arXiv subjects

Fabrizio Genovese

Publications and source records attributed to Fabrizio Genovese.

At least 19 recordsLinked to original sources

arXiv2025

A Note on Publicly Verifiable Quantum Money with Low Quantum Computational Resources

In this work we present a publicly verifiable quantum money protocol which assumes close to no quantum computational capabilities. We rely on one-time memories which in turn can be built from quantum conjugate coding and hardware-based assumptions. Specifically, our scheme allows for a limited number of verifications and also allows for quantum tokens for digital signatures. Double spending is prevented by the no-cloning principle of conjugate coding states. An implementation of the concepts presented in this work can be found at https://github.com/neverlocal/otm_billz.

quant-ph↗

arXiv2024

Escrows are optics

We provide a categorical interpretation for escrows, i.e. trading protocols in trustless environment, where the exchange between two agents is mediated by a third party where the buyer locks the money until they receive the goods they want from the seller. A simplified escrow system can be modeled as a certain kind of morphism in the category of optics on a monoidal category. When objects in the base category have monoid and comonoid structures, more involved kinds of escrows `with intermediaries' can be modelled as morphisms with action-like properties.

math.CT↗

arXiv2023

Obstructions to Compositionality

Compositionality is at the heart of computer science and several other areas of applied category theory such as computational linguistics, categorical quantum mechanics, interpretable AI, dynamical systems, compositional game theory, and Petri nets. However, the meaning of the term seems to vary across the many different applications. This work contributes to understanding, and in particular qualifying, different kinds of compositionality. Formally, we introduce invariants of categories that we call zeroth and first homotopy posets, generalising in a precise sense the pi0 and pi1 of a groupoid. These posets can be used to obtain a qualitative description of how far an object is from being terminal and a morphism is from being iso. In the context of applied category theory, this formal machinery gives us a way to qualitatively describe the "failures of compositionality", seen as failures of certain (op)lax functors to be strong, by classifying obstructions to the (op)laxators being isomorphisms. Failure of compositionality, for example for the interpretation of a categorical syntax in a semantic universe, can both be a bad thing and a good thing, which we illustrate by respective examples in graph theory and quantum theory.

math.CT↗

arXiv2023

Fibrational Linguistics (FibLang): Language Acquisition

In this work we show how FibLang, a category-theoretic framework concerned with the interplay between language and meaning, can be used to describe vocabulary acquisition, that is the process with which a speaker acquires new vocabulary (through experience or interaction). We model two different kinds of vocabulary acquisition, which we call 'by example' and 'by paraphrasis'. The former captures the idea of acquiring the meaning of a word by being shown a witness representing that word, as in 'understanding what a cat is, by looking at a cat'. The latter captures the idea of acquiring meaning by listening to some other speaker rephrasing the word with others already known to the learner. We provide a category-theoretic model for vocabulary acquisition by paraphrasis based on the construction of free promonads. We draw parallels between our work and Wittgenstein's dynamical approach to language, commonly known as 'language games'.

math.CT↗

arXiv2023

The Essence of Petri Net Gluings

Many categorical frameworks have been proposed to formalize the idea of gluing Petri nets with each other. Such frameworks model net gluings in terms of sharing of resources or synchronization of transitions. Interpretations given to these gluings are more or less satisfactory when we consider Petri nets with a semantics attached to them. In this work, we define a framework to compose Petri nets together in such a way that their semantics is respected. In addition to this, we show how our framework generalizes the previously defined ones.

