## Typability is undecidable for F+eta (1995)

Citations: | 9 - 6 self |

### BibTeX

@TECHREPORT{Wells95typabilityis,

author = {J. B. Wells},

title = {Typability is undecidable for F+eta},

institution = {},

year = {1995}

}

### Years of Citing Articles

### OpenURL

### Abstract

System F is the well-known polymorphically-typed-calculus with universal quanti ers (\8"). F+ is System F extended with the eta rule, which says that if term M can be given type and M-reduces to N, then N can also be given the type. Adding the eta rule to System F is equivalent to adding the subsumption rule using the subtyping (\containment") relation that Mitchell de ned and axiomatized [Mit88]. The subsumption rule says that if M can be given type and is a subtype of type,thenMcan be given type. Mitchell's subtyping relation involves no extensions to the syntaxoftypes, i.e., no bounded polymorphism and no supertype of all types, and is thus unrelated to the system F (\F-sub"). Typability for F+ is the problem of determining for any termMwhether there is any type that can be given to it using the type inference rules of F+. Typability has been proven undecidable for System F [Wel94] (without the eta rule), but the decidability oftypability has been an open problem for F+. Mitchell's subtyping relation has recently been proven undecidable [TU95, Wel95b], implying the undecidability of\type checking " for F+. This paper reduces the problem of subtyping to the problem of typability for F+,thus proving the undecidability oftypability. The proof methods are similar in outline to those used to prove the undecidability oftypability for System F, but the ne details di er greatly. 1

### Citations

364 |
Towards a theory of type structure
- Reynolds
- 1974
(Show Context)
Citation Context ... similar in outline to those used to prove the undecidability of typability for System F, but the fine details differ greatly. 1 Introduction 1.1 Background and Motivation Girard [Gir72] and Reynolds =-=[Rey74] independe-=-ntly formulated the type system of the second-order, parametrically-polymorphic -calculus about twenty years ago. Girard developed his system (named by chance "System F") to prove properties... |

238 |
Interprétation fonctionnelle et elimination des coupures de l’arithmétique d’ordre supérieur. Thèse d’état, Université de Paris 7
- Girard
- 1972
(Show Context)
Citation Context ...The proof methods are similar in outline to those used to prove the undecidability of typability for System F, but the fine details differ greatly. 1 Introduction 1.1 Background and Motivation Girard =-=[Gir72] and Reyno-=-lds [Rey74] independently formulated the type system of the second-order, parametrically-polymorphic -calculus about twenty years ago. Girard developed his system (named by chance "System F"... |

154 |
The Lambda Calculus: its Syntax and Semantics. NorthHolland, revised edition
- Barendregt
- 1984
(Show Context)
Citation Context ..."R +1 " denotes R. If R is denoted by a directional symbol (such as "!"), the reversed symbol (such as "?") denotes R \Gamma1 . 2.2 Terms Our notation for the -calculus g=-=enerally follows Barendregt's [Bar84]-=-. The set of all -termssis built from the countably infinite set of -term variables V using application and abstraction as specified by this grammar:s::= V j ( ) j (V :) Small Roman letters from the b... |

102 |
Polymorphic type inference and containment
- Mitchell
- 1988
(Show Context)
Citation Context ...N , then N can also be given the types. Adding the eta rule to System F is equivalent to adding the subsumption rule using the subtyping ("containment") relation that Mitchell defined and ax=-=iomatized [Mit88]-=-. The subsumption rule says that if M can be given typesandsis a subtype of type oe, then M can be given type oe. Mitchell's subtyping relation involves no extensions to the syntax of types, i.e., no ... |

102 | Bounded quantification is undecidable - Pierce - 1994 |

85 |
Polymorphic Type Inference
- Leivant
- 1983
(Show Context)
Citation Context ...olds wanted to express polymorphic typing in programming explicitly. Both Girard and Reynolds formulated F in the "Church style", but we deal with the "Curry style" formulation fir=-=st given by Leivant [Lei83]-=-. In the Church style, types are embedded in terms and the termformation rules are also the typing rules, while in the Curry style, types are given to pure terms of the -calculus. The inference rules ... |

25 |
Typability and Type Checking in the Second-Order - Calculus Are Equivalent and Undecidable. submitted to APAP
- Wells
- 1996
(Show Context)
Citation Context ...ility for F+j is the problem of determining for any term M whether there is any typesthat can be given to it using the type inference rules of F+j. Typability has been proven undecidable for System F =-=[Wel94]-=- (without the eta rule), but the decidability of typability has been an open problem for F+j. Mitchell's subtyping relation has recently been proven undecidable [TU95, Wel95b], implying the undecidabi... |

20 | The subtyping problem for second-order types is undecidable - Tiuryn, Urzyczyn - 1996 |

14 | A logic of subtyping
- Longo, Milsted, et al.
- 1995
(Show Context)
Citation Context ...problem whether the subtyping relation is decidable. Longo, Milsted, and Soloviev recently devised a new axiomatization of the subtyping relation which avoids having an explicit rule for transitivity =-=[LMS95]-=-. Using this new axiomatization, Tiuryn and Urzyczyn recently proved the undecidability of the subtyping relation by a reduction from the halting problem for 2-counter automata [TU95]. Also using this... |

13 |
The undecidability of Mitchell’s subtyping relation
- Wells
- 1995
(Show Context)
Citation Context ...ed rules, I proved the subtyping relation to be undecidable by technique totally different from Tiuryn and Urzyczyn's and significantly simpler, namely a reduction from the problem of semiunification =-=[Wel95b]. The unde-=-cidability of subtyping implies the undecidability of type checking for F+j, because there is a trivial reduction from subtyping to type checking where the subtyping question "oes" becomes t... |

9 | Bounded quanti cation is undecidable - Pierce - 1992 |

8 | Equational axiomatization of bicoercibility for polymorphic types
- Tiuryn
- 1995
(Show Context)
Citation Context ...r a type system, then the overall outline of our methods can probably be applied, but the details will likely differ. 1.3 Acknowledgements Trevor Jim made me aware of Tiuryn's paper on bicoercibility =-=[Tiu95]-=-, which was essential for my understanding of the subtyping relation. Assaf Kfoury provided vital support and encouragement. 2 Definitions and Foundation This section introduces basic definitions, not... |

2 |
Type inference in System F plus subtyping
- Jim
- 1995
(Show Context)
Citation Context ...cture that the full problem of type checking can be reduced to typability in F+j. T. Jim has recently reduced typability in F+j to subtyping using principal typings of terms in distinct operator form =-=[Jim95]-=-. Since typability is reducible to subtyping, and since subtyping is reducible to type checking, our conjecture would imply that all three problems are equivalent. There are other systems related to S... |

2 |
The undecidability ofMitchell's subtyping relation
- Wells
- 1995
(Show Context)
Citation Context ...ected rules, I proved the subtyping relation to be undecidable by technique totally di erent from Tiuryn and Urzyczyn's and signi cantly simpler, namely a reduction from the problem of semiuni cation =-=[Wel95b]-=-. The undecidability of subtyping implies the undecidability oftype checking for F+ , because there is a trivial reduction from subtyping to type checking where the subtyping question \ " becomes the ... |