International audience

We study row polymorphism for records types in systems with set-theoretic types, specifically, union, intersection, and negation types. We consider record types that embed row variables and define a subtyping relation by interpreting record types into sets of record values, and row variables into sets of rows, that is, “chunks” of record values where some record keys are left out: subtyping is then containment of the interpretations. We define a λ -calculus equipped with operations for field extension, selection, and deletion, its operational semantics, and a type system that we prove to be sound. We provide algorithms for deciding the typing and subtyping relations, and to decide whether two types can be instantiated to make one subtype of the other. This research is motivated by the current trend of defining static type systems for dynamic languages and, in our case, by an ongoing effort of endowing the Elixir programming language with a gradual type system.

We define and study a term calculus implementing higher-order node replication. It is used to specify two different (weak) evaluation strategies: call-by-name and fully lazy call-by-need, that are shown to be observationally equivalent by using type theoretical technical tools.

We introduce a call-by-name lambda-calculus $\lambda Jn$ with generalized applications which is equipped with distant reduction. This allows to unblock $\beta$-redexes without resorting to the standard permutative conversions of generalized applications used in the original $\Lambda J$-calculus with generalized applications of Joachimski and Matthes. We show strong normalization of simply-typed terms, and we then fully characterize strong normalization by means of a quantitative (i.e. non-idempotent intersection) typing system. This characterization uses a non-trivial inductive definition of strong normalization --related to others in the literature--, which is based on a weak-head normalizing strategy. We also show that our calculus $\lambda Jn$ relates to explicit substitution calculi by means of a faithful translation, in the sense that it preserves strong normalization. Moreover, our calculus $\lambda Jn$ and the original $\Lambda J$-calculus determine equivalent notions of strong normalization. As a consequence, $\lambda J$ inherits a faithful translation into explicit substitutions, and its strong normalization can also be characterized by the quantitative typing system designed for $\lambda Jn$, despite the fact that quantitative subject reduction fails for permutative conversions.

Abstract We introduce a call-by-name lambda-calculus $$\lambda J$$ λJ with generalized applications which integrates a notion of distant reduction that allows to unblock $$\beta $$ β -redexes without resorting to the permutative conversions of generalized applications. We show strong normalization of simply typed terms, and we then fully characterize strong normalization by means of a quantitative typing system. This characterization uses a non-trivial inductive definition of strong normalization –that we relate to others in the literature–, which is based on a weak-head normalizing strategy. Our calculus relates to explicit substitution calculi by means of a translation between the two formalisms which is faithful, in the sense that it preserves strong normalization. We show that our calculus $$\lambda J$$ λJ and the well-know calculus $$\varLambda J$$ ΛJ determine equivalent notions of strong normalization. As a consequence, $$\varLambda J$$ ΛJ inherits a faithful translation into explicit substitutions, and its strong normalization can be characterized by the quantitative typing system designed for $$\lambda J$$ λJ , despite the fact that quantitative subject reduction fails for permutative conversions.

Solvability is a key notion in the theory of call-by-name lambda-calculus, used in particular to identify meaningful terms. However, adapting this notion to other call-by-name calculi, or extending it to different models of computation - such as call-by-value - , is not straightforward. In this paper, we study solvability for call-by-name and call-by-value lambda-calculi with generalized applications, both variants inspired from von Plato’s natural deduction with generalized elimination rules. We develop an operational as well as a logical theory of solvability for each of them. The operational characterization relies on a notion of solvable reduction for generalized applications, and the logical characterization is given in terms of typability in an appropriate non-idempotent intersection type system. Finally, we show that solvability in generalized applications and solvability in the lambda-calculus are equivalent notions.

Abstract We define and study a term calculus implementing higher-order node replication. It is used to specify two different (weak) evaluation strategies: call-by-name and fully lazy call-by-need, that are shown to be observationally equivalent by using type theoretical technical tools.