Intelligence-biological, artificial, or collective-requires structural coherence across recursive reasoning processes to scale effectively. As complex systems grow, coherence becomes fragile unless a higher-order structure ensures semantic consistency. This paper introduces the Recursive Coherence Principle (RCP): a foundational constraint stating that for any reasoning system of order N, composed of systems operating over conceptual spaces of order N-1, semantic coherence is preserved only by a recursively evaluable generalization operator that spans and aligns those lower-order conceptual spaces. Crucially, this coherence enables structural alignment. Without recursive coherence, no system can reliably preserve goals, meanings, or reasoning consistency at scale. We formally define the Functional Model of Intelligence (FMI) as the only known operator capable of satisfying the RCP at any scale. The FMI is a minimal, composable architecture with internal functions (evaluation, modeling, adaptation, stability, decomposition, bridging) and external functions (storage, recall, System 1 and System 2 reasoning) vital for preserving semantic structure across inference and coordination layers. We prove that any system lacking the FMI will experience recursive coherence breakdown as it scales, arguing that common AI issues like misalignment, hallucination, and instability are symptoms of this structural coherence loss. Unlike other foundational principles, RCP uniquely captures the internal, recursive dynamics needed for coherent, alignable intelligence, modeling semantic coherence under recursion. This work significantly impacts AI alignment, advocating a shift from behavioral constraints to structural coherence, and offers a pathway for safely generalizable, robustly coherent AI at scale.

Causal modelling frameworks link observable correlations to causal explanations, which is a crucial aspect of science. These models represent causal relationships through directed graphs, with vertices and edges denoting systems and transformations within a theory. Most studies focus on acyclic causal graphs, where well-defined probability rules and powerful graph-theoretic properties like the d-separation theorem apply. However, understanding complex feedback processes and exotic fundamental scenarios with causal loops requires cyclic causal models, where such results do not generally hold. While progress has been made in classical cyclic causal models, challenges remain in uniquely fixing probability distributions and identifying graph-separation properties applicable in general cyclic models. In cyclic quantum scenarios, existing frameworks have focussed on a subset of possible cyclic causal scenarios, with graph-separation properties yet unexplored. This work proposes a framework applicable to all consistent quantum and classical cyclic causal models on finite-dimensional systems. We address these challenges by introducing a robust probability rule and a novel graph-separation property, p-separation, which we prove to be sound and complete for all such models. Our approach maps cyclic causal models to acyclic ones with post-selection, leveraging the post-selected quantum teleportation protocol. We characterize these protocols and their success probabilities along the way. We also establish connections between this formalism and other classical and quantum frameworks to inform a more unified perspective on causality. This provides a foundation for more general cyclic causal discovery algorithms and to systematically extend open problems and techniques from acyclic informational networks (e.g., certification of non-classicality) to cyclic causal structures and networks.

Functional causal models (fCMs) specify functional dependencies between random variables associated to the vertices of a graph. In directed acyclic graphs (DAGs), fCMs are well-understood: a unique probability distribution on the random variables can be easily specified, and a crucial graph-separation result called the d-separation theorem allows one to characterize conditional independences between the variables. However, fCMs on cyclic graphs pose challenges due to the absence of a systematic way to assign a unique probability distribution to the fCM's variables, the failure of the d-separation theorem, and lack of a generalization of this theorem that is applicable to all consistent cyclic fCMs. In this work, we develop a causal modeling framework applicable to all cyclic fCMs involving finite-cardinality variables, except inconsistent ones admitting no solutions. Our probability rule assigns a unique distribution even to non-uniquely solvable cyclic fCMs and reduces to the known rule for uniquely solvable fCMs. We identify a class of fCMs, called averagely uniquely solvable, that we show to be the largest class where the probabilities admit a Markov factorization. Furthermore, we introduce a new graph-separation property, p-separation, and prove this to be sound and complete for all consistent finite-cardinality cyclic fCMs while recovering the d-separation theorem for DAGs. These results are obtained by considering classical post-selected teleportation protocols inspired by analogous protocols in quantum information theory. We discuss further avenues for exploration, linking in particular problems in cyclic fCMs and in quantum causality.

We propose a novel approach to nonlinear functional regression, called the Mapping-to-Parameter function model, which addresses complex and nonlinear functional regression problems in parameter space by employing any supervised learning technique. Central to this model is the mapping of function data from an infinite-dimensional function space to a finite-dimensional parameter space. This is accomplished by concurrently approximating multiple functions with a common set of B-spline basis functions by any chosen order, with their knot distribution determined by the Iterative Local Placement Algorithm, a newly proposed free knot placement algorithm. In contrast to the conventional equidistant knot placement strategy that uniformly distributes knot locations based on a predefined number of knots, our proposed algorithms determine knot location according to the local complexity of the input or output functions. The performance of our knot placement algorithms is shown to be robust in both single-function approximation and multiple-function approximation contexts. Furthermore, the effectiveness and advantage of the proposed prediction model in handling both function-on-scalar regression and function-on-function regression problems are demonstrated through several real data applications, in comparison with four groups of state-of-the-art methods.

