The computational content of atomic polymorphism

Ferreira, Gilda; Vasconcelos, Vasco T.

doi:10.1093/jigpal/jzy076

Cited by 2 publications

(2 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, as we show below, the numerical functions which can be typed in F at are precisely the simply typable ones (i.e. the so-called extended polynomials [54,16]).…”

Section: Typability and Type-checkingmentioning

confidence: 99%

“…Could one tell exactly at which point, in the range from the simply typed λ-calculus and ML to full System F, the type-related properties of polymorphism become undecidable? Atomic Polymorphism In more recent times Ferreira et al have undertaken the investigation of what can be seen as the least expressive predicative fragment of F, System F at , or atomic System F [12,11,13,15,16,10,9]. The predicative restriction of F at is such that a universally quantified type @X.A can be instantiated solely with an atomic type, i.e.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

What's Decidable about (Atomic) Polymorphism

Pistone¹,

Tranchini²

2021

Preprint

View full text Add to dashboard Cite

Due to the undecidability of most type-related properties of System F like type inhabitation or type checking, restricted polymorphic systems have been widely investigated (the most wellknown being ML-polymorphism). In this paper we investigate System Fat, or atomic System F, a very weak predicative fragment of System F whose typable terms coincide with the simply typable ones. We show that the type-checking problem for Fat is decidable and we propose an algorithm which sheds some new light on the source of undecidability in full System F. Moreover, we investigate free theorems and contextual equivalence in this fragment, and we show that the latter, unlike in the simply typed lambda-calculus, is undecidable.

show abstract

“…In particular, as we show below, the numerical functions which can be typed in F at are precisely the simply typable ones (i.e. the so-called extended polynomials [54,16]).…”

Section: Typability and Type-checkingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

What's Decidable about (Atomic) Polymorphism

Pistone¹,

Tranchini²

2021

Preprint

View full text Add to dashboard Cite

show abstract

Type inhabitation of atomic polymorphism is undecidable

Protin

2021

Journal of Logic and Computation

View full text Add to dashboard Cite

Atomic polymorphism $\mathbf{F_{at}}$ is a restriction of Girard and Reynold’s system $\mathbf{F} $(or $\lambda 2$) which was first introduced in Ferreira [ 2] in the context of a philosophical commentary on predicativism. $\lambda 2$ is a well-known and powerful formal tool for studying polymorphic functional programming languages and formal methods in program specification and development, but its computational power far exceeds the recursive level of interest in applications. Hence, the interest of studying subsystems of $\lambda 2$ with weaker computational power. $\mathbf{F_{at}}$ is defined by restricting instantiation to atomic variables only. It turns out that the type system is still sufficiently powerful to possess embeddings of full intuitionistic propositional calculus [ 3, 4], and since the calculus has fewer connectives and strong normalizability is simple to prove [ 3], this result allows us to circumvent many of the extra computational complexities present when dealing with the proof theory of IPC. It is natural to inquire whether type inhabitation, i.e. provability in the corresponding fragment of second-order intuitionistic propositional logic, is decidable or not and in general to see whether the negative results involving the undecidability of type inhabitation, typability and type-checking for $\mathbf{F}$ still hold in this fragment. A further theme would be to study the result of adding type constructors, recursors or even dependent types to $\mathbf{F_{at}}$. In this paper, we show that type inhabitation for $\mathbf{F_{at}}$ is undecidable by codifying within it an undecidable fragment of first-order intuitionistic predicate calculus, adapting and modifying the technique of Urzyczyn’s [ 1, 7] purely syntactic proof of the undecidability of type inhabitation for $\mathbf{F}$.

show abstract

The computational content of atomic polymorphism

Cited by 2 publications

References 13 publications

What's Decidable about (Atomic) Polymorphism

What's Decidable about (Atomic) Polymorphism

Type inhabitation of atomic polymorphism is undecidable

Contact Info

Product

Resources

About