Nondeterminism introduced by race conditions and message reorderings makes parallel and distributed programming hard. Nevertheless, promising approaches such as LVars and CRDTs address this problem by introducing a partial order structure on shared state that describes how the state evolves over time. Monotone programs that respect the order are deterministic. Datalog-inspired languages incorporate this idea of monotonicity in a first-class way but they are not general-purpose. We would like parallel and distributed languages to be as natural to use as any functional language, without sacrificing expressivity, and with a formal basis of study as appealing as the lambda calculus. This paper presents λ ∨ , a core language for deterministic parallelism that embodies the ideas above. In λ ∨ , values may increase over time according to a streaming order and all computations are monotone with respect to that order. The streaming order coincides with the approximation order found in Scott semantics and so unifies the foundations of functional programming with the foundations of deterministic distributed computation. The resulting lambda calculus has a computationally adequate model rooted in domain theory. It integrates the compositionality and power of abstraction characteristic of functional programming with the declarative nature of Datalog.

The typed merge operator offers the promise of a compositional style of statically-typed programming in which solutions to the expression problem arise naturally. This approach, dubbed compositional programming , has recently been demonstrated by Zhang et al. Unfortunately, the merge operator is an unwieldy beast. Merging values from overlapping types may be ambiguous, so disjointness relations have been introduced to rule out undesired nondeterminism and obtain a well-behaved semantics. Past type systems using a disjoint merge operator rely on intersection types, but extending such systems to include union types or overloaded functions is problematic: naively adding either reintroduces ambiguity. In a nutshell: the elimination forms of unions and overloaded functions require values to be distinguishable by case analysis, but the merge operator can create exotic values that violate that requirement. This paper presents F ⋈ , a core language that demonstrates how unions, intersections, and overloading can all coexist with a tame merge operator. The key is an underlying design principle that states that any two inhabited types can support either the deterministic merging of their values, or the ability to distinguish their values, but never both. To realize this invariant, we decompose previously studied notions of disjointness into two new, dual relations that permit the operation that best suits each pair of types. This duality respects the polarization of the type structure, yielding an expressive language that we prove to be both type safe and deterministic.

Focusing is a technique from proof theory that exploits type information to prune inessential nondeterminism from proof search procedures. Viewed through the lens of the Curry-Howard correspondence, a focused typing derivation yields terms in normal form. This paper explores how to exploit focusing for reasoning about contextual equivalences and full abstraction. We present a focused polymorphic call-by-push-value calculus and prove a computational completeness result: for every well-typed term, there exists a focused term that is βη-equivalent to it. This completeness result yields a powerful way to refine the context lemmas for establishing contextual equivalences, cutting down the set that must be considered to just focused contexts. The paper demonstrates the application of focusing to establish program equivalences, including free theorems. It also uses focusing to prove full abstraction of a translation of the pure, total call-by-push-value language into a language with divergence and simple effect types, yielding a novel solution to a simple-to-state, but hitherto difficult to solve problem.

Instead of a monolithic programming language trying to cover all features of interest, some programming systems are designed by combining together simpler languages that cooperate to cover the same feature space. This can improve usability by making each part simpler than the whole, but there is a risk of abstraction leaks from one language to another that would break expectations of the users familiar with only one or some of the involved languages. We propose a formal specification for what it means for a given language in a multi-language system to be usable without leaks: it should embed into the multi-language in a fully abstract way, that is, its contextual equivalence should be unchanged in the larger system. To demonstrate our proposed design principle and formal specification criterion, we design a multi-language programming system that combines an ML-like statically typed functional language and another language with linear types and linear state. Our goal is to cover a good part of the expressiveness of languages that mix functional programming and linear state (ownership), at only a fraction of the complexity. We prove that the embedding of ML into the multi-language system is fully abstract: functional programmers should not fear abstraction leaks. We show examples of combined programs demonstrating in-place memory updates and safe resource handling, and an implementation extending OCaml with our linear language. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Dependently typed languages such as Coq are used to specify and prove functional correctness of source programs, but what we ultimately need are guarantees about correctness of compiled code. By preserving dependent types through each compiler pass, we could preserve source-level specifications and correctness proofs into the generated target-language programs. Unfortunately, type-preserving compilation of dependent types is hard. In 2002, Barthe and Uustalu showed that type-preserving CPS is not possible for languages such as Coq. Specifically, they showed that for strong dependent pairs (Σ types), the standard typed call-by-name CPS is not type preserving . They further proved that for dependent case analysis on sums, a class of typed CPS translations—including the standard translation—is not possible . In 2016, Morrisett noticed a similar problem with the standard call-by-value CPS translation for dependent functions (Π types). In essence, the problem is that the standard typed CPS translation by double-negation, in which computations are assigned types of the form ( A → ⊥) → ⊥, disrupts the term/type equivalence that is used during type checking in a dependently typed language. In this paper, we prove that type-preserving CPS translation for dependently typed languages is not not possible. We develop both call-by-name and call-by-value CPS translations from the Calculus of Constructions with both Π and Σ types (CC) to a dependently typed target language, and prove type preservation and compiler correctness of each translation. Our target language is CC extended with an additional equivalence rule and an additional typing rule, which we prove consistent by giving a model in the extensional Calculus of Constructions. Our key observation is that we can use a CPS translation that employs answer-type polymorphism , where CPS-translated computations have type ∀ α. ( A → α) → α. This type justifies, by a free theorem , the new equality rule in our target language and allows us to recover the term/type equivalences that CPS translation disrupts. Finally, we conjecture that our translation extends to dependent case analysis on sums, despite the impossibility result, and provide a proof sketch.