219 research outputs found
Toward Isomorphism of Intersection and Union types
This paper investigates type isomorphism in a lambda-calculus with
intersection and union types. It is known that in lambda-calculus, the
isomorphism between two types is realised by a pair of terms inverse one each
other. Notably, invertible terms are linear terms of a particular shape, called
finite hereditary permutators. Typing properties of finite hereditary
permutators are then studied in a relevant type inference system with
intersection and union types for linear terms. In particular, an isomorphism
preserving reduction between types is defined. Type reduction is confluent and
terminating, and induces a notion of normal form of types. The properties of
normal types are a crucial step toward the complete characterisation of type
isomorphism. The main results of this paper are, on one hand, the fact that two
types with the same normal form are isomorphic, on the other hand, the
characterisation of the isomorphism between types in normal form, modulo
isomorphism of arrow types.Comment: In Proceedings ITRS 2012, arXiv:1307.784
On Isomorphism of "Functional" Intersection and Union Types
Type isomorphism is useful for retrieving library components, since a
function in a library can have a type different from, but isomorphic to, the
one expected by the user. Moreover type isomorphism gives for free the coercion
required to include the function in the user program with the right type. The
present paper faces the problem of type isomorphism in a system with
intersection and union types. In the presence of intersection and union,
isomorphism is not a congruence and cannot be characterised in an equational
way. A characterisation can still be given, quite complicated by the
interference between functional and non functional types. This drawback is
faced in the paper by interpreting each atomic type as the set of functions
mapping any argument into the interpretation of the type itself. This choice
has been suggested by the initial projection of Scott's inverse limit
lambda-model. The main result of this paper is a condition assuring type
isomorphism, based on an isomorphism preserving reduction.Comment: In Proceedings ITRS 2014, arXiv:1503.0437
Program representation size in an intermediate language with intersection and union types
The CIL compiler for core Standard ML compiles whole programs using a novel typed intermediate language (TIL) with intersection and union types and flow labels on both terms and types. The CIL term representation duplicates portions of the program where intersection types are introduced and union types are eliminated. This duplication makes it easier to represent type information and to introduce customized data representations. However, duplication incurs compile-time space costs that are potentially much greater than are incurred in TILs employing type-level abstraction or quantification. In this paper, we present empirical data on the compile-time space costs of using CIL as an intermediate language. The data shows that these costs can be made tractable by using sufficiently fine-grained flow analyses together with standard hash-consing techniques. The data also suggests that non-duplicating formulations of intersection (and union) types would not achieve significantly better space complexity.National Science Foundation (CCR-9417382, CISE/CCR ESS 9806747); Sun grant (EDUD-7826-990410-US); Faculty Fellowship of the Carroll School of Management, Boston College; U.K. Engineering and Physical Sciences Research Council (GR/L 36963, GR/L 15685
Preciseness of Subtyping on Intersection and Union Types
Abstract. The notion of subtyping has gained an important role both in theoretical and applicative domains: in lambda and concurrent calculi as well as in programming languages. The soundness and the complete-ness, together referred to as the preciseness of subtyping, can be consid-ered from two different points of view: denotational and operational. The former preciseness is based on the denotation of a type which is a math-ematical object that describes the meaning of the type in accordance with the denotations of other expressions from the language. The latter preciseness has been recently developed with respect to type safety, i.e. the safe replacement of a term of a smaller type when a term of a bigger type is expected. We propose a technique for formalising and proving operational pre-ciseness of the subtyping relation in the setting of a concurrent lambda calculus with intersection and union types. The key feature is the link between typings and the operational semantics. We then prove sound-ness and completeness getting that the subtyping relation of this calculus enjoys both denotational and operational preciseness.
Session Types = Intersection Types + Union Types
We propose a semantically grounded theory of session types which relies on
intersection and union types. We argue that intersection and union types are
natural candidates for modeling branching points in session types and we show
that the resulting theory overcomes some important defects of related
behavioral theories. In particular, intersections and unions provide a native
solution to the problem of computing joins and meets of session types. Also,
the subtyping relation turns out to be a pre-congruence, while this is not
always the case in related behavioral theories.Comment: In Proceedings ITRS 2010, arXiv:1101.410
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
Types as Resources for Classical Natural Deduction
We define two resource aware typing systems for the lambda-mu-calculus based on non-idempotent intersection and union types. The
non-idempotent approach provides very simple combinatorial arguments - based on decreasing measures of type derivations - to characterize head and strongly normalizing terms. Moreover, typability provides upper bounds for the length of head-reduction sequences and maximal reduction sequences
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