Mirco A. Mannucci

Mirco A. Mannucci

In 2006 we proposed Quantum Fuzzy Sets, observing that states of a quantum register could serve as characteristic functions of fuzzy subsets, embedding Zadeh's unit interval into the Bloch sphere. That paper was deliberately preliminary. In the two decades since, the idea has been taken up by researchers working on quantum annealers, intuitionistic fuzzy connectives, and quantum machine learning, while parallel developments in categorical quantum mechanics have reshaped the theoretical landscape. The present paper revisits that programme and introduces two main extensions. First, we move from pure states to density matrices, so that truth values occupy the entire Bloch ball rather than its surface; this captures the phenomenon of semantic decoherence that pure-state semantics cannot express. Second, we introduce the Q-Matrix, a global density matrix from which individual quantum fuzzy sets emerge as local sections via partial trace. We define a category QFS of quantum fuzzy sets, establish basic structural properties (monoidal structure, fibration over Set), characterize the classical limit as simultaneous diagonalizability, and exhibit an obstruction to a fully internal Frobenius-algebra treatment.

Mirco A. Mannucci

A fundamental question in search-guided AI: what topology should guide Monte Carlo Tree Search (MCTS) in puzzle solving? Prior work applied topological features to guide MCTS in ARC-style tasks using grid topology -- the Laplacian spectral properties of cell connectivity -- and found no benefit. We identify the root cause: grid topology is constant across all instances. We propose measuring \emph{solution space topology} instead: the structure of valid color assignments constrained by detected pattern rules. We build this via compatibility graphs where nodes are $(cell, color)$ pairs and edges represent compatible assignments under pattern constraints. Our method: (1) detect pattern rules automatically with 100\% accuracy on 5 types, (2) construct compatibility graphs encoding solution space structure, (3) extract topological features (algebraic connectivity, rigidity, color structure) that vary with task difficulty, (4) integrate these features into MCTS node selection via sibling-normalized scores. We provide formal definitions, a rigorous selection formula, and comprehensive ablations showing that algebraic connectivity is the dominant signal. The work demonstrates that topology matters for search -- but only the \emph{right} topology. For puzzle solving, this is solution space structure, not problem space structure.

Mirco A. Mannucci

GANs promise indistinguishability, logic explains it. We put the two on a budget: a discriminator that can only ``see'' up to a logical depth $k$, and a generator that must look correct to that bounded observer. \textbf{LOGAN} (LOGical GANs) casts the discriminator as a depth-$k$ Ehrenfeucht--Fraïssé (EF) \emph{Opponent} that searches for small, legible faults (odd cycles, nonplanar crossings, directed bridges), while the generator plays \emph{Builder}, producing samples that admit a $k$-round matching to a target theory $T$. We ship a minimal toolkit -- an EF-probe simulator and MSO-style graph checkers -- and four experiments including real neural GAN training with PyTorch. Beyond verification, we score samples with a \emph{logical loss} that mixes budgeted EF round-resilience with cheap certificate terms, enabling a practical curriculum on depth. Framework validation demonstrates $92\%$--$98\%$ property satisfaction via simulation (Exp.~3), while real neural GAN training achieves $5\%$--$14\%$ improvements on challenging properties and $98\%$ satisfaction on connectivity (matching simulation) through adversarial learning (Exp.~4). LOGAN is a compact, reproducible path toward logic-bounded generation with interpretable failures, proven effectiveness (both simulated and real training), and dials for control.

Mirco A. Mannucci, Deborah Tylor

In this article we describe a new approach for detecting changes in rapidly evolving large-scale graphs. The key notion involved is local alertness: nodes monitor change within their neighborhoods at each time step. Here we propose a financial local alertness application for cointegrated stock pairs