graph-generation environment and a permutation-invariant network architecture for learning to search in the space of graphs.

Multi-agent reasoning systems adopt a "generate-then-transfer" paradigm that forces end-to-end latency to scale linearly with pipeline depth. We introduce StreamMA, a multi-agent reasoning system that streams each reasoning step to downstream agents as soon as it is generated, pipelining adjacent agents and thus reducing latency. Surprisingly, this pipelining also improves effectiveness: because multi-step reasoning quality is non-uniform and early steps are more reliable than later ones, working with these reliable early steps instead of the full chain prevents error-prone late steps from misleading downstream agents. We formalize both advantages with the first closed-form joint analysis of stream, serial, and single protocols, deriving the effectiveness ordering, speedup upper bound, and cost ratio. Across eight reasoning benchmarks spanning mathematics, science, and code, two frontier LLMs (Claude Opus 4.6 and GPT-5.4), and three topologies (Chain, Tree, Graph), StreamMA outperforms both baselines (avg. +7.3 pp, max +22.4 pp on HMMT 2026; Claude Opus 4.6-high). Beyond these contributions, we discover a "step-level scaling law": increasing per-agent steps consistently improves both effectiveness and efficiency, a new scaling dimension orthogonal to and composable with agent-count scaling.

Multi-agent systems (MAS) have emerged as a promising paradigm for solving complex tasks. Recent work has explored self-evolving MAS that automatically optimize agent capabilities or communication topologies. However, existing methods either learn a topology that remains fixed at inference time or adapt only the topology or capability during inference. We empirically and theoretically show that effective test-time evolution requires jointly adapting both axes, but on different time scales: capabilities should update rapidly to handle emerging subtasks, while the topology should evolve more slowly to preserve coordination stability. We then introduce TacoMAS, a test-time co-evolution framework for dynamic MAS. TacoMAS formulates MAS inference as a task of online graph adaptation, where nodes represent agents with role-specific capabilities and edges define their communication topology. During inference, a fast capability loop updates agent expertise using trajectory-level feedback, while a slow meta-LLM-driven topology loop performs agents' birth-death operations on MAS, including edge edit, agent addition, and agent removal. We further show that this fast-slow design drives MAS evolution toward a task-conditioned stable equilibrium. Experiments on four benchmarks demonstrate that TacoMAS outperforms nearly 20 multi-agent baselines, achieving an average improvement of 13.3% over the strongest baseline. The codes are released at https://github.com/chenxu2-gif/TacoMAS-MultiAgent.

Graphs are an essential part of many machine learning problems such as analysis of parse trees, social networks, knowledge graphs, transportation systems, and molecular structures. Applying machine learning in these areas typically involves learning the graph structure and the relationship between the nodes of the graph. However, learning the graph structure is often complex, particularly when the graph is cyclic, and the transitions from one node to another are conditioned such as graphs used to represent a finite state machine. To solve this problem, we propose to extend the memory based Neural Turing Machine (NTM) with two novel additions. We allow for transitions between nodes to be influenced by information received from external environments, and we let the NTM learn the context of those transitions. We refer to this extension as the Conditional Neural Turing Machine (CNTM). We show that the CNTM can infer conditional transition graphs by empirically verifiying the model on two data sets: a large set of randomly generated graphs, and a graph modeling the information retrieval process during certain crisis situations. The results show that the CNTM is able to reproduce the paths inside the graph with accuracy ranging from 82,12% for 10 nodes graphs to 65,25% for 100 nodes graphs.

Message passing graph neural networks (GNNs) would appear to be powerful tools to learn distributed algorithms via gradient descent, but generate catastrophically incorrect predictions when nodes update asynchronously during inference. This failure under asynchrony effectively excludes these architectures from many potential applications, such as learning local communication policies between resource-constrained agents in, e.g., robotic swarms or sensor networks. In this work we explore why this failure occurs in common GNN architectures, and identify "implicitly-defined" GNNs as a class of architectures which is provably robust to partially asynchronous "hogwild" inference, adapting convergence guarantees from work in asynchronous and distributed optimization, e.g., Bertsekas (1982); Niu et al. (2011). We then propose a novel implicitly-defined GNN architecture, which we call an energy GNN. We show that this architecture outperforms other GNNs from this class on a variety of synthetic tasks inspired by multi-agent systems, and achieves competitive performance on real-world datasets.

