Calculi with disjoint intersection types support a symmetric merge operator with subtyping. The merge operator generalizes record concatenation to any type, enabling expressive forms of object composition, and simple solutions to hard modularity problems. Unfortunately, recent calculi with disjoint intersection types and the merge operator lack a (direct) operational semantics with expected properties such as determinism and subject reduction, and only account for terminating programs. This paper proposes a type-directed operational semantics (TDOS) for calculi with intersection types and a merge operator. We study two variants of calculi in the literature. The first calculus, called λ i , is a variant of a calculus presented by Oliveira et al. (2016) and closely related to another calculus by Dunfield (2014). Although Dunfield proposes a direct small-step semantics for her calculus, her semantics lacks both determinism and subject reduction. Using our TDOS, we obtain a direct semantics for λ i that has both properties. The second calculus, called λ i + , employs the well-known subtyping relation of Barendregt, Coppo and Dezani-Ciancaglini (BCD). Therefore, λ i + extends the more basic subtyping relation of λ i , and also adds support for record types and nested composition (which enables recursive composition of merged components). To fully obtain determinism, both λ i and λ i + employ a disjointness restriction proposed in the original λ i calculus. As an added benefit the TDOS approach deals with recursion in a straightforward way, unlike previous calculi with disjoint intersection types where recursion is problematic. We relate the static and dynamic semantics of λ i to the original version of the calculus and the calculus by Dunfield. Furthermore, for λ i + , we show a novel formulation of BCD subtyping, which is algorithmic, has a very simple proof of transitivity and allows for the modular addition of distributivity rules (i.e. without affecting other rules of subtyping). All results have been fully formalized in the Coq theorem prover.

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Subtyping with intersection and union types is nowadays common in many programming languages. From the perspective of logic, the subtyping problem is essentially the problem of determining logical entailment : does a logical statement follow from another one? Unfortunately, algorithms for deciding subtyping and logical entailment with intersections, unions and various distributivity laws can be highly non-trivial. This functional pearl presents a novel algorithmic formulation for subtyping (and logical entailment) in the presence of various distributivity rules between intersections, unions and implications (i.e. function types). Unlike many existing algorithms which first normalize types and then apply a subtyping algorithm on the normalized types, our new subtyping algorithm works directly on source types. Our algorithm is based on two recent ideas: a generalization of subtyping based on the duality of language constructs called duotyping ; and splittable types , which characterize types that decompose into two simpler types. We show that our algorithm is sound, complete and decidable with respect to a declarative formulation of subtyping based on the minimal relevant logic B + . Moreover, it leads to a simple and compact implementation in under 50 lines of functional code.

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.

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