arXiv subjects

Fabio Gadducci

Publications and source records attributed to Fabio Gadducci.

At least 19 recordsLinked to original sources

arXiv2026

Verification of Quantum Protocols Adopting Physically Admissible Schedulers

Reliable verification techniques for quantum communication protocols are of paramount importance, given their high implementation cost and critical contexts of application. Extensions of process calculi have been proposed, together with various notions of behavioural equivalence. However, their standard probabilistic models turn out to introduce some non-deterministic capabilities not aligned with the observational properties of physical quantum systems, leading to bisimilarity notions that distinguish physically equivalent processes. Nonetheless, non-deterministic features are fundamental to account for inputs, environments and adversarial behaviour. To address this issue, we propose lqCCS, a process calculus that integrates concurrency, non-determinism and quantum capabilities. We introduce a novel semantics in terms of distributions, where explicit physically admissible schedulers constrain probabilistic composition and forbid ill-defined non-deterministic moves, while preserving the expressivity needed to model real-world protocols. We investigate a scheduled version of saturated bisimilarity, pairing two processes if no observer can tell them apart, and we verify its adequacy by lifting a known result from quantum mechanics to lqCCS: equivalent processes acting on indistinguishable mixtures of quantum states are correctly recognized as bisimilar. Finally, we give an alternative semantics and a labelled bisimilarity based on a quantum generalization of probability distributions. This characterizes our behavioural equivalence as a congruence with respect to the parallel operator, enabling compositional reasoning without the need to explicitly check all possible contexts. We describe a rich class of lqCCS processes for which equivalence is decidable using standard techniques, and we analyse real-world quantum communication protocols.

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arXiv2025

Between Markov and restriction. Two more monads on categories for relations

The study of categories abstracting the structural properties of relations has been extensively developed over the years, resulting in a rich and diverse body of work. In a previous paper we offered a survey providing a modern presentation of these ``categories for relations'' as instances of gs-monoidal categories, showing how they arise as Kleisli categories of suitable symmetric monoidal monads. The end result was a taxonomy that organised numerous related concepts in the literature, including in particular Markov and restriction categories. This paper further enriches the taxonomy: it proposes two categories that are once more instances of gs-monoidal categories, yet more abstract than Markov and restriction categories. They are characterised by an axiomatic notion of mass and domain of an arrow, the latter one of the key ingredients of restriction categories, which generalises the domain of partial functions. The paper then introduces mass and domain preserving monads, proving that the associated Kleisli categories in fact preserve the corresponding equations and that these monads arise naturally for the categories of semiring-weighted relations.

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arXiv2025

EGGs are adhesive!

The use of rewriting-based visual formalisms is on the rise. In the formal methods community, this is due also to the introduction of adhesive categories, where most properties of classical approaches to graph transformation, such as those on parallelism and confluence, can be rephrased and proved in a general and uniform way.E-graphs (EGGs) are a formalism for program optimisation via an efficient implementation of equality saturation. In short, EGGs can be defined as (acyclic) term graphs with an additional notion of equivalence on nodes that is closed under the operators of the signature. Instead of replacing the components of a program, the optimisation step is performed by adding new components and linking them to the existing ones via an equivalence relation, until an optimal program is reached. This work describes EGGs via adhesive categories. Besides the benefits in itself of a formal presentation, which renders the properties of the data structure precise, the description of the addition of equivalent program components using standard graph transformation tools offers the advantages of the adhesive framework in modelling, for example, concurrent updates.

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arXiv2025

A Constraint Opinion Model

This paper introduces a generalised opinion model that extends the standard DeGroot model by representing agents' opinions and influences as soft constraints rather than single real values. This allows for modelling scenarios beyond the scope of the DeGroot model, such as agents sharing partial information and preferences, engaging in discussions on multiple topics simultaneously, and representing opinions with different degrees of uncertainty. By considering soft constraints as influences, the proposed model captures also situations where agents impose conditions on how others' opinions are integrated during belief revision. Finally, the flexibility offered by soft constraints allows us to introduce a novel polarisation measure that takes advantage of this generalised framework.

