Jesse Hughes

Modal operators and the formal dual of Birkhoff's completeness theorem

We present the dual to Birkhoffs variety theorem in terms of predicates over the carrier of a cofree coalgebra (that is, in terms of ‘coequations’). We then discuss the dual to Birkhoffs completeness theorem, showing how closure under deductive rules dualises to yield two modal operators acting on coequations. We discuss the properties of these operators and show that they commute. We prove as our main result the invariance theorem, which is the formal dual of Birkhoffs completeness theorem.

Simulations in Coalgebra

Abstract A new approach to simulations is proposed within the theory of coalgebras by taking a notion of order on a functor as primitive. Such an order forms a basic building block for a “lax relation lifting”, or “relator” as used by other authors. Simulations appear as coalgebras of this lifted functor, and similarity as greatest simulation. Two-way similarity is then similarity in both directions. In general, it is different from bisimilarity (in the usual coalgebraic sense), but a sufficient condition is formulated (and illustrated) to ensure that bisimilarity and two-way similarity coincide. Also, a distributive law is identified which ensures that similarity on a final coalgebra forms a dcpo structure.

Factorization systems and fibrations: Toward a fibred Birkhoff variety theorem

Abstract It is well-known that a factorization system on a category (with sufficient pullbacks) gives rise to a fibration. This paper characterizes the fibrations that arise in such a way, by making precise the logical structure that is given by factorization systems. The underlying motivation is to obtain general Birkhoff results in a fibred setting.

Distributivity of categories of coalgebras

For any Set-endofunctor F, the category SetF of F-coalgebras has preimages, i.e. pullbacks along an injective map. If F preserves preimages, then SetF is distributive, and the converse holds, whenever SetF has finite products.

The Coinductive Approach to Verifying Cryptographic Protocols

We look at a new way of specifying and verifying cryptographic protocols using the Coalgebraic Class Specification Language. Protocols are specified into CCSL (with temporal operators for ”free”) and translated by the CCSL compiler into theories for the theorem prover PVS. Within PVS, the desired security conditions can then be (dis)proved.

Some Co-Birkhoff Type Theorems

Abstract We consider the dual of Theorem 1 from [33], relating closure conditions on subcategories with projectivity classes for collections of discrete cocones. We extend these results by adding a new operator for closure under domains of epis and show that this corresponds to taking projectivity classes for cocones with vertex a subobject of the terminal object. We show that these theorems apply to categories of coalgebras under reasonable assumptions on the base category and endofunctor. Lastly, we discuss cofree for V coalgebras in this setting and give examples of so-called Horn covarieties of coalgebras.

Concise Graphs and Functional Bisimulations

We investigate the conditions under which least bisimulations exist with respect to set inclusion. In particular, we describe a natural way to remove redundant pairs from a given bisimulation. We then introduce the conciseness property on process graphs, which characterizes the existence of least bisimulations under the aforementioned method. Subsequently, we consider the category of process graphs and functional bisimulations. This category has all coequalizers. Binary products and coproducts can be constructed with some further assumptions. Moreover, the full subcategory of concise graphs is a reectiv e subcategory.