9,092 research outputs found
The Algebraic Intersection Type Unification Problem
The algebraic intersection type unification problem is an important component
in proof search related to several natural decision problems in intersection
type systems. It is unknown and remains open whether the algebraic intersection
type unification problem is decidable. We give the first nontrivial lower bound
for the problem by showing (our main result) that it is exponential time hard.
Furthermore, we show that this holds even under rank 1 solutions (substitutions
whose codomains are restricted to contain rank 1 types). In addition, we
provide a fixed-parameter intractability result for intersection type matching
(one-sided unification), which is known to be NP-complete.
We place the algebraic intersection type unification problem in the context
of unification theory. The equational theory of intersection types can be
presented as an algebraic theory with an ACI (associative, commutative, and
idempotent) operator (intersection type) combined with distributivity
properties with respect to a second operator (function type). Although the
problem is algebraically natural and interesting, it appears to occupy a
hitherto unstudied place in the theory of unification, and our investigation of
the problem suggests that new methods are required to understand the problem.
Thus, for the lower bound proof, we were not able to reduce from known results
in ACI-unification theory and use game-theoretic methods for two-player tiling
games
Principal Typings in a Restricted Intersection Type System for Beta Normal Forms with De Bruijn Indices
The lambda-calculus with de Bruijn indices assembles each alpha-class of
lambda-terms in a unique term, using indices instead of variable names.
Intersection types provide finitary type polymorphism and can characterise
normalisable lambda-terms through the property that a term is normalisable if
and only if it is typeable. To be closer to computations and to simplify the
formalisation of the atomic operations involved in beta-contractions, several
calculi of explicit substitution were developed mostly with de Bruijn indices.
Versions of explicit substitutions calculi without types and with simple type
systems are well investigated in contrast to versions with more elaborate type
systems such as intersection types. In previous work, we introduced a de Bruijn
version of the lambda-calculus with an intersection type system and proved that
it preserves subject reduction, a basic property of type systems. In this paper
a version with de Bruijn indices of an intersection type system originally
introduced to characterise principal typings for beta-normal forms is
presented. We present the characterisation in this new system and the
corresponding versions for the type inference and the reconstruction of normal
forms from principal typings algorithms. We briefly discuss the failure of the
subject reduction property and some possible solutions for it
The involutions-as-principal types/ application-as-unification analogy
In 2005, S. Abramsky introduced various universal models of computation based on Affine Combinatory Logic, consisting of partial involutions over a suitable formal language of moves, in order to discuss reversible computation in a game-theoretic setting. We investigate Abramsky\u2019s models from the point of view of the model theory of \u3bb-calculus, focusing on the purely linear and affine fragments of Abramsky\u2019s Combinatory Algebras. Our approach stems from realizing a structural analogy, which had not been hitherto pointed out in the literature, between the partial involution interpreting a combinator and the principal type of that term, with respect to a simple types discipline for \u3bb-calculus. This analogy allows for explaining as unification between principal types the somewhat awkward linear application of involutions arising from Geometry of Interaction (GoI). Our approach provides immediately an answer to the open problem, raised by Abramsky, of characterising those finitely describable partial involutions which are denotations of combinators, in the purely affine fragment. We prove also that the (purely) linear combinatory algebra of partial involutions is a (purely) linear \u3bb-algebra, albeit not a combinatory model, while the (purely) affine combinatory algebra is not. In order to check the complex equations involved in the definition of affine \u3bb-algebra, we implement in Erlang the compilation of \u3bb-terms as involutions, and their execution
Light Logics and the Call-by-Value Lambda Calculus
The so-called light logics have been introduced as logical systems enjoying
quite remarkable normalization properties. Designing a type assignment system
for pure lambda calculus from these logics, however, is problematic. In this
paper we show that shifting from usual call-by-name to call-by-value lambda
calculus allows regaining strong connections with the underlying logic. This
will be done in the context of Elementary Affine Logic (EAL), designing a type
system in natural deduction style assigning EAL formulae to lambda terms.Comment: 28 page
The Intersection Type Unification Problem
The intersection type unification problem is an important component in
proof search related to several natural decision problems in
intersection type systems. It is unknown and remains open whether the
unification problem is decidable. We give the first nontrivial lower
bound for the problem by showing (our main result) that it is
exponential time hard. Furthermore, we show that this holds even under
rank 1 solutions (substitutions whose codomains are restricted to
contain rank 1 types). In addition, we provide a fixed-parameter
intractability result for intersection type matching (one-sided
unification), which is known to be NP-complete.
We place the intersection type unification problem in the context of
unification theory. The equational theory of intersection types can
be presented as an algebraic theory with an ACI (associative,
commutative, and idempotent) operator (intersection type) combined
with distributivity properties with respect to a second operator
(function type). Although the problem is algebraically natural and
interesting, it appears to occupy a hitherto unstudied place in the
theory of unification, and our investigation of the problem suggests
that new methods are required to understand the problem. Thus, for the
lower bound proof, we were not able to reduce from known results in
ACI-unification theory and use game-theoretic methods for two-player
tiling games
Polymorphism and Inference in Database Programming
The polymorphic type system of ML can be extended in two ways to make it the appropriate basis of a database programming language. The first is an extension to the language of types that captures the polymorphic nature of field selection; the second is a technique that generalizes relational operators to arbitrary data structures. The combination provides a statically typed language in which relational databases may be cleanly represented as typed structures. As in ML types are inferred, which relieves the programmer of making the rather complicated type assertions that may be required to express the most general type of a program that involving field selection and generalized relational operators. These extensions may also be used to provide static polymorphic typechecking in object-oriented languages and databases. A problem that arises with object-oriented databases is the apparent need for dynamic typechecking when dealing with queries on heterogeneous collections of objects. An extension of the type system needed for generalized relational operations can also be used for manipulating collections of dynamically typed values in a statically typed language. A prototype language based on these ideas has been implemented. While it lacks a proper treatment of persistent data, it demonstrates that a wide variety of database structures can be cleanly represented in a polymorphic programming language
- …