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Consistent subtyping is employed in some gradual type systems to validate type conversions. The original definition by Siek and Taha serves as a guideline for designing gradual type systems with subtyping. Polymorphic types à la System F also induce a subtyping relation that relates polymorphic types to their instantiations. However Siek and Taha’s definition is not adequate for polymorphic subtyping. The first goal of this paper is to propose a generalization of consistent subtyping that is adequate for polymorphic subtyping, and subsumes the original definition by Siek and Taha. The new definition of consistent subtyping provides novel insights with respect to previous polymorphic gradual type systems, which did not employ consistent subtyping. The second goal of this paper is to present a gradually typed calculus for implicit (higher-rank) polymorphism that uses our new notion of consistent subtyping. We develop both declarative and (bidirectional) algorithmic versions for the type system. We prove that the new calculus satisfies all static aspects of the refined criteria for gradual typing, which are mechanically formalized using the Coq proof assistant.

We propose HOBiT, a higher-order bidirectional programming language, in which users can write bidirectional programs in the familiar style of conventional functional programming, while enjoying the full expressiveness of lenses. A bidirectional transformation, or a lens, is a pair of mappings between source and view data objects, one in each direction. When the view is modified, the source is updated accordingly with respect to some laws—a pattern that is found in databases, model-driven development, compiler construction, and so on. The most common way of programming lenses is with lens combinators, which are lens-to-lens functions that compose simpler lenses to form more complex ones. Lens combinators preserve the bidirectionality of lenses and are expressive; but they compel programmers to a specialised point-free style—i.e., no naming of intermediate computation results—limiting the scalability of bidirectional programming. To address this issue, we propose a new bidirectional programming language HOBiT, in which lenses are represented as standard functions, and combinators are mapped to language constructs with binders. This design transforms bidirectional programming, enabling programmers to write bidirectional programs in a flexible functional style and at the same time access the full expressiveness of lenses. We formally define the syntax, type system, and the semantics of the language, and then show that programs in HOBiT satisfy bidirectionality. Additionally, we demonstrate HOBiT ’s programmability with examples.

We characterize the relation between generalized algebraic datatypes (GADTs) with pattern matching on their constructors one hand, and generalized algebraic co-datatypes (GAcoDTs) with copattern matching on their destructors on the other hand: GADTs can be converted mechanically to GAcoDTs by refunctionalization, GAcoDTs can be converted mechanically to GADTs by defunctionalization, and both defunctionalization and refunctionalization correspond to a transposition of the matrix in which the equations for each constructor/destructor pair of the (co-)datatype are organized. We have defined a calculus, , which unifies GADTs and GAcoDTs in such a way that GADTs and GAcoDTs are merely different ways to partition the program.

We have formalized the type system and operational semantics of in the Coq proof assistant and have mechanically verified the following results: (1) The type system of is sound, (2) defunctionalization and refunctionalization can translate GADTs to GAcoDTs and back, (3) both transformations are type- and semantics-preserving and are inverses of each other, (4) (co-)datatypes can be represented by matrices in such a way the aforementioned transformations correspond to matrix transposition, (5) GADTs are extensible in an exactly dual way to GAcoDTs; we thereby clarify folklore knowledge about the “expression problem”.

We believe that the identification of this relationship can guide future language design of “dual features” for data and codata.

Synchronous Programming () is a universal computational principle that provides deterministic concurrency. The same input sequence with the same timing always results in the same externally observable output sequence, even if the internal behaviour generates uncertainty in the scheduling of concurrent memory accesses. Consequently, languages have always been strongly founded on mathematical semantics that support formal program analysis. So far, however, communication has been constrained to a set of primitive clock-synchronised shared memory () data types, such as data-flow registers, streams and signals with restricted read and write accesses that limit modularity and behavioural abstractions.

This paper proposes an extension to the theory which retains the advantages of deterministic concurrency, but allows communication to occur at higher levels of abstraction than currently supported by data types. Our approach is as follows. To avoid data races, each type publishes a for specifying the admissibility and precedence of its access methods. Each instance of the type has to be policy-coherent, meaning it must behave deterministically under its own policy—a natural requirement if the goal is to build deterministic systems that use these types. In a policy-constructive system, all access methods can be scheduled in a policy-conformant way for all the types without deadlocking. In this paper, we show that a policy-constructive program exhibits deterministic concurrency in the sense that all policy-conformant interleavings produce the same input-output behaviour. Policies are conservative and support the types existing in current languages. Technically, we introduce a kernel language that uses arbitrary policy-driven types. A big-step fixed-point semantics for this language is developed for which we prove determinism and termination of constructive programs.

