# Intersection Types and Counting

## TL;DR

This paper introduces a novel type system approach to analyze the finiteness of trees generated by nondeterministic higher-order recursion schemes, linking tree properties to derivation properties for decision-making.

## Contribution

It presents a new type system that characterizes the finiteness of generated trees through derivation size, enabling decidability of the finiteness problem for HORSes.

## Key findings

- Type system accurately characterizes tree finiteness.
- Decidability of the finiteness problem for nondeterministic HORSes.
- Framework links tree properties to derivation properties for analysis.

## Abstract

We present a new approach to the following meta-problem: given a quantitative property of trees, design a type system such that the desired property for the tree generated by an infinitary ground lambda-term corresponds to some property of a derivation of a type for this lambda-term, in this type system. Our approach is presented in the particular case of the language finiteness problem for nondeterministic higher-order recursion schemes (HORSes): given a nondeterministic HORS, decide whether the set of all finite trees generated by this HORS is finite. We give a type system such that the HORS can generate a tree of an arbitrarily large finite size if and only if in the type system we can obtain derivations that are arbitrarily large, in an appropriate sense; the latter condition can be easily decided.