The Keras Functional API provides a way to build flexible and complex neural networks in TensorFlow. The Functional API is used to design networks that are not linear. We used the Sequential API in the CNN tutorial to build an image classification model with Keras and TensorFlow. The Sequential API involves stacking layers. One layer is followed by another layer until the final dense layer.

When causal quantities cannot be point identified, researchers often pursue partial identification to quantify the range of possible values. However, the peculiarities of applied research conditions can make this analytically intractable. We present a general and automated approach to causal inference in discrete settings. We show causal questions with discrete data reduce to polynomial programming problems, and we present an algorithm to automatically bound causal effects using efficient dual relaxation and spatial branch-and-bound techniques. The user declares an estimand, states assumptions, and provides data (however incomplete or mismeasured). The algorithm then searches over admissible data-generating processes and outputs the most precise possible range consistent with available information -- i.e., sharp bounds -- including a point-identified solution if one exists. Because this search can be computationally intensive, our procedure reports and continually refines non-sharp ranges that are guaranteed to contain the truth at all times, even when the algorithm is not run to completion. Moreover, it offers an additional guarantee we refer to as $\epsilon$-sharpness, characterizing the worst-case looseness of the incomplete bounds. Analytically validated simulations show the algorithm accommodates classic obstacles, including confounding, selection, measurement error, noncompliance, and nonresponse.

The inexpressive Description Logic (DL) $\mathcal{FL}_0$, which has conjunction and value restriction as its only concept constructors, had fallen into disrepute when it turned out that reasoning in $\mathcal{FL}_0$ w.r.t. general TBoxes is ExpTime-complete, i.e., as hard as in the considerably more expressive logic $\mathcal{ALC}$. In this paper, we rehabilitate $\mathcal{FL}_0$ by presenting a dedicated subsumption algorithm for $\mathcal{FL}_0$, which is much simpler than the tableau-based algorithms employed by highly optimized DL reasoners. Our experiments show that the performance of our novel algorithm, as prototypically implemented in our $\mathcal{FL}_o$wer reasoner, compares very well with that of the highly optimized reasoners. $\mathcal{FL}_o$wer can also deal with ontologies written in the extension $\mathcal{FL}_{\bot}$ of $\mathcal{FL}_0$ with the top and the bottom concept by employing a polynomial-time reduction, shown in this paper, which eliminates top and bottom. We also investigate the complexity of reasoning in DLs related to the Horn-fragments of $\mathcal{FL}_0$ and $\mathcal{FL}_{\bot}$.

We present a scalable nonparametric Bayesian method to perform network reconstruction from observed functional behavior, that at the same time infers the communities present in the network. We show that the joint reconstruction with community detection has a synergistic effect, where the edge correlations used to inform the existence of communities are inherently also used to improve the accuracy of the reconstruction, which in turn can better inform the uncovering of communities. We illustrate the use of our method with observations arising from epidemic models and the Ising model, both on synthetic and empirical networks, as well as on data containing only functional information.

It first reviews the core issues in experiencebased design, for example, (1) the content of a design experience (or case), (2) the internal organization of design cases, (3) the language for indexing the cases, (4) the mechanism for retrieving a case relevant to a given design task, (5) the mechanism for adapting a retrieved design to satisfy the constraints of the design task, (6) the mechanism for evaluating a design against the specification of the design task, (7) the mechanism for redesigning a failed design, (8) the mechanism for acquiring new design knowledge, (9) the mechanism for chunking information about a design into a new case, and (10) the mechanism for storing a new case in memory for potential reuse in the future. It then proposes that decisions about these issues might lie in the designer's comprehension of the designs of artifacts he/she has encountered in the past, that is, in his/her mental models of how the designs achieve the functions and satisfy the constraints of the artifacts. To elaborate and evaluate this proposal, the dissertation analyzes the design of physical devices such as simple electric circuits, heat exchangers, and angular momentum controllers. It develops a theory of designers' comprehension of device designs in terms of functional models of how devices work. The functional model of a device provides a causal explanation of how the structure of the device produces its functions.

This post is the second in a series of three posts, each of which discuss the fundamental concepts of Artificial Intelligence. In our first post we discussed AI definitions, helping our readers to understand the basic concepts behind AI, giving them the tools required to sift through the many AI articles out there and form their own opinion. In this second post, we will discuss several notions which are important in understanding the limits of AI. Figure 1: How intelligent can Artificial Intelligence get? When we speak about how far AI can go, there are two "philosophies": strong AI and weak AI. The most commonly followed philosophy is that of weak AI, which means that machines can manifest certain intelligent behavior to solve specific (hard) tasks, but that they will never equal the human mind.