arXiv subjects

Daniele Palombi

Publications and source records attributed to Daniele Palombi.

6 recordsLinked to original sources

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.

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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.

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arXiv2023

Classifying topoi in synthetic guarded domain theory

Several different topoi have played an important role in the development and applications of synthetic guarded domain theory (SGDT), a new kind of synthetic domain theory that abstracts the concept of guarded recursion frequently employed in the semantics of programming languages. In order to unify the accounts of guarded recursion and coinduction, several authors have enriched SGDT with multiple "clocks" parameterizing different time-streams, leading to more complex and difficult to understand topos models. Until now these topoi have been understood very concretely qua categories of presheaves, and the logico-geometrical question of what theories these topoi classify has remained open. We show that several important topos models of SGDT classify very simple geometric theories, and that the passage to various forms of multi-clock guarded recursion can be rephrased more compositionally in terms of the lower bagtopos construction of Vickers and variations thereon due to Johnstone. We contribute to the consolidation of SGDT by isolating the universal property of multi-clock guarded recursion as a modular construction that applies to any topos model of single-clock guarded recursion.

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arXiv2022

A Categorical Semantics for Bounded Petri Nets

We provide a categorical semantics for bounded Petri nets, both in the collective- and individual-token philosophy. In both cases, we describe the process of bounding a net internally, by just constructing new categories of executions of a net using comonads, and externally, using lax-monoidal-lax functors. Our external semantics is non-local, meaning that tokens are endowed with properties that say something about the global state of the net. We then prove, in both cases, that the internal and external constructions are equivalent, by using machinery built on top of the Grothendieck construction. The individual-token case is harder, as it requires a more explicit reliance on abstract methods.

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arXiv2021

A Categorical Semantics for Hierarchical Petri Nets

We show how a particular variety of hierarchical nets, where the firing of a transition in the parent net must correspond to an execution in some child net, can be modelled utilizing a functorial semantics from a free category -- representing the parent net -- to the category of sets and spans between them. This semantics can be internalized via Grothendieck construction, resulting in the category of executions of a Petri net representing the semantics of the overall hierarchical net. We conclude the paper by giving an engineering-oriented overview of how our model of hierarchical nets can be implemented in a transaction-based smart contract environment.

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arXiv2021

Nets with Mana: A Framework for Chemical Reaction Modelling

We use categorical methods to define a new flavor of Petri nets where transitions can only fire a limited number of times, specified by a quantity that we call mana. We do so with chemistry in mind, looking at ways of modelling the behavior of chemical reactions that depend on enzymes to work. We prove that such nets can be either obtained as a result of a comonadic construction, or by enriching them with extra information encoded into a functor. We then use a well-established categorical result to prove that the two constructions are equivalent, and generalize them to the case where the firing of some transitions can "regenerate" the mana of others. This allows us to represent the action of catalysts and also of biochemical processes where the byproducts of some chemical reaction are exactly the enzymes that another reaction needs to work.

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