An agent trained within a closed system can master any desired capability, as long as the following three conditions hold: (a) it receives sufficiently informative and aligned feedback, (b) its coverage of experience/data is broad enough, and (c) it has sufficient capacity and resource. In this position paper, we justify these conditions, and consider what limitations arise from (a) and (b) in closed systems, when assuming that (c) is not a bottleneck. Considering the special case of agents with matching input and output spaces (namely, language), we argue that such pure recursive self-improvement, dubbed "Socratic learning", can boost performance vastly beyond what is present in its initial data or knowledge, and is only limited by time, as well as gradual misalignment concerns. Furthermore, we propose a constructive framework to implement it, based on the notion of language games.

Reasoning system dynamics is one of the most important analytical approaches for many scientific studies. With the initial state of a system as input, the recent graph neural networks (GNNs)-based methods are capable of predicting the future state distant in time with high accuracy. Although these methods have diverse designs in modeling the coordinates and interacting forces of the system, we show that they actually share a common paradigm that learns the integration of the velocity over the interval between the initial and terminal coordinates. However, their integrand is constant w.r.t. time. Inspired by this observation, we propose a new approach to predict the integration based on several velocity estimations with Newton-Cotes formulas and prove its effectiveness theoretically. Extensive experiments on several benchmarks empirically demonstrate consistent and significant improvement compared with the state-of-the-art methods.

In this paper, we study the ideas of composition and decomposition in the context of vector spaces, graphs and matroids. For vector spaces V_{AB}, treated as collection of row vectors, with specified column set Auplus B, we define V_{SP}lrarv V_{PQ}, Scap Q= emptyset, to be the collection of all vectors (f_S,f_Q) such that (f_S,f_P)in V_{SP}, (f_P,f_Q)in V_{PQ}. An analogous operation G_{SP}lrarg G_{PQ}equivd G_{PQ} can be defined in relation to graphs G_{SP}, G_{PQ}, on edge sets Suplus P, Puplus Q, respectively in terms of an overlapping subgraph G_P which gets deleted in the right side graph (see for instance the notion of k-sum oxley). For matroids we define the `linking' M_{SP}lrarm M_{PQ} equivd (M_{SP}vee M_{PQ})times (Suplus Q), denoting the contraction operation by 'times'. In each case, we examine how to minimize the size of the `overlap' set P, without affecting the right side entity. In the case of vector spaces, there is a polynomial time algorithm for achieving the minimum, which we present. Similar ideas work for graphs and for matroids under appropriate conditions. Next we consider the problem of decomposition. Here, in the case of vector spaces, the problem is to decompose V_{SQ} as V_{SP}lrarv V_{PQ}, with minimum size P. We give a polynomial time algorithm for this purpose. In the case of graphs and matroids we give a solution to this problem under certain restrictions.

We incorporate discrete and continuous time Markov processes as building blocks into probabilistic graphical models with latent and observed variables. We introduce the automatic Backward Filtering Forward Guiding (BFFG) paradigm (Mider et al., 2021) for programmable inference on latent states and model parameters. Our starting point is a generative model, a forward description of the probabilistic process dynamics. We backpropagate the information provided by observations through the model to transform the generative (forward) model into a pre-conditional model guided by the data. It approximates the actual conditional model with known likelihood-ratio between the two. The backward filter and the forward change of measure are suitable to be incorporated into a probabilistic programming context because they can be formulated as a set of transformation rules. The guided generative model can be incorporated in different approaches to efficiently sample latent states and parameters conditional on observations. We show applicability in a variety of settings, including Markov chains with discrete state space, interacting particle systems, state space models, branching diffusions and Gamma processes.