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arXiv2025

A taxonomy of categories for relations

The study of categories that abstract the structural properties of relations has been extensively developed over the years, resulting in a rich and diverse body of work. This paper strives to provide a modern presentation of these ``categories for relations'', including their enriched version, further showing how they arise as Kleisli categories of symmetric monoidal monads. The resulting taxonomy aims at bringing clarity and organisation to the many related concepts and frameworks occurring in the literature.

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arXiv2024

Quantum Bisimilarity is a Congruence under Physically Admissible Schedulers

The development of quantum algorithms and protocols calls for adequate modelling and verification techniques, which requires abstracting and focusing on the basic features of quantum concurrent systems, like CCS and CSP have done for their classical counterparts. So far, an equivalence relation is still missing that is a congruence for parallel composition and adheres to the limited discriminating power implied by quantum theory. In fact, defining an adequate bisimilarity for quantum-capable, concurrent systems proved a difficult task, because unconstrained non-determinism allows to spuriously discriminate indistinguishable quantum systems. We investigate this problem by enriching a linear quantum extension of CCS with simple physically admissible schedulers. We show that our approach suffices for deriving a well-behaved bisimilarity that satisfies the aforementioned desiderata.

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arXiv2024

Left-Linear Rewriting in Adhesive Categories

When can two sequential steps performed by a computing device be considered (causally) independent? This is a relevant question for concurrent and distributed systems, since independence means that they could be executed in any order, and potentially in parallel. Equivalences identifying rewriting sequences which differ only for independent steps are at the core of the theory of concurrency of many formalisms. We investigate the issue in the context of the double pushout approach to rewriting in the general setting of adhesive categories. While a consolidated theory exists for linear rules,which can consume, preserve and generate entities, this paper focuses on left-linear rules which may also "merge" parts of the state. This is an apparently minimal, yet technically hard enhancement,since a standard characterisation of independence that - in the linear case - allows one to derive a number of properties, essential in the development of a theory of concurrency, no longer holds. The paper performs an in-depth study of the notion of independence for left-linear rules: it introduces a novel characterisation of independence, identifies well-behaved classes of left-linear rewriting systems,and provides some fundamental results including a Church-Rosser property and the existence of canonical equivalence proofs for concurrent computations. These results properly extends the class of formalisms that can be modelled in the adhesive framework

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arXiv2023

Quantum Bisimilarity via Barbs and Contexts: Curbing the Power of Non-Deterministic Observers

Past years have seen the development of a few proposals for quantum extensions of process calculi. The rationale is clear: with the development of quantum communication protocols, there is a need to abstract and focus on the basic features of quantum concurrent systems, like CCS has done for its classical counterpart. So far, though, no accepted standard has emerged, neither for the syntax nor for the behavioural semantics. Indeed, the various proposals do not agree on what should be the observational properties of quantum values, and as a matter of fact, the soundness of such properties has never been validated against the prescriptions of quantum theory. To this aim, we introduce a new calculus, Linear Quantum CCS, and investigate the features of behavioural equivalences based on barbs and contexts. Our calculus can be thought of as an asynchronous, linear version of qCCS (based on value-passing CCS). Linearity ensures that each qubit is sent exactly once, precisely specifying which qubits of a process interact with the context. We exploit contexts to examine how bisimilarities relate to quantum theory. We show that the observational power of general contexts is incompatible with quantum theory: roughly, they can perform non-deterministic moves depending on quantum values without measuring (hence perturbing) them. Therefore, we refine the operational semantics in order to prevent contexts from performing unfeasible non-deterministic choices. This induces a coarser bisimilarity that better fits the quantum setting: (i) it lifts the indistinguishability of quantum states to the distributions of processes and, despite the additional constraints, (ii) it preserves the expressivity of non-deterministic choices based on classical information. To the best of our knowledge, our semantics is the first one that satisfies the two properties above.

