## Second-order unification and type inference for Church-style polymorphism (1998)

Venue: | In Conference Record of POPL 98: The 25TH ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages |

Citations: | 14 - 0 self |

### BibTeX

@INPROCEEDINGS{Schubert98second-orderunification,

author = {Aleksy Schubert},

title = {Second-order unification and type inference for Church-style polymorphism},

booktitle = {In Conference Record of POPL 98: The 25TH ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages},

year = {1998},

pages = {279--288},

publisher = {ACM Press}

}

### OpenURL

### Abstract

We present a proof of the undecidability of type inference for the Church-style system F --- an abstraction of polymorphism. A natural reduction from the second-order unification problem to type inference leads to strong restriction on instances --- arguments of variables cannot contain variables. This requires another proof of the undecidability of the second-order unification since known results use variables in arguments of other variables. Moreover, our proof uses elementary techniques, which is important from the methodological point of view, because Goldfarb's proof [Gol81] highly relies on the undecidability of the tenth Hilbert's problem. 1 1 Introduction The Church-style system F was independently introduced by Girard [Gir72] and Reynolds [Rey74] as an extension of the simply-typed -calculus a type system introduced of H. B. Curry [Cur69]. As usual for type systems, the decidability of so called sequent decision problems was considered. A sequent decision problem in some ty...

### Citations

4242 |
Introduction to Automata Theory, Languages, and Computation
- Hopcroft, Ullman
- 1979
(Show Context)
Citation Context ... ;. Note that these restrictions do not affect the following property Theorem 2.2 (the undecidability of the halting problem) The "halting problem" for two-counter automata is undecidable. P=-=roof: See [HU79]-=-. We write TC1; TC2;q 1 ! ffi A1; A2;q 2 where TC1, TC2 2 T ; q 1 ; q 2 2 Q, and A1, A2 2 O, to denote action of the automaton according to ffi . We assume existence of some linear order on rules in f... |

303 |
Interprétation fonctionelle et élimination des coupures dans l’arithmétique d’ordre supérieur
- Girard
- 1972
(Show Context)
Citation Context ...l point of view, because Goldfarb's proof [Gol81] highly relies on the undecidability of the tenth Hilbert's problem. 1 1 Introduction The Church-style system F was independently introduced by Girard =-=[Gir72]-=- and Reynolds [Rey74] as an extension of the simply-typed -calculus a type system introduced of H. B. Curry [Cur69]. As usual for type systems, the decidability of so called sequent decision problems ... |

83 |
Lambda calculi with types". in: Handbook of logic in computer science
- Barendregt
- 1993
(Show Context)
Citation Context ...-style polymorphic -calculus is often defined as followssF = V jsFsF jsF T j V : T F j VsF T = V j T ! T j 8V T where V is the set of type variables and V is the set of individual variables (see e.g. =-=[Bar92]-=-, section 5.1, called there 2-Church). Since individual variables have no type annotations, the type reconstruction problem is no 1 A proof of this fact using syntactic methods is in [Urz96], too. 2 l... |

83 | Partial Polymorphic Type Inference and Higher-Order Unification - Pfenning - 1988 |

27 | On the undecidability of partial polymorphic type reconstruction. Fundarnenta Informaticae - Pfenning - 1993 |

25 |
Typability and Type Checking in the Second-Order - Calculus Are Equivalent and Undecidable. submitted to APAP
- Wells
- 1996
(Show Context)
Citation Context ... not. Table 1 presents which of those non-trivial ones were solvedfor the Church and Curry version of the system F . As it \Gamma ` M : oe ? \Gamma ` M : ? \Gamma `? : oe ? ` M : oe ? ` M : ? Curry-F =-=[Wel94]-=- an easy reduction from [Wel94] [Lob76] 1 an easy reduction from [Wel94] [Wel94] Church-F a formulation for normalising PTSs [vBJ93] an easy reduction from [vBJ93] [Lob76] 1 Table 1: Decidability of n... |

24 |
Simple second-order languages for which unification is undecidable
- Farmer
- 1991
(Show Context)
Citation Context ...nce the Goldfarb's proof relies on the undecidability of tenth Hilbert's problem which relies heavily on sophisticated number theoretic properties. We conjecture that the techniques used by Farmer in =-=[Far91]-=- apply to our case (narrowing down the number of function variables and function constants). An appropriate modification of the approach would give the undecidability of the most restricted case of se... |

