A Realizability Interpretation for Intersection and Union Types

International audienceProof-functional logical connectives allow reasoning about the structure of logical proofs, in this way giving to the latter the status of first-class objects. This is in contrast to classical truth-functional con- nectives where the meaning of a compound formula is dependent only on the truth value of its subformulas.In this paper we present a typed lambda calculus, enriched with strong products, strong sums, and a related proof-functional logic. This cal- culus, directly derived from a typed calculus previously defined by two of the current authors, has been proved isomorphic to the well-known Barbanera-Dezani-Ciancaglini-de’Liguoro type assignment system. We present a logic L∩∪ featuring two proof-functional connectives, namely strong conjunction and strong disjunction. We prove the typed calculus to be isomorphic to the logic L∩∪ and we give a realizability semantics using Mints’ realizers [Min89] and a completeness theorem. A prototype implementation is also described

Towards a Logical Framework with Intersection and Union Types

International audienceWe present an ongoing implementation of a dependent-type theory (∆-framework) based on the Edinburgh Logical Framework LF, extended with Proof-functional logical connectives such as intersection , union, and strong (or minimal relevant) implication. Proof-functional connectives take into account the shape of logical proofs, thus allowing to reflect polymorphic features of proofs in formulae. This is in contrast to classical Truth-functional connec-tives where the meaning of a compound formula is only dependent on the truth value of its subformulas. Both Logical Frameworks and proof functional logics consider proofs as first class citizens. But they do it differently namely, explicitly in the former while implicitly in the latter. Their combination opens up new possibilites of formal reasoning on proof-theoretic semantics. We provide some examples in the extended type theory and we outline a type checker. The theory of the system is under investigation. Once validated in vitro, the proof-functional type theory can be successfully plugged in existing truth-functional proof assistants

A Type Checker for a Logical Framework with Union and Intersection Types

International audienceWe present the syntax, semantics, typing, subtyping, unification, refinement, and REPL of Bull, a prototype theorem prover based on the ∆-Framework, i.e. a fully-typed Logical Framework à la Edinburgh LF decorated with union and intersection types, as described in previous papers by the authors. Bull also implements a subtyping algorithm for the Type Theory Ξ of Barbanera-Dezani-de'Liguoro. Bull has a command-line interface where the user can declare axioms, terms, and perform computations and some basic terminal-style features like error pretty-printing, subexpressions highlighting, and file loading. Moreover, it can typecheck a proof or normalize it. These terms can be incomplete, therefore the typechecking algorithm uses unification to try to construct the missing subterms. Bull uses the syntax of Berardi's Pure Type Systems to improve the compactness and the modularity of the kernel. Abstract and concrete syntax are mostly aligned and similar to the concrete syntax of Coq. Bull uses a higher-order unification algorithm for terms, while typechecking and partial type inference are done by a bidirectional refinement algorithm, similar to the one found in Matita and Beluga. The refinement can be split into two parts: the essence refinement and the typing refinement. Binders are implemented using commonly-used de Bruijn indices. We have defined a concrete language syntax that will allow user to write ∆-terms. We have defined the reduction rules and an evaluator. We have implemented from scratch a refiner which does partial typechecking and type reconstruction. We have experimented Bull with classical examples of the intersection and union literature, such as the ones formalized by Pfenning with his Refinement Types in LF and by Pierce. We hope that this research vein could be useful to experiment, in a proof theoretical setting, forms of polymorphism alternatives to Girard's parametric one