Unboundedness for Recursion Schemes: A Simpler Type System

TL;DR

This paper introduces a simpler, more intuitive type system to decide unboundedness in higher-order recursion schemes, effectively handling safe schemes and extending to simultaneous unboundedness, with an implemented automatic checker.

Contribution

A new simplified type system for unboundedness in recursion schemes, applicable to safe schemes and extendable to simultaneous unboundedness, with an automatic checking algorithm.

Findings

The type system effectively solves unboundedness for safe recursion schemes.

The algorithm automatically checks unboundedness efficiently.

Soundness is established for unsafe schemes, with completeness remaining open.

Abstract

Decidability of the problems of unboundedness and simultaneous unboundedness (aka. the diagonal problem) for higher-order recursion schemes was established by Clemente, Parys, Salvati, and Walukiewicz (2016). Then a procedure of optimal complexity was presented by Parys (2017); this procedure used a complicated type system, involving multiple flags and markers. We present here a simpler and much more intuitive type system serving the same purpose. We prove that this type system allows to solve the unboundedness problem for a widely considered subclass of recursion schemes, called safe schemes. For unsafe recursion schemes we only have soundness of the type system: if one can establish a type derivation claiming that a recursion scheme is unbounded then it is indeed unbounded. Completeness of the type system for unsafe recursion schemes is left as an open question. Going further, we discuss an extension of the type system that allows to handle the simultaneous unboundedness problem. We also design and implement an algorithm that fully automatically checks unboundedness of a given recursion scheme, completing in a short time for a wide variety of inputs.