23 |
Benthem Jutting. Typing in pure type systems
- van
- 1993
(Show Context)
Citation Context ...M : oe ? \Gamma ` M : ? \Gamma `? : oe ? ` M : oe ? ` M : ? Curry-F [Wel94] an easy reduction from [Wel94] [Lob76] 1 an easy reduction from [Wel94] [Wel94] Church-F a formulation for normalising PTSs =-=[vBJ93]-=- an easy reduction from [vBJ93] [Lob76] 1 Table 1: Decidability of non-trivial sequent decision problems for the system F . is seen there are two missing entries in the Table 1. These two problems are... |

21 |
Partial polymorphic type inference is undecidable
- Boehm
- 1985
(Show Context)
Citation Context ...l94] applies to the Leivant's calculus in the Curry-style. A problem similar to the type inference problem for the Church version of the system F has been proposed, called partial type reconstruction =-=[Boe85]-=-. The problem consist in reconstructing of type information for polymorphic terms with partially erased type annotations 0 F = V js0 Fs0 F js0 F T js0 F [ ] j V : T F 0 j V : [ ] F 0 j V F 0 This prob... |

21 |
Finitely stratified polymorphism
- Leivant
- 1991
(Show Context)
Citation Context ...96], too. 2 longer trivial. In fact, contrary to the common belief, it occurs that the problem is undecidable. Our proof concerning type inference applies to the restricted system proposed by Leivant =-=[Lei91]-=-, too. The restriction considered there consists in forbidding instantiation of general quantifier bound variables by types that are roughly speaking greater than the type with this quantifier. This r... |

16 |
Modi basic functionality in combinatory logic
- Curry
- 1969
(Show Context)
Citation Context ...m. 1 1 Introduction The Church-style system F was independently introduced by Girard [Gir72] and Reynolds [Rey74] as an extension of the simply-typed -calculus a type system introduced of H. B. Curry =-=[Cur69]-=-. As usual for type systems, the decidability of so called sequent decision problems was considered. A sequent decision problem in some type system A is a decision problem --- given some part of a jud... |

6 |
Embedding first order predicate logic in fragments of intuitionistic logic, this JOURNAL
- LOB
(Show Context)
Citation Context ...on-trivial ones were solvedfor the Church and Curry version of the system F . As it \Gamma ` M : oe ? \Gamma ` M : ? \Gamma `? : oe ? ` M : oe ? ` M : ? Curry-F [Wel94] an easy reduction from [Wel94] =-=[Lob76]-=- 1 an easy reduction from [Wel94] [Wel94] Church-F a formulation for normalising PTSs [vBJ93] an easy reduction from [vBJ93] [Lob76] 1 Table 1: Decidability of non-trivial sequent decision problems fo... |

6 |
Inhabitation in typed lambda-calculi (a syntactic approach
- Urzyczyn
- 1997
(Show Context)
Citation Context ...(see e.g. [Bar92], section 5.1, called there 2-Church). Since individual variables have no type annotations, the type reconstruction problem is no 1 A proof of this fact using syntactic methods is in =-=[Urz96]-=-, too. 2 longer trivial. In fact, contrary to the common belief, it occurs that the problem is undecidable. Our proof concerning type inference applies to the restricted system proposed by Leivant [Le... |

1 |
The undecidability of the second-order unification problem, TCS
- Goldfarb
- 1981
(Show Context)
Citation Context ...n since known results use variables in arguments of other variables. Moreover, our proof uses elementary techniques, which is important from the methodological point of view, because Goldfarb's proof =-=[Gol81]-=- highly relies on the undecidability of the tenth Hilbert's problem. 1 1 Introduction The Church-style system F was independently introduced by Girard [Gir72] and Reynolds [Rey74] as an extension of t... |

1 |
Mathematical foundations of software development, (Ehring et
- Reynolds
- 1974
(Show Context)
Citation Context ...use Goldfarb's proof [Gol81] highly relies on the undecidability of the tenth Hilbert's problem. 1 1 Introduction The Church-style system F was independently introduced by Girard [Gir72] and Reynolds =-=[Rey74]-=- as an extension of the simply-typed -calculus a type system introduced of H. B. Curry [Cur69]. As usual for type systems, the decidability of so called sequent decision problems was considered. A seq... |