Typability and Type Checking in the Second-Order lambda-Calculus Are Equivalent and Undecidable

Typability and Type Checking in the Second-Order lambda-Calculus Are Equivalent and Undecidable (1993)

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by J. B. Wells

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TITLE	Typability andType Checking in the Second-Order-Calculus Are Equivalent and Undecidable (Preliminary Draft)	SVM HeaderParse 0.2
AUTHOR NAME	J. B. Wells	SVM HeaderParse 0.2
AUTHOR AFFIL	; Dept. of Computer Science; Boston University	SVM HeaderParse 0.2
AUTHOR ADDR	; Boston, MA 02215, USA	SVM HeaderParse 0.2
ABSTRACT	We consider the problems of typability 1 and type checking 2 in the Girard/Reynolds secondorder polymorphic typed-calculus, for which we use the short name \System F " andwhichwe use in the \Curry style " where types are assigned to pure-terms. These problems have been considered and proven to be decidable or undecidable for various restrictions and extensions of System F and other related systems, and lower-bound complexity results for System F have been achieved, but they have remained \embarrassing open problems " 3 for System F itself. We rst prove that type checking in System F is undecidable by a reduction from semi-uni cation. We then prove typability in System F is undecidable by a reduction from type checking. Since the reverse reduction is already known, this implies the two problems are equivalent. The second reduction uses a novel method of constructing-terms such that in all type derivations, speci c bound variables must always be assigned a speci c type. Using this technique, we can require that speci c subterms must be typable using a speci c, xed type assignment in order for the entire term to be typable at all. Any desired type assignment maybe simulated. We develop this method, which we call \constants for free", for both the K and I calculi. This work is partly supported by NSF grant CCR{9113196. 1 Typability is also called type reconstruction.	SVM HeaderParse 0.2
YEAR	1993	SVM HeaderParse 0.2
CITATIONS	2 found	ParsCit 1.0