Typability and type checking in the second-order /spl lambda/-calculus are equivalent and undecidable

@article{Wells1994TypabilityAT,
  title={Typability and type checking in the second-order /spl lambda/-calculus are equivalent and undecidable},
  author={Joe B. Wells},
  journal={Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science},
  year={1994},
  pages={176-185}
}
  • J. B. Wells
  • Published 4 July 1994
  • Mathematics
  • Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science
The problems of typability and type checking exist for the Girard/Reynolds second-order polymorphic typed /spl lambda/-calculus (also known as "system F") when it is considered in the "Curry style" (where types are derived for pure /spl lambda/-terms). Until now the decidability of these problems for F itself has remained unknown. We first prove that type checking in F is undecidable by a reduction from semi-unification. We then prove typability in F is undecidable by a reduction from type… 

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References

SHOWING 1-10 OF 50 REFERENCES
Typability is Undecidable for F+Eta
TLDR
The problem of subtyping is reduced to the problem of typability for F+eta, thus proving the undecidability ofTypability, a well-known polymorphically-typed lambda calculus with universal quantifiers.
The complexity of type inference for higher-order lambda calculi
TLDR
The results are the first nontrivial lower bounds on type inference for the Girard/Reynolds system as well as its higher-order extensions, based on generic simulation of Turing Machines.
A direct algorithm for type inference in the rank-2 fragment of the second-order λ-calculus
TLDR
A new notion of λ-term reduction is developed and used to prove that the problem of typability at rank 2 is reducible to the problemof acyclic semi-unification, which is an undecidable problem at every rank k≥3.
Partial polymorphic type inference is undecidable
  • H. Boehm
  • Computer Science
    26th Annual Symposium on Foundations of Computer Science (sfcs 1985)
  • 1985
TLDR
It is shown here that a natural formalization of the automatic inference of omitted type information is undecidable, and the proof is directly applicable to some practical situations, and provides a partial explanation of the difficulties encountered in other cases.
On the Undecidability of Partial Polymorphic Type Reconstruction
We prove that partial type reconstruction for the pure polymorphic lambda-calculus is undecidable by a reduction from the second-order unification problem, extending a previous result by H.-J. Boehm.
Characterization of typings in polymorphic type discipline
  • P. Giannini, S. D. Rocca
  • Mathematics
    [1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science
  • 1988
TLDR
Using the set of constraints, some properties of polymorphic type discipline are proved.
Finitely Stratified Polymorphism
The undecidability of the semi-unification problem
TLDR
It is shown that SUP in general is undecidable, by reducing what is called the "boundedness problem of Turing machines to SUP", which is a natural generalization of both first-order unification and matching.
Recursive Unsolvability of Post's Problem of "Tag" and other Topics in Theory of Turing Machines
TLDR
The main results of this paper show that the same notions of computability can be realized within the highly restricted monogenic formal systems called by Post the "Tag" systems, and within a peculiarly restricted variant of Turing machine which has two tapes, but can neither write on nor erase these tapes.
Modified basic functionality in combinatory logic
...
...