We present two seemingly different definitions of constructive ordinal exponentiation, where an ordinal is taken to be a transitive, extensional, and wellfounded order on a set. The first definition is abstract, uses suprema of ordinals, and is solely motivated by the expected equations. The second is more concrete, based on decreasing lists, and can be seen as a constructive version of a classical construction by Sierpiński based on functions with finite support. We show that our two approaches are equivalent (whenever it makes sense to ask the question), and use this equivalence to prove algebraic laws and decidability properties of the exponential. Our work takes place in the framework of homotopy type theory, and all results are formalized in the proof assistant Agda.

The ability to exchange data while maintaining sovereignty is fundamental to emerging decentralized data-driven ecosystems. Data sovereignty refers to the entity's capability to be self-determined concerning data usage. As such, a data usage control system (UCON) is critical for sovereignty. UCON, a generalization of attribute-based access control, enforces continuous authorization, allowing attribute mutability after access is granted. In theory, UCON comprises a policy language to express constraints and obligations of data usage, and a technology to evaluate and enforce them. In practice, realizing the above is challenging and poses trust concerns. Partly, this is due to the complexity of UCON (continuous authorization, obligations) and the advanced usage constraints (stemming from, e.g., regulations or business contracts) combined with the decentralized nature of data ecosystems that allow different actors (e.g., data provider, security engineers) to author policies, and operate UCON. To that end, we propose to aid actors with automated policy analysis and verification methods. We present a new policy analysis method based on the combination of symbolic execution for policy evaluation and SMT solving to compute concrete scenarios answering queries on the policies. Our approach supports symbolic queries, where attribute values may be concrete values, a range of values, or symbolic variables. We also propose a monitoring approach using RTLola tool to verify the correctness of UCON's behavior in terms of decisions, obligations, and user-specified properties. To monitor obligations, we define their essential parameters and show how to monitor their fulfillment based on the configuration. We also present eight templates that allow users to generate the most important properties for monitoring UCON.

In constructive set theory, an ordinal is a hereditarily transitive set. In homotopy type theory (HoTT), an ordinal is a type with a transitive, wellfounded, and extensional binary relation. We show that the two definitions are equivalent if we use (the HoTT refinement of) Aczel’s interpretation of constructive set theory into type theory. Following this, we generalize the notion of a type-theoretic ordinal to capture all sets in Aczel’s interpretation rather than only the ordinals. This leads to a natural class of ordered structures which contains the type-theoretic ordinals and realizes the higher inductive interpretation of set theory. All our results are formalized in Agda.

Modern system architectures require sophisticated access and usage control mechanisms. The need stems from demanding requirements for security, data sovereignty and privacy regulations, as well as the challenges presented by architectural approaches like zero trust networking. Usage control systems provide one approach to encapsulate and manage the complexities related to access and usage control. In order to trust a usage control system, it is essential to ensure that usage control policies express the intended properties and are enforced correctly. To achieve this, we need a precise specification of the intended behavior of a usage control system. For attribute-based access control, the XACML standard is a sufficient specification of the behavior of policies. Usage control models, such as UCON, extend access control with features for continuous authorization based on mutability of attribute values. This adds significant complexity to the problem of specifying the intended behavior. In this paper, we identify challenges with specifying a practical usage control system regarding continuous control, obligations, and concurrency aspects. We describe an approach to specifying the UCON+ model of Dimitrakos et al. and outline an implementation of the specification with Answer Set Programming.

We introduce a generic type system for Featherweight Java (FJ) that is parametrized with a monad-like structure, and prove a uniform soundness theorem. Its instances include some region type systems studied by Martin Hofmann et al. as well as a new one that performs more precise analysis of trace-based properties. Their soundness is guaranteed by the uniform theorem. We only need to verify some natural conditions. Instead of refining the FJ type system as in the previous work, our region type system is separate from the FJ type system, making it simpler and also easier to move to larger fragments of Java. Moreover, the uniform framework helps to avoid redundant work on the meta-theory when extending the system to cover other language features such as exception handling.

In classical set theory, there are many equivalent ways to introduce ordinals. In a constructive setting, however, the different notions split apart, with different advantages and disadvantages for each. We consider three different notions of ordinals in homotopy type theory, and show how they relate to each other: A notation system based on Cantor normal forms, a refined notion of Brouwer trees (inductively generated by zero, successor and countable limits), and wellfounded extensional orders. For Cantor normal forms, most properties are decidable, whereas for wellfounded extensional transitive orders, most are undecidable. Formulations for Brouwer trees are usually partially decidable. We demonstrate that all three notions have properties expected of ordinals: their order relations, although defined differently in each case, are all extensional and wellfounded, and the usual arithmetic operations can be defined in each case. We connect these notions by constructing structure preserving embeddings of Cantor normal forms into Brouwer trees, and of these in turn into wellfounded extensional orders. We have formalised most of our results in cubical Agda.

A common approach to improve software quality is to use programming guidelines to avoid common kinds of errors. In this paper, we consider the problem of enforcing guidelines for Featherweight Java (FJ). We formalize guidelines as sets of finite or infinite execution traces and develop a region-based type and effect system for FJ that can enforce such guidelines. We build on the work by Erbatur, Hofmann and Zălinescu, who presented a type system for verifying the finite event traces of terminating FJ programs. We refine this type system, separating region typing from FJ typing, and use ideas of Hofmann and Chen to extend it to capture also infinite traces produced by non-terminating programs. Our type and effect system can express properties of both finite and infinite traces and can compute information about the possible infinite traces of FJ programs. Specifically, the set of infinite traces of a method is constructed as the greatest fixed point of the operator which calculates the possible traces of method bodies. Our type inference algorithm is realized by working with the finitary abstraction of the system based on Büchi automata.

We present three ordinal notation systems representing ordinals below ε0 in type theory, using recent type-theoretical innovations such as mutual inductive-inductive definitions and higher inductive types. We show how ordinal arithmetic can be developed for these systems, and how they admit a transfinite induction principle. We prove that all three notation systems are equivalent, so that we can transport results between them using the univalence principle. All our constructions have been implemented in cubical Agda.

We introduce a syntactic translation of Gödel’s System 𝖳 parametrized by a weak notion of a monad, and prove a corresponding fundamental theorem of logical relation. Our translation structurally corresponds to Gentzen’s negative translation of classical logic. By instantiating the monad and the logical relation, we reveal the well-known properties and structures of 𝖳-definable functionals including majorizability, continuity and bar recursion. Our development has been formalized in the Agda proof assistant.

We give a new proof of the well-known fact that all functions $(\mathbb{N} \to \mathbb{N}) \to \mathbb{N}$ which are definable in G\"odel's System T are continuous via a syntactic approach. Differing from the usual syntactic method, we firstly perform a translation of System T into itself in which natural numbers are translated to functions $(\mathbb{N} \to \mathbb{N}) \to \mathbb{N}$. Then we inductively define a continuity predicate on the translated elements and show that the translation of any term in System T satisfies the continuity predicate. We obtain the desired result by relating terms and their translations via a parametrized logical relation. Our constructions and proofs have been formalized in the Agda proof assistant. Because Agda is also a programming language, we can execute our proof to compute moduli of continuity of T-definable functions.