math.CT↗

arXiv2022

Fibrational linguistics: First concepts

We define a general mathematical framework for linguistics based on the theory of fibrations, called FibLang. We start by modelling the interaction between linguistics and cognition in the most general way possible, with a heavy focus on conceptually motivating any assumption we make. The advantage is that FibLang remains agnostic with respect to any particular axiomatization of grammar one may choose. As such, it is compatible with already existing categorical models of language (such as for example, DisCoCat), providing a formally sound framework to apply mathematical tools developed in the context of category theory, mainly categorical logic, to the study of language

math.CT↗

arXiv2022

Yoneda Hacking: The Algebra of Attacker Actions

Our work focuses on modeling the security of systems from their component-level designs. Towards this goal, we develop a categorical formalism to model attacker actions. Equipping the categorical formalism with algebras produces two interesting results for security modeling. First, using the Yoneda lemma, we can model attacker reconnaissance missions. In this context, the Yoneda lemma shows us that if two system representations, one being complete and the other being the attacker's incomplete view, agree at every possible test, they behave the same. The implication is that attackers can still successfully exploit the system even with incomplete information. Second, we model the potential changes to the system via an exploit. An exploit either manipulates the interactions between system components, such as providing the wrong values to a sensor, or changes the components themselves, such as controlling a global positioning system (GPS). One additional benefit of using category theory is that mathematical operations can be represented as formal diagrams, helpful in applying this analysis in a model-based design setting. We illustrate this modeling framework using an unmanned aerial vehicle (UAV) cyber-physical system model. We demonstrate and model two types of attacks (1) a rewiring attack, which violates data integrity, and (2) a rewriting attack, which violates availability.

cs.CR↗

arXiv2021

Categories of Nets

We present a unified framework for Petri nets and various variants, such as pre-nets and Kock's whole-grain Petri nets. Our framework is based on a less well-studied notion that we call $Σ$-nets, which allow finer control over whether tokens are treated using the collective or individual token philosophy. We describe three forms of execution semantics in which pre-nets generate strict monoidal categories, $Σ$-nets (including whole-grain Petri nets) generate symmetric strict monoidal categories, and Petri nets generate commutative monoidal categories, all by left adjoint functors. We also construct adjunctions relating these categories of nets to each other, in particular showing that all kinds of net can be embedded in the unifying category of $Σ$-nets, in a way that commutes coherently with their execution semantics.

math.CT↗

arXiv2020

idris-ct: A Library to do Category Theory in Idris

We introduce idris-ct, a Idris library providing verified type definitions of categorical concepts.idris-ct strives to be a bridge between academy and industry, catering both to category theorists who want to implement and try their ideas in a practical environment and to businesses and engineers who care about formalization with category theory: It is inspired by similar libraries developed for theorem proving but remains very practical, being aimed at software production in business. Nevertheless, the use of dependent types allows for a formally correct implementation of categorical concepts, so that guarantees can be made on software properties.

cs.LO↗

arXiv2020

A Categorical Semantics for Guarded Petri Nets

We build on the correspondence between Petri nets and free symmetric strict monoidal categories already investigated in the literature, and present a categorical semantics for Petri nets with guards. This comes in two flavors: Deterministic and with side-effects. Using the Grothendieck construction, we show how the guard semantics can be internalized in the net itself.

math.CT↗

arXiv2019

Mapping finite state machines to zk-SNARKS Using Category Theory

We provide a categorical procedure to turn graphs corresponding to state spaces of finite state machines into boolean circuits, leveraging on the fact that boolean circuits can be easily turned into zk-SNARKS. Our circuits verify that a given sequence of edges and nodes is indeed a path in the graph they represent. We then generalize to circuits verifying paths in arbitrary graphs. We prove that all of our correspondences are pseudofunctorial, and behave nicely with respect to each other.

cs.CR↗

arXiv2019

The Mathematical Specification of the Statebox Language

This document defines the mathematical backbone of the Statebox programming language. In the simplest way possible, Statebox can be seen as a clever way to tie together different theoretical structures to maximize their benefits and limit their downsides. Since consistency and correctness are central requisites for our language, it became clear from the beginning that such tying could not be achieved by just hacking together different pieces of code representing implementations of the structures we wanted to leverage: Rigorous mathematics is employed to ensure both conceptual consistency of the language and reliability of the code itself. The mathematics presented here is what guided the implementation process, and we deemed very useful to release it to the public to help people wanting to audit our work to better understand the code itself.

