Definable functions in the simply typed lambdacalculus
Abstract
It is a common knowledge that the integer functions definable in simply typed lambdacalculus are exactly the extended polynomials. This is indeed the case when one interprets integers over the type (p>p)>p>p where p is a base type and/or equality is taken as betaconversion. It is commonly believed that the same holds for betaeta equality and for integers represented over any fixed type of the form (t>t)>t>t. In this paper we show that this opinion is not quite true. We prove that the class of functions strictly definable in simply typed lambdacalculus is considerably larger than the extended polynomials. Namely, we define F as the class of strictly definable functions and G as a class that contains extended polynomials and two additional functions, or more precisely, two function schemas, and is closed under composition. We prove that G is a subset of F. We conjecture that G exactly characterizes strictly definable functions, i.e. G=F, and we gather some evidence for this conjecture proving, for example, that every skewly representable finite range function is strictly representable over (t>t)>t>t for some t.
 Publication:

arXiv eprints
 Pub Date:
 January 2007
 DOI:
 10.48550/arXiv.cs/0701022
 arXiv:
 arXiv:cs/0701022
 Bibcode:
 2007cs........1022Z
 Keywords:

 Computer Science  Logic in Computer Science
 EPrint:
 Submitted to TLCA 2007