This paper presents a reformulation in topos logic of a safety result arising in an abstract presentation of blockchain consensus protocols. That is, in a high-level template for "correct-by-construction" consensus protocols, it is shown that a proposition and its negation cannot both be safe in protocol states that have executions to some common state. This is in fact true for any inconsistent propositions and the proof requires only intuitionistic reasoning. This opens the door for work on consensus protocols in the internal language of a topos. As a first pass on such a program, the main contribution of this paper is the formulation of estimate safety in abstract correct-by-construction protocols as a forcing statement in the internal logic of a given topos. This is illustrated first in the setting of copresheaf toposes. It is also seen there that safety can be viewed as a modal statement. For these interpretations, some extensions and adaptations of results in the literature on modal operators in toposes are presented. The final reformulation of estimate safety is a completely elementary version in the language of an arbitrary topos where it is seen that estimate safety is equivalent to a certain forcing statement.
Fabrizio Romano Genovese, Jelle Herold, Fosco Loregian, Daniele Palombi
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.
This paper introduces an approach to constructing ledger structures for cryptocurrency systems with basic category theory. Compositional theories of resource convertibility allow us to express the material history of virtual goods, and ownership is modelled by a free construction. Our notion of ownership admits an intuitive graphical representation through string diagrams for monoidal functors.