Graph neural networks are commonly described through family-specific equations whose notation obscures shared computations and structural differences. We introduce a common layer equation that represents covered architectures through seven components: an update domain, channel set, propagation bank, per-channel message maps, channel-fusion operator, ego/residual map, and update map. The central factorization separates where information moves, encoded by the propagation bank, from what moves, encoded by the message maps. Function-valued fillings extend the same equation across local message passing, attention, spectral filtering, global communication, relation-specific channels, higher-order domains, and geometric messages. We make this unification explicit and checkable through worked reductions of canonical layers and component assignments spanning seven nonexclusive architectural families. A fixed slot discipline assigns operations by computational role and defines the framework's coverage boundary. The decomposition also yields component-level theoretical insights: under endpoint-local messages and node-local updates, operator support bounds one-layer dependencies, and one-layer global mixing requires a full effective operator row under the stated hypotheses. The resulting framework organizes more than 200 architectures in a common design space, enables component-wise comparison and generation of structurally consistent architectures, and connects propagation choices to oversmoothing, oversquashing, heterophily, and expressivity. It further exposes the empirical inverse problem of mapping measurable graph and task properties to validated component choices.

This paper introduces a new metamodel-based knowledge representation that significantly improves autonomous learning and adaptation. While interest in hybrid machine learning / symbolic AI systems leveraging, for example, reasoning and knowledge graphs, is gaining popularity, we find there remains a need for both a clear definition of knowledge and a metamodel to guide the creation and manipulation of knowledge. Some of the benefits of the metamodel we introduce in this paper include a solution to the symbol grounding problem, cumulative learning, and federated learning. We have applied the metamodel to problems ranging from time series analysis, computer vision, and natural language understanding and have found that the metamodel enables a wide variety of learning mechanisms ranging from machine learning, to graph network analysis and learning by reasoning engines to interoperate in a highly synergistic way. Our metamodel-based projects have consistently exhibited unprecedented accuracy, performance, and ability to generalize. This paper is inspired by the state-of-the-art approaches to AGI, recent AGI-aspiring work, the granular computing community, as well as Alfred Korzybski's general semantics. One surprising consequence of the metamodel is that it not only enables a new level of autonomous learning and optimal functioning for machine intelligences, but may also shed light on a path to better understanding how to improve human cognition.

Recent multi-agent LLM systems increasingly rely on graph-structured communication to coordinate specialized agents. We revisit multi-agent orchestration from a graph-engineering perspective: rather than optimizing a static topology, we synthesize a task-conditioned temporal workflow graph that jointly specifies agent connectivity and edge-level communication semantics. We introduce ReActNet, a training-free framework that compiles a query and a set of role-specialized agents into a sequence of directed communication graphs. Each graph snapshot corresponds to one reasoning stage, and each edge carries a natural-language instruction specifying the message that a source agent should provide to a target agent. The compiled temporal graph is then executed through structured message passing: agents update their reasoning states by integrating their previous states with messages from controller-assigned neighbors, and a final aggregator synthesizes the resulting states into the answer. This design separates graph compilation from graph execution, making multi-agent coordination explicit, inspectable, and task-conditioned without requiring reinforcement learning or gradient-based topology optimization. Across knowledge reasoning, mathematical problem solving, code generation, and GAIA-style assistant tasks, ReActNet consistently improves over fixed-topology and learned-topology baselines while maintaining competitive inference cost. These results suggest that effective multi-agent orchestration depends not only on which agents communicate, but also on engineering executable workflow graphs that encode when, why, and how information should flow during reasoning.

Theory of mind (ToM; Premack & Woodruff, 1978) broadly refers to humans' ability to represent the mental states of others, including their desires, beliefs, and intentions. We propose to train a machine to build such models too. We design a Theory of Mind neural network -- a ToMnet -- which uses meta-learning to build models of the agents it encounters, from observations of their behaviour alone. Through this process, it acquires a strong prior model for agents' behaviour, as well as the ability to bootstrap to richer predictions about agents' characteristics and mental states using only a small number of behavioural observations. We apply the ToMnet to agents behaving in simple gridworld environments, showing that it learns to model random, algorithmic, and deep reinforcement learning agents from varied populations, and that it passes classic ToM tasks such as the "Sally-Anne" test (Wimmer & Perner, 1983; Baron-Cohen et al., 1985) of recognising that others can hold false beliefs about the world. We argue that this system -- which autonomously learns how to model other agents in its world -- is an important step forward for developing multi-agent AI systems, for building intermediating technology for machine-human interaction, and for advancing the progress on interpretable AI.

