2007/01/04 by Mateusz Zakrzewski, Zakrzewski, Mateusz
Computer Science · #Computability, Logic, AI Algorithms #Logic, Reasoning, and Knowledge #Logic, programming, and type systems #cs.LO
paper · pdf · doi:10.48550/arxiv.cs/0701022
Submitted to TLCA 2007
arxiv created 2007/01/04 · arxiv updated 2009/12/01
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.