Mathematical logic and computer science have driven the design of a growing number of logics and related formalisms such as set theories and type theories. In response to this population explosion, logical frameworks have been developed as formal meta-languages in which to represent, structure, relate and reason about logics.

Research on logical frameworks has diverged into separate communities, often with conflicting backgrounds and philosophies. In particular, two of the most important logical frameworks are the framework of institutions, from the area of model theory based on category theory, and the Edinburgh Logical Framework LF, from the area of proof theory based on dependent type theory. Even though their ultimate motivations overlap – for example in applications to software verification – they have fundamentally different perspectives on logic.

In the current paper, we design a logical framework that integrates the frameworks of institutions and LF in a way that combines their complementary advantages while retaining the elegance of each of them. In particular, our framework takes a balanced approach between model theory and proof theory, and permits the representation of logics in a way that comprises all major ingredients of a logic: syntax, models, satisfaction, judgments and proofs. This provides a theoretical basis for the systematic study of logics in a comprehensive logical framework. Our framework has been applied to obtain a large library of structured and machine-verified encodings of logics and logic translations.

We use constructions in monoid and group theory to exhibit an adjunction between the category of partially ordered monoids and lazy monoid homomorphisms and the category of partially ordered groups and group homomorphisms such that the unit of the adjunction is injective. We also prove a similar result for sets acted on by monoids and groups.

We introduce the new notion of a lazy homomorphism for a function f between partially ordered monoids such that f(m ○ m′) ≤ f(m) ○ f(m′).

Every monoid can be endowed with the discrete partial ordering (m ≤ m′ if and only if m=m′), so our constructions provide a way of embedding monoids into groups. A simple counterexample (the two-element monoid with a non-trivial idempotent) and some calculations show that one can never hope for such an embedding to be a monoid homomorphism, so the price paid for injecting a monoid into a group is that we must weaken the notion of a homomorphism to this new notion of a lazy homomorphism.

The computational significance of this is that a monoid is an abstract model of computation – or at least of ‘operations’ – and, similarly, a group models reversible computations/operations. With this reading, the adjunction with its injective unit gives a systematic high-level way of faithfully translating an irreversible system into a ‘lazy’ reversible one.

Informally, but perhaps informatively, we can describe this work as follows: we give an abstract analysis of how we can sensibly add ‘undo’ (in the sense of ‘control-Z’).

Non-interference guarantees the absence of illicit information flow throughout program execution. It can be enforced by appropriate information flow type systems. Much of the previous work on type systems for non-interference has focused on calculi or high-level programming languages, and existing type systems for low-level languages typically omit objects, exceptions and method calls. We define an information flow type system for a sequential JVM-like language that includes all these programming features, and we prove, in the Coq proof assistant, that it guarantees non-interference. An additional benefit of the formalisation is that we have extracted from our proof a certified lightweight bytecode verifier for information flow. Our work provides, to the best of our knowledge, the first sound and certified information flow type system for such an expressive fragment of the JVM.

The relative realisability toposes introduced by Awodey, Birkedal and Scott in Awodey et al. (2002) satisfy a universal property involving regular functors to other categories. We use this universal property to define what relative realisability categories are when they are based on categories other than the topos of sets. This paper explains the property and gives a construction for relative realisability categories that works for arbitrary base Heyting categories. The universal property also provides some new geometric morphisms to relative realisability toposes.