179 research outputs found
Typability and type checking in the second-order Λ-calculus are equivalent and undecidable (Preliminary Draft)
We consider the problems of typability[1] and type checking[2] in the Girard/Reynolds second-order polymorphic typed λ-calculus, for which we use the short name "System F" and which we use in the "Curry style" where types are assigned to pure λ -terms. These problems have been considered and proven to be decidable or undecidable for various restrictions and extensions of System F and other related systems, and lower-bound complexity results for System F have been achieved, but they have remained "embarrassing open problems"[3] for System F itself. We first prove that type checking in System F is undecidable by a reduction from semi-unification. We then prove typability in System F is undecidable by a reduction from type checking. Since the reverse reduction is already known, this implies the two problems are equivalent. The second reduction uses a novel method of constructing λ-terms such that in all type derivations, specific bound variables must always be assigned a specific type. Using this technique, we can require
that specific subterms must be typable using a specific, fixed type assignment in order for the entire term to be typable at all. Any desired type assignment may be simulated. We develop this method, which we call "constants for free", for both the λK and λI calculi.NSF (CCR-9113196
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
Decidable structures between Church-style and Curry-style
It is well-known that the type-checking and type-inference problems are undecidable for second order lambda-calculus in Curry-style, although those for Church-style are decidable. What causes the differences in decidability and undecidability on the problems? We examine crucial conditions on terms for the (un)decidability property from the viewpoint of partially typed terms, and what kinds of type annotations are essential for (un)decidability of type-related problems. It is revealed that there exists an intermediate structure of second order lambda-terms, called a style of hole-application, between Church-style and Curry-style, such that the type-related problems are decidable under the structure. We also extend this idea to the omega-order polymorphic calculus F-omega, and show that the type-checking and type-inference problems then become undecidable
Typability and type inference in atomic polymorphism
It is well-known that typability, type inhabitation and type inference are
undecidable in the Girard-Reynolds polymorphic system F. It has recently been proven
that type inhabitation remains undecidable even in the predicative fragment of system F
in which all universal instantiations have an atomic witness (system Fat). In this paper we
analyze typability and type inference in Curry style variants of system Fat and show that
typability is decidable and that there is an algorithm for type inference which is capable of
dealing with non-redundancy constraints.The second author acknowledges the support of FCT — Fundação para a Ciência e a Tecnologia under the
projects UIDB/04561/2020, UIDB/00408/2020 and UIDP/00408/2020, and she is also grateful to CMAFcIO —
Centro de Matemática, Aplicações Fundamentais e Investigação Operacional and to LASIGE — Computer
Science and Engineering Research Centre (Universidade de Lisboa).info:eu-repo/semantics/publishedVersio
Type Inference for Bimorphic Recursion
This paper proposes bimorphic recursion, which is restricted polymorphic
recursion such that every recursive call in the body of a function definition
has the same type. Bimorphic recursion allows us to assign two different types
to a recursively defined function: one is for its recursive calls and the other
is for its calls outside its definition. Bimorphic recursion in this paper can
be nested. This paper shows bimorphic recursion has principal types and
decidable type inference. Hence bimorphic recursion gives us flexible typing
for recursion with decidable type inference. This paper also shows that its
typability becomes undecidable because of nesting of recursions when one
removes the instantiation property from the bimorphic recursion.Comment: In Proceedings GandALF 2011, arXiv:1106.081
A Direct Algorithm for the Type Interference in the Rank 2 Fragment of the Second--Order λ-Calculus
We study the problem of type inference for a family of polymorphic type disciplines containing the power of Core-ML. This family comprises all levels of the stratification of the second-order lambda-calculus by "rank" of types. We show that typability is an undecidable problem at every rank k ≥ 3 of this stratification. While it was already known that typability is decidable at rank ≤ 2, no direct and easy-to-implement algorithm was available. To design such an algorithm, we develop a new notion of reduction and show how to use it to reduce the problem of typability at rank 2 to the problem of acyclic semi-unification. A by-product of our analysis is the publication of a simple solution procedure for acyclic semi-unification
Constructive Many-One Reduction from the Halting Problem to Semi-Unification
Semi-unification is the combination of first-order unification and
first-order matching. The undecidability of semi-unification has been proven by
Kfoury, Tiuryn, and Urzyczyn in the 1990s by Turing reduction from Turing
machine immortality (existence of a diverging configuration). The particular
Turing reduction is intricate, uses non-computational principles, and involves
various intermediate models of computation. The present work gives a
constructive many-one reduction from the Turing machine halting problem to
semi-unification. This establishes RE-completeness of semi-unification under
many-one reductions. Computability of the reduction function, constructivity of
the argument, and correctness of the argument is witnessed by an axiom-free
mechanization in the Coq proof assistant. Arguably, this serves as
comprehensive, precise, and surveyable evidence for the result at hand. The
mechanization is incorporated into the existing, well-maintained Coq library of
undecidability proofs. Notably, a variant of Hooper's argument for the
undecidability of Turing machine immortality is part of the mechanization.Comment: CSL 2022 - LMCS special issu
Typability and type inference in atomic polymorphism
It is well-known that typability, type inhabitation and type inference are
undecidable in the Girard-Reynolds polymorphic system F. It has recently been proven
that type inhabitation remains undecidable even in the predicative fragment of system F
in which all universal instantiations have an atomic witness (system Fat). In this paper we
analyze typability and type inference in Curry style variants of system Fat and show that
typability is decidable and that there is an algorithm for type inference which is capable of
dealing with non-redundancy constraints.The second author acknowledges the support of FCT — Fundação para a Ciência e a Tecnologia under the
projects UIDB/04561/2020, UIDB/00408/2020 and UIDP/00408/2020, and she is also grateful to CMAFcIO —
Centro de Matemática, Aplicações Fundamentais e Investigação Operacional and to LASIGE — Computer
Science and Engineering Research Centre (Universidade de Lisboa).info:eu-repo/semantics/publishedVersio
The undecidability of Mitchell's subtyping relationship
Mitchell defined and axiomatized a subtyping relationship (also known as containment, coercibility, or subsumption) over the types of System F (with "→" and "∀"). This subtyping relationship is quite simple and does not involve bounded quantification. Tiuryn and Urzyczyn quite recently proved this subtyping relationship to be undecidable. This paper supplies a new undecidability proof for this subtyping relationship. First, a new syntax-directed axiomatization of the subtyping relationship is defined. Then, this axiomatization is used to prove a reduction from the undecidable problem of semi-unification to subtyping. The undecidability of subtyping implies the undecidability of type checking for System F extended with Mitchell's subtyping, also known as "F plus eta".National Science Foundation (CCR-9113196, CCR-9417382
- …