We present , a sound and relatively complete assertion-based program logic, and demonstrate its expressivity by verifying several classical examples of randomized algorithms using an implementation in the proof assistant. features new proof rules for loops and adversarial code, and supports richer assertions than existing program logics. We also show that allows convenient reasoning about complex probabilistic concepts by developing a new program logic for probabilistic independence and distribution law, and then smoothly embedding it into .

Probabilistic programming is an emerging technique for modeling processes involving uncertainty. Thus, it is important to ensure these programs are assigned precise formal semantics that also cleanly handle typical exceptions such as non-termination or division by zero. However, existing semantics of probabilistic programs do not fully accommodate different exceptions and their interaction, often ignoring some or conflating multiple ones into a single exception state, making it impossible to distinguish exceptions or to study their interaction.

In this paper, we provide an expressive probabilistic programming language together with a fine-grained measure-theoretic denotational semantics that handles and distinguishes non-termination, observation failures and error states. We then investigate the properties of this semantics, focusing on the interaction of different kinds of exceptions. Our work helps to better understand the intricacies of probabilistic programs and ensures their behavior matches the intended semantics.

Bayesian networks (BNs) are probabilistic graphical models for describing complex joint probability distributions. The main problem for BNs is inference: Determine the probability of an event given observed evidence. Since exact inference is often infeasible for large BNs, popular approximate inference methods rely on sampling.

We study the problem of determining the expected time to obtain a single valid sample from a BN. To this end, we translate the BN together with observations into a probabilistic program. We provide proof rules that yield the exact expected runtime of this program in a fully automated fashion. We implemented our approach and successfully analyzed various real–world BNs taken from the Bayesian network repository.

We extend the simply-typed guarded -calculus with discrete probabilities and endow it with a program logic for reasoning about relational properties of guarded probabilistic computations. This provides a framework for programming and reasoning about infinite stochastic processes like Markov chains. We demonstrate the logic sound by interpreting its judgements in the topos of trees and by using probabilistic couplings for the semantics of relational assertions over distributions on discrete types.

The program logic is designed to support syntax-directed proofs in the style of relational refinement types, but retains the expressiveness of higher-order logic extended with discrete distributions, and the ability to reason relationally about expressions that have different types or syntactic structure. In addition, our proof system leverages a well-known theorem from the coupling literature to justify better proof rules for relational reasoning about probabilistic expressions. We illustrate these benefits with a broad range of examples that were beyond the scope of previous systems, including shift couplings and lump couplings between random walks.

We define the , a syntactic translation of the Calculus of Inductive Constructions (CIC) into itself, that covers full dependent elimination. The new resulting type theory features call-by-name exceptions with decidable type-checking and canonicity, but at the price of inconsistency. Then, noticing parametricity amounts to Kreisel’s realizability in this setting, we provide an additional layer on top of the exceptional translation in order to tame exceptions and ensure that all exceptions used locally are caught, leading to the which fully preserves consistency. This way, we can consistently extend the logical expressivity of CIC with independence of premises, Markov’s rule, and the negation of function extensionality while retaining -expansion. As a byproduct, we also show that Markov’s principle is not provable in CIC. Both translations have been implemented in a plugin, which we use to formalize the examples.

Bi-directional type checking has proved to be an extremely useful and versatile tool for type checking and type inference. The conventional presentation of bi-directional type checking consists of two modes: mode and mode. In traditional bi-directional type-checking, type annotations are used to guide (via the checked mode) the type inference/checking procedure to determine the type of an expression, and .

This paper presents a variant of bi-directional type checking where the . This variant retains the inference mode, but adds a so-called mode. Such design can remove annotations that basic bi-directional type checking cannot, and is useful when type information from arguments is required to type-check the functions being applied. We present two applications and develop the meta-theory (mostly verified in Coq) of the application mode.