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arXiv2023

From Gs-monoidal to Oplax Cartesian Categories: Constructions and Functorial Completeness

Originally introduced in the context of the algebraic approach to term graph rewriting, the notion of gs-monoidal category has surfaced a few times under different monikers in the last decades. They can be thought of as symmetric monoidal categories whose arrows are generalised relations, with enough structure to talk about domains and partial functions, but less structure than cartesian bicategories. The aim of this paper is threefold. The first goal is to extend the original definition of gs-monoidality by enriching it with a preorder on arrows, giving rise to what we call oplax cartesian categories. Second, we show that (preorder-enriched) gs-monoidal categories naturally arise both as Kleisli categories and as span categories, and the relation between the resulting formalisms is explored. Finally, we present two theorems concerning Yoneda embeddings on the one hand and functorial completeness on the other, the latter inducing a completeness result also for lax functors from oplax cartesian categories to $\mathbf{Rel}$.

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arXiv2023

Weakly Markov categories and weakly affine monads

Introduced in the 1990s in the context of the algebraic approach to graph rewriting, gs-monoidal categories are symmetric monoidal categories where each object is equipped with the structure of a commutative comonoid. They arise for example as Kleisli categories of commutative monads on cartesian categories, and as such they provide a general framework for effectful computation. Recently proposed in the context of categorical probability, Markov categories are gs-monoidal categories where the monoidal unit is also terminal, and they arise for example as Kleisli categories of commutative affine monads, where affine means that the monad preserves the monoidal unit. The aim of this paper is to study a new condition on the gs-monoidal structure, resulting in the concept of weakly Markov categories, which is intermediate between gs-monoidal categories and Markov ones. In a weakly Markov category, the morphisms to the monoidal unit are not necessarily unique, but form a group. As we show, these categories exhibit a rich theory of conditional independence for morphisms, generalising the known theory for Markov categories. We also introduce the corresponding notion for commutative monads, which we call weakly affine, and for which we give two equivalent characterisations. The paper argues that these monads are relevant to the study of categorical probability. A case at hand is the monad of finite non-zero measures, which is weakly affine but not affine. Such structures allow to investigate probability without normalisation within an elegant categorical framework.

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arXiv2023

Specification and verification of a linear-time logic for graph transformation

We present a first-order linear-time temporal logic for reasoning about the evolution of directed graphs. Its semantics is based on the counterpart paradigm, thus allowing our logic to represent the creation, duplication, merging, and deletion of elements of a graph as well as how its topology changes over time. We then introduce a positive normal forms presentation, thus simplifying the actual process of verification. We provide the syntax and semantics of our logics with a computer-assisted formalisation using the proof assistant Agda, and we round up the paper by highlighting the crucial aspects of our formalisation and the practical use of quantified temporal logics in a constructive proof assistant.

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arXiv2022

String Diagram Rewrite Theory II: Rewriting with Symmetric Monoidal Structure

Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thoughts as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras.

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arXiv2022

A new criterion for $\mathcal{M}, \mathcal{N}$-adhesivity, with an application to hierarchical graphs

Adhesive categories provide an abstract framework for the algebraic approach to rewriting theory, where many general results can be recast and uniformly proved. However, checking that a model satisfies the adhesivity properties is sometimes far from immediate. In this paper we present a new criterion giving a sufficient condition for $\mathcal{M}, \mathcal{N}$-adhesivity, a generalisation of the original notion of adhesivity. We apply it to several existing categories, and in particular to hierarchical graphs, a formalism that is notoriously difficult to fit in the mould of algebraic approaches to rewriting and for which various alternative definitions float around.

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arXiv2022

String Diagram Rewrite Theory III: Confluence with and without Frobenius

In this paper we address the problem of proving confluence for string diagram rewriting, which was previously shown to be characterised combinatorically as double-pushout rewriting with interfaces (DPOI) on (labelled) hypergraphs. For standard DPO rewriting without interfaces, confluence for terminating rewrite systems is, in general, undecidable. Nevertheless, we show here that confluence for DPOI, and hence string diagram rewriting, is decidable. We apply this result to give effective procedures for deciding local confluence of symmetric monoidal theories with and without Frobenius structure by critical pair analysis. For the latter, we introduce the new notion of path joinability for critical pairs, which enables finitely many joins of a critical pair to be lifted to an arbitrary context in spite of the strong non-local constraints placed on rewriting in a generic symmetric monoidal theory.