cs.PL↗

arXiv2019

Computational Petri Nets: Adjunctions Considered Harmful

We review some of the endeavors in trying to connect Petri nets with free symmetric monoidal categories. We give a list of requirement such connections should respect if they are meant to be useful for practical/implementation purposes. We show how previous approaches do not satisfy them, and give compelling evidence that this depends on trying to make the correspondence functorial in the direction from nets to free symmetric monoidal categories, in order to produce an adjunction. We show that dropping this immediately honors our desiderata, and conclude by introducing an Idris library which implements them.

math.CT↗

arXiv2019

Executions in (Semi-)Integer Petri Nets are Compact Closed Categories

In this work, we analyse Petri nets where places are allowed to have a negative number of tokens. For each net we build its correspondent category of executions, which is compact closed, and prove that this procedure is functorial. We moreover exhibit a procedure to recover the original net from its category of executions, show that it is again functorial, and that this gives rise to an adjoint pair. Finally, we use compact closeness to infer that allowing negative tokens in a Petri net makes the causal relations between transition firings non-trivial, and we use this to model interesting phenomena in economics and computer science.

math.CT↗

arXiv2019

Quantum Field Theory in Categorical Quantum Mechanics

We use tools from non-standard analysis to formulate the building blocks of quantum field theory within the framework of categorical quantum mechanics. Building upon previous work, we construct an object of *Hilb having quantum fields as states and we show that the usual ladder and field operators can be defined as suitable endomorphisms. We deal with relativistic normalisation and we obtain the Lorentz invariant Heisenberg picture operators. By moving to a coherent perspective -- where the classical time and momentum parameters are replaced by wavefunctions over the parameter spaces -- we show that ladder operators and field operators can be obtained by applying the same morphism to plane waves and delta functions respectively. Finally, we formulate the commutation relations diagrammatically and we use them to derive the propagator.

quant-ph↗

arXiv2018

Uniqueness of Composition in Quantum Theory and Linguistics

We derive a uniqueness result for non-Cartesian composition of systems in a large class of process theories, with important implications for quantum theory and linguistics. Specifically, we consider theories of wavefunctions valued in commutative involutive semirings -- as modelled by categories of free finite-dimensional modules -- and we prove that the only bilinear compact-closed symmetric monoidal structure is the canonical one (up to linear monoidal equivalence). Our results apply to conventional quantum theory and other toy theories of interest in the literature, such as real quantum theory, relational quantum theory, hyperbolic quantum theory and modal quantum theory. In computational linguistics they imply that linear models for categorical compositional distributional semantics (DisCoCat) -- such as vector spaces, sets and relations, and sets and histograms -- admit an (essentially) unique compatible pregroup grammar.

quant-ph↗

arXiv2018

Towards Quantum Field Theory in Categorical Quantum Mechanics

In this work, we use tools from non-standard analysis to introduce infinite-dimensional quantum systems and quantum fields within the framework of Categorical Quantum Mechanics. We define a dagger compact category *Hilb suitable for the algebraic manipulation of unbounded operators, Dirac deltas and plane-waves. We cover in detail the construction of quantum systems for particles in boxes with periodic boundary conditions, particles on cubic lattices, and particles in real space. Not quite satisfied with this, we show how certain non-separable Hilbert spaces can also be modelled in our non-standard framework, and we explicitly treat the cases of quantum fields on cubic lattices and quantum fields in real space.

quant-ph↗

arXiv2017

Interacting Conceptual Spaces I : Grammatical Composition of Concepts

The categorical compositional approach to meaning has been successfully applied in natural language processing, outperforming other models in mainstream empirical language processing tasks. We show how this approach can be generalized to conceptual space models of cognition. In order to do this, first we introduce the category of convex relations as a new setting for categorical compositional semantics, emphasizing the convex structure important to conceptual space applications. We then show how to construct conceptual spaces for various types such as nouns, adjectives and verbs. Finally we show by means of examples how concepts can be systematically combined to establish the meanings of composite phrases from the meanings of their constituent parts. This provides the mathematical underpinnings of a new compositional approach to cognition.

cs.LO↗