Dependent Object Types

A scalable programming language is one in which the same concepts can describe small as well as large parts. Towards this goal, Scala unifies concepts from object and module systems. In particular, objects can contain type members, which can be selected as types, called path-dependent types. Focusing on path-dependent types, we develop a type-theoretic foundation for Scala: the calculus of Dependent Object Types (DOT). We derive DOT from System F, we add a lower bound to each type variable, in addition to its usual upper bound, (2) in System D, we turn each type variable into a regular term variable containing a type, (3) for a full subtyping lattice, we add intersection and union types, (4) for objects, we consolidate all values into records, (5) for objects that close over a self, we introduce a recursive type, binding a self term variable, (6) for recursive types, we first extend the theory in typing and then also in subtyping. Through this bottom-up exploration, we discover a sound, uniform yet powerful design for DOT. We devise strategies and techniques for proving soundness that scale through this iterative step-by-step process: (1) "pushback" of subtyping transitivity or subsumption, to concisely capture inversion of subtyping or typing, (2) distinction between concrete vs. abstract context variables, to resolve tension between preservation of types vs. preservation of type abstractions, (3) and, specifically for big-step semantics, a type that closes over an environment, to relate context-dependent types across closures. While ultimately, we have developed sound models of DOT in both big-step and small-step operational semantics, historically, the shift to big-step semantics has been helpful in focusing the requirements. In particular, by developing a novel big-step soundness proof for System F<:, calculi like System D<: emerge as straightforward generalizations, almost like removing artificial restrictions. Interesting in their own right, our type soundness techniques for definitional interpreters extend to mutable references without use of co-induction. The DOT calculus finally grounds languages like Scala in firm theory. The DOT calculus helps in finding bugs in Scala, and in understanding feature interaction better as well as requirements. The DOT calculus serves as a good basis for future work which studies extensions or encodings on top of the core, bridging the gap from DOT to Dotty / Scala

A Dependently Typed Language with Nontermination

We propose a full-spectrum dependently typed programming language, Zombie, which supports general recursion natively. The Zombie implementation is an elaborating typechecker. We prove type saftey for a large subset of the Zombie core language, including features such as computational irrelevance, CBV-reduction, and propositional equality with a heterogeneous, completely erased elimination form. Zombie does not automatically beta-reduce expressions, but instead uses congruence closure for proof and type inference. We give a specification of a subset of the surface language via a bidirectional type system, which works up-to-congruence, and an algorithm for elaborating expressions in this language to an explicitly typed core language. We prove that our elaboration algorithm is complete with respect to the source type system. Zombie also features an optional termination-checker, allowing nonterminating programs returning proofs as well as external proofs about programs

Programming Languages and Systems

This open access book constitutes the proceedings of the 29th European Symposium on Programming, ESOP 2020, which was planned to take place in Dublin, Ireland, in April 2020, as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The actual ETAPS 2020 meeting was postponed due to the Corona pandemic. The papers deal with fundamental issues in the specification, design, analysis, and implementation of programming languages and systems

Algebraic Verification of Probabilistic and Concurrent Systems

This thesis provides an algebraic modelling and verification of probabilistic concurrent systems in the style of Kleene algebra. Without concurrency, it is shown that the equational theory of continuous probabilistic Kleene algebra is complete with respect to an automata model under standard simulation equivalence. This yields a minimisation-based decision procedure for the algebra. Without probability, an event structure model of Hoare et al.'s concurrent Kleene algebra is constructed. These two algebras are then ``merged" to provide probabilistic concurrent Kleene algebra which is used to discover and prove development rules for probabilistic concurrent systems (e.g. rely/guarantee calculus). Soundness of the new algebra is ensured by models based on probabilistic automata (interleaving) and probabilistic bundle event structures (true concurrency) quotiented with the respective simulation equivalences. Lastly, event structures with implicit probabilities are constructed to provide a state based model for the soundness of the probabilistic rely/guarantee rules