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arXiv2022

String Diagram Rewrite Theory I: Rewriting with Frobenius Structure

String diagrams are a powerful and intuitive graphical syntax, originated in the study of symmetric monoidal categories. In the last few years, they have found application in the modelling of various computational structures, in fields as diverse as Computer Science, Physics, Control Theory, Linguistics, and Biology. In many such proposals, the transformations of the described systems are modelled as rewrite rules of diagrams. These developments demand a mathematical foundation for string diagram rewriting: whereas rewrite theory for terms is well-understood, the two-dimensional nature of string diagrams poses additional challenges. This work systematises and expands a series of recent conference papers laying down such foundation. As first step, we focus on the case of rewrite systems for string diagrammatic theories which feature a Frobenius algebra. This situation ubiquitously appear in various approaches: for instance, in the algebraic semantics of linear dynamical systems, Frobenius structures model the wiring of circuits; in categorical quantum mechanics, they model interacting quantum observables. Our work introduces a combinatorial interpretation of string diagram rewriting modulo Frobenius structures, in terms of double-pushout hypergraph rewriting. Furthermore, we prove this interpretation to be sound and complete. In the last part, we also see that the approach can be generalised to model rewriting modulo multiple Frobenius structures. As a proof of concept, we show how to derive from these results a termination strategy for Interacting Bialgebras, an important rewrite theory in the study of quantum circuits and signal flow graphs.

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arXiv2022

A Presheaf Semantics for Quantified Temporal Logics

Temporal logics stands for a widely adopted family of formalisms for the verification of computational devices, enriching propositional logics by operators predicating on the step-wise behaviour of a system. Its quantified extensions allow to reason on the properties of the individual components of the system at hand. The expressiveness of the resulting logics poses problems in correctly identifying a semantics that exploit its features without resorting to the imposition of restrictions on the acceptable behaviours. In this paper we address this issue by means of counterpart models and relational presheaves.

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arXiv2021

Residuation for Soft Constraints: Lexicographic Orders and Approximation Techniques

Residuation theory concerns the study of partially ordered algebraic structures, most often monoids, equipped with a weak inverse for the monoidal operator. One of its area of application has been constraint programming, whose key requirement is the presence of an aggregator operator for combining preferences. Given a residuated monoid of preferences, the paper first shows how to build a new residuated monoid of (possibly infinite) tuples, which is based on the lexicographic order. Second, it introduces a variant of an approximation technique (known as Mini-bucket) that exploits the presence of the weak inverse.

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arXiv2020

Domains and Event Structures for Fusions

Stable event structures, and their duality with prime algebraic domains arising as partial orders of configurations, are a landmark of concurrency theory, providing a clear characterisation of causality in computations. They have been used for defining a concurrent semantics of several formalisms, from Petri nets to (linear) graph rewriting systems, which in turn lay at the basis of many visual modelling frameworks. Stability however is restrictive when dealing with formalisms with "fusion", i.e., where a computational step can not only consume and produce but also merge parts of the state. This happens, e.g., for graph rewriting systems with non-linear rules, which are needed to cover some relevant applications (such as the graphical encoding of calculi with name passing). Guided by the need of capturing the semantics of formalisms with fusion we leave aside stability and we characterise, as a natural generalisation of prime algebraic domains, a class of domains, referred to as weak prime domains. We then identify a corresponding class of event structures, that we call connected event structures, via a duality result formalised as an equivalence of categories. We show that connected event structures are exactly the class of event structures that arise as the semantics of non-linear graph rewriting systems. Interestingly, the category of general unstable event structures coreflects into our category of weak prime domains, so that our result provides a characterisation of the partial orders of configurations of such event structures.

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