arXiv subjects

Leo Lobski

Publications and source records attributed to Leo Lobski.

9 recordsLinked to original sources

arXiv2026

Modulation-Reaction Networks

Biochemical systems involve both the flow of matter, in which entities transform into one another via reactions, and the flow of information, in which entities regulate which reactions may occur. Boolean networks capture the latter; reaction networks capture the former. Yet no unified qualitative formalism treats regulated reactions as its principal objects of study, despite their prominence in standards such as the Systems Biology Graphical Notation Process Description (SBGN-PD) language. We introduce modulation-reaction networks (MR-networks), a mathematical framework in which entities modulate reactions through activations and inhibitions, and study their synchronous Boolean semantics. To reason about MR-networks we develop Modulation-Reaction Logic (MRL), a hybrid modal $\mu$-calculus whose modalities reason about the structure of the network and whose fixed-point operators capture temporal evolution of the computation. We establish a collection of validities, including a complete characterisation of the one-step update rule, and demonstrate the expressive power of MRL by formalising properties of biological interest such as reachability, sustained production, and presence of attractors. We show that MRL admits model-checking via an evaluation game, and introduce a bisimulation relation for MR-networks, which is proved to be invariant for all MRL-formulas. As a step towards a biologically more realistic computational model, we sketch the asynchronous semantics of MR-networks, and outline how the developments for the synchronous case transfer to the study of the asynchronous one.

cs.LO↗

arXiv2026

Layered Monoidal Theories II: Fibrational Semantics

Layered monoidal theories provide a categorical framework for studying scientific theories at different levels of abstraction, via string diagrammatic algebra. We introduce models for three closely related classes of layered monoidal theories: fibrational, opfibrational and deflational theories. We prove soundness and completeness of these theories for the respective models. Our work reveals connections between layered monoidal theories and well-known categorical structures such as Grothendieck fibrations and displayed categories.

math.CT↗

arXiv2026

Layered Monoidal Theories I: Diagrammatic Algebra and Applications

We develop layered monoidal theories -- a generalisation of monoidal theories combining formal descriptions of a system at different levels of abstraction. Via their representation as string diagrams, monoidal theories provide a graphical formalism to reason algebraically about information flow in models across different fields of science. Layered monoidal theories allow mixing several monoidal theories (together with translations between them) within the same string diagram, while retaining mathematical precision and semantic interpretability. We develop the mathematical foundations of layered monoidal theories, as well as providing several instances of our approach, including digital and electrical circuits, quantum processes, chemical reactions, concurrent processes, and probability theory.

cs.LO↗

arXiv2025

Layered Monoidal Theories

In the first part, we develop layered monoidal theories - a generalisation of monoidal theories combining descriptions of a system at several levels. Via their representation as string diagrams, monoidal theories provide a graphical syntax with a visually intuitive notions of information flow and composition. Layered monoidal theories allow mixing several monoidal theories (together with translations between them) within the same string diagram, while retaining mathematical precision and semantic interpretability. We define three flavours of layered monoidal theories, provide a recursively generated syntax for each, and construct a free-forgetful adjunction with respect to three closely related semantics: opfibrations, fibrations and deflations. We motivate the general theory by providing several examples from existing literature. In the second part, we develop a formal approach to retrosynthesis - the process of backwards reaction search in synthetic chemistry. Chemical processes are treated at three levels of abstraction: (1) (formal) reactions encode all chemically feasible combinatorial rearrangements of molecules, (2) reaction schemes encode transformations applicable to 'patches' of molecules (including the functional groups), and (3) disconnection rules encode local chemical rewrite rules applicable to a single bond or atom at a time. We show that the three levels are tightly linked: the reactions are generated by the reaction schemes, while there is a functorial translation from the disconnection rules to the reactions. Moreover, the translation from the disconnection rules to the reactions is shown to be sound, complete and universal - allowing one to treat the disconnection rules as a formal syntax with the semantics provided by the reactions. We tie together the two parts by providing a formalisation of retrosynthesis within a certain layered monoidal theory.

cs.LO↗

arXiv2024

Disconnection Rules are Complete for Chemical Reactions

We provide a category theoretical framework capturing two approaches to graph-based models of chemistry: formal reactions and disconnection rules. We model a translation from the latter to the former as a functor, which is faithful, and full up to isomorphism. This allows us to state, as our main result, that the disconnection rules are sound, complete and universal with respect to the reactions. Concretely, this means that every reaction can be decomposed into a sequence of disconnection rules in an essentially unique way. This provides a uniform way to store reaction data, and gives an algorithmic interface between (forward) reaction prediction and (backward) reaction search or retrosynthesis.

cs.LO↗

arXiv2024

Learning Closed Signal Flow Graphs

We develop a learning algorithm for closed signal flow graphs - a graphical model of signal transducers. The algorithm relies on the correspondence between closed signal flow graphs and weighted finite automata on a singleton alphabet. We demonstrate that this procedure results in a genuine reduction of complexity: our algorithm fares better than existing learning algorithms for weighted automata restricted to the case of a singleton alphabet.

cs.LO↗

arXiv2023

A Categorical Model for Retrosynthetic Reaction Analysis

We introduce a mathematical framework for retrosynthetic analysis, an important research method in synthetic chemistry. Our approach represents molecules and their interaction using string diagrams in layered props - a recently introduced categorical model for partial explanations in scientific reasoning. Such principled approach allows one to model features currently not available in automated retrosynthesis tools, such as chirality, reaction environment and protection-deprotection steps.

cs.LO↗

arXiv2023

String Diagrams for Layered Explanations

We propose a categorical framework to reason about scientific explanations: descriptions of a phenomenon meant to translate it into simpler terms, or into a context that has been already understood. Our motivating examples come from systems biology, electrical circuit theory, and concurrency. We demonstrate how three explanatory models in these seemingly diverse areas can be all understood uniformly via a graphical calculus of layered props. Layered props allow for a compact visual presentation of the same phenomenon at different levels of precision, as well as the translation between these levels. Notably, our approach allows for partial explanations, that is, for translating just one part of a diagram while keeping the rest of the diagram untouched. Furthermore, our approach paves the way for formal reasoning about counterfactual models in systems biology.

cs.LO↗

arXiv2021

There and back again: A circuit extraction tale

Translations between the quantum circuit model and the measurement-based one-way model are useful for verification and optimisation of quantum computations. They make crucial use of a property known as gflow. While gflow is defined for one-way computations allowing measurements in three different planes of the Bloch sphere, most research so far has focused on computations containing only measurements in the XY-plane. Here, we give the first circuit-extraction algorithm to work for one-way computations containing measurements in all three planes and having gflow. The algorithm is efficient and the resulting circuits do not contain ancillae. One-way computations are represented using the ZX-calculus, hence the algorithm also represents the most general known procedure for extracting circuits from ZX-diagrams. In developing this algorithm, we generalise several concepts and results previously known for computations containing only XY-plane measurements. We bring together several known rewrite rules for measurement patterns and formalise them in a unified notation using the ZX-calculus. These rules are used to simplify measurement patterns by reducing the number of qubits while preserving both the semantics and the existence of gflow. The results can be applied to circuit optimisation by translating circuits to patterns and back again.

quant-ph↗