Recurrent neural networks (RNNs) are widely used throughout neuroscience as models of local neural activity. Many properties of single RNNs are well characterized theoretically, but experimental neuroscience has moved in the direction of studying multiple interacting areas, and RNN theory needs to be likewise extended. We take a constructive approach towards this problem, leveraging tools from nonlinear control theory and machine learning to characterize when combinations of stable RNNs will themselves be stable. Importantly, we derive conditions which allow for massive feedback connections between interacting RNNs. We parameterize these conditions for easy optimization using gradient-based techniques, and show that stability-constrained "networks of networks" can perform well on challenging sequential-processing benchmark tasks. Altogether, our results provide a principled approach towards understanding distributed, modular function in the brain.

While machine learning has advanced through massive parallelization, we identify a critical blind spot: some problems are fundamentally sequential. These "inherently serial" problems-from mathematical reasoning to physical simulations to sequential decision-making-require dependent computational steps that cannot be parallelized. Drawing from complexity theory, we formalize this distinction and demonstrate that current parallel-centric architectures face fundamental limitations on such tasks. We argue that recognizing the serial nature of computation holds profound implications on machine learning, model design, hardware development. As AI tackles increasingly complex reasoning, deliberately scaling serial computation-not just parallel computation-is essential for continued progress.

Agentic AI denotes an architectural transition from stateless, prompt-driven generative models toward goal-directed systems capable of autonomous perception, planning, action, and adaptation through iterative control loops. This paper examines this transition by connecting foundational intelligent agent theories, including reactive, deliberative, and Belief-Desire-Intention models, with contemporary LLM-centric approaches such as tool invocation, memory-augmented reasoning, and multi-agent coordination. The paper presents three primary contributions: (i) a reference architecture for production-grade LLM agents that separates cognitive reasoning from execution using typed tool interfaces; (ii) a taxonomy of multi-agent topologies, together with their associated failure modes and mitigation approaches; and (iii) an enterprise hardening checklist that incorporates governance, observability, and reproducibility considerations. Through an analysis of emerging industry platforms, including Kore.ai, Salesforce Agentforce, TrueFoundry, ZenML, and LangChain, the study identifies a convergence toward standardized agent loops, registries, and auditable control mechanisms. It is argued that the subsequent phase of agentic AI development will parallel the maturation of web services, relying on shared protocols, typed contracts, and layered governance structures to support scalable and composable autonomy. The persistent challenges related to verifiability, interoperability, and safe autonomy remain key areas for future research and practical deployment.

We consider the problem of how a trusted, but computationally bounded agent (a 'verifier') can learn to interact with one or more powerful but untrusted agents ('provers') in order to solve a given task. More specifically, we study the case in which agents are represented using neural networks and refer to solutions of this problem as neural interactive proofs. First we introduce a unifying framework based on prover-verifier games, which generalises previously proposed interaction protocols. We then describe several new protocols for generating neural interactive proofs, and provide a theoretical comparison of both new and existing approaches. Finally, we support this theory with experiments in two domains: a toy graph isomorphism problem that illustrates the key ideas, and a code validation task using large language models. In so doing, we aim to create a foundation for future work on neural interactive proofs and their application in building safer AI systems.

The performance of a language model has been shown to be effectively modeled as a power-law in its parameter count. Here we study the scaling behaviors of Routing Networks: architectures that conditionally use only a subset of their parameters while processing an input. For these models, parameter count and computational requirement form two independent axes along which an increase leads to better performance. In this work we derive and justify scaling laws defined on these two variables which generalize those known for standard language models and describe the performance of a wide range of routing architectures trained via three different techniques. Afterwards we provide two applications of these laws: first deriving an Effective Parameter Count along which all models scale at the same rate, and then using the scaling coefficients to give a quantitative comparison of the three routing techniques considered. Our analysis derives from an extensive evaluation of Routing Networks across five orders of magnitude of size, including models with hundreds of experts and hundreds of billions of parameters.

Large language models generate plausible code but cannot verify correctness. Existing multi-agent systems simulate execution or leave verification optional. We introduce execution-grounded verification as a first-class principle: every code change must survive sandboxed execution before propagation. We instantiate this principle in AGENTFORGE, a multi-agent framework where Planner, Coder, Tester, Debugger, and Critic agents coordinate through shared memory and a mandatory Docker sandbox. We formalize software engineering with LLMs as an iterative decision process over repository states, where execution feedback provides a stronger supervision signal than next-token likelihood. AGENTFORGE achieves 40.0\% resolution on SWE-BENCH Lite, outperforming single-agent baselines by 26--28 points. Ablations confirm that execution feedback and role decomposition each independently drive performance. The framework is open-source at https://github.com/raja21068/AutoCodeAI.

