## Typability and Type Checking in the Second-Order lambda-Calculus Are Equivalent and Undecidable (1996)

Venue: | In Proceedings of the Ninth Annual IEEE Symposium on Logic in Computer Science (LICS |

Citations: | 12 - 1 self |

### BibTeX

@INPROCEEDINGS{Wells96typabilityand,

author = {J. B. Wells},

title = {Typability and Type Checking in the Second-Order lambda-Calculus Are Equivalent and Undecidable},

booktitle = {In Proceedings of the Ninth Annual IEEE Symposium on Logic in Computer Science (LICS},

year = {1996},

pages = {176--185},

publisher = {Society Press}

}

### OpenURL

### Abstract

Girard and Reynolds independently invented the second-order polymorphically typed lambda calculus, known as System F, to handle problems in logic and computer programming language design, respectively. Viewing F in the Curry style, which associates types with untyped lambda terms, raises the questions of typability and type checking. Typability asks for a term whether there exists some type it can be given. Type checking asks whether a term can be given a particular type. The decidability of these problems has been settled for restrictions and extensions of F and related systems and complexity lower-bounds have been determined for typability in F, but until now their decidability for F has remained unknown. This report proves that type checking in F is undecidable, by a reduction from semiunification, and that typability in F is undecidable, by a reduction from type checking. Since there is an easy reduction from typability to type checking, the two problems are equivalent. ...