Definable functions in the simply typed lambda-calculus
<|reference_start|>Definable functions in the simply typed lambda-calculus: It is a common knowledge that the integer functions definable in simply typed lambda-calculus 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 beta-conversion. It is commonly believed that the same holds for beta-eta 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 lambda-calculus 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.<|reference_end|>
@article{zakrzewski2007definable,
    title={Definable functions in the simply typed lambda-calculus},
    author={Mateusz Zakrzewski},
    journal={arXiv preprint arXiv:cs/0701022},
    year={2007},
    archivePrefix={arXiv},
    eprint={cs/0701022},
    primaryClass={cs.LO}
}