The shift from monolithic LLMs to distributed multi-agent architectures demands new frameworks for verifying and securing autonomous coordination. Unlike traditional multi-agent systems focused on cooperative state alignment, modern LLM patterns: multi-agent debate, constitutional oversight, helper-critic loops-rely on adversarial critique for error correction and reasoning refinement. Since LLMs are dynamical systems whose latent states are imperfectly observable from verbalized outputs, securing these networks requires understanding both macroscopic topology and microscopic agent observability. This paper establishes a rigorous graph-theoretic framework for analyzing consensus in signed, directed interaction networks, bridging graph theory and LLM reasoning by formally mapping Transformer cross-entropy log-odds to the signed Laplacian. We characterize agreement stability through structural balance theory, showing how unbalanced critique cycles produce logical frustration and persistent reasoning oscillations, and prove that unobservable latent states from hidden system prompts act as topological Trojan horses that destabilize cooperative consensus. To resolve unobservable deadlocks, we restrict interaction topologies to chordal graphs and apply matrix decomposition with Gram-Schmidt orthogonalization, proving that rank-one spectral edge perturbations deterministically break expertise symmetry by shifting eigenvalues into the stable left-half plane. Core contributions include consensus theorems, polynomial-time Perfect Elimination Ordering verification algorithms, and large-scale empirical validation on clustered ensembles of LLaMA-3, Mistral, and Gemma agents.

Reasoning is a fundamental substrate for solving novel and complex problems. Deliberate efforts in learning and developing frameworks around System 2 reasoning have made great strides, yet problems of sufficient complexity remain largely out of reach for open models. To address this gap, we examine the potential of Generative Flow Networks as a fine-tuning method for LLMs to unlock advanced reasoning capabilities. In this paper, we present a proof of concept in the domain of formal reasoning, specifically in the Neural Theorem Proving (NTP) setting, where proofs specified in a formal language such as Lean can be deterministically and objectively verified. Unlike classical reward-maximization reinforcement learning, which frequently over-exploits high-reward actions and fails to effectively explore the state space, GFlowNets have emerged as a promising approach for sampling compositional objects, improving generalization, and enabling models to maintain diverse hypotheses. Our early results demonstrate GFlowNet fine-tuning's potential for enhancing model performance in a search setting, which is especially relevant given the paradigm shift towards inference time compute scaling and "thinking slowly."

We develop a learning-based framework for constructing shrinking disturbance-invariant tubes under state- and input-dependent uncertainty, intended as a building block for tube Model Predictive Control (MPC), and certify safety via a lifted, isotone (order-preserving) fixed-point map. Gaussian Process (GP) posteriors become (1-α) credible ellipsoids, then polytopic outer sets for deterministic set operations. A two-time-scale scheme separates learning epochs, where these polytopes are frozen, from an inner, outside-in iteration that converges to a compact fixed point Z^star!subseteq!mathcal G; its state projection is RPI for the plant. As data accumulate, disturbance polytopes tighten, and the associated tubes nest monotonically, resolving the circular dependence between the set to be verified and the disturbance model while preserving hard constraints. A double-integrator study illustrates shrinking tube cross-sections in data-rich regions while maintaining invariance.

Deep neural networks, despite their success in numerous applications, often function without established theoretical foundations. In this paper, we bridge this gap by drawing parallels between deep learning and classical numerical analysis. By framing neural networks as operators with fixed points representing desired solutions, we develop a theoretical framework grounded in iterative methods for operator equations. Under defined conditions, we present convergence proofs based on fixed point theory. We demonstrate that popular architectures, such as diffusion models and AlphaFold, inherently employ iterative operator learning. Empirical assessments highlight that performing iterations through network operators improves performance. We also introduce an iterative graph neural network, PIGN, that further demonstrates benefits of iterations. Our work aims to enhance the understanding of deep learning by merging insights from numerical analysis, potentially guiding the design of future networks with clearer theoretical underpinnings and improved performance.