Probabilistic Termination by Monadic Affine Sized Typing (Long Version)

TL;DR

This paper presents a novel type system using monadic affine sized types that can type probabilistic higher-order programs, ensuring they terminate almost surely, and extends traditional typing methods with quantitative and affinity constraints.

Contribution

It introduces a new type system that generalizes sized types for probabilistic programs, enabling almost sure termination proofs for complex higher-order examples.

Findings

Successfully types probabilistic programs like random walks

Extends reducibility techniques for probabilistic termination proofs

Demonstrates the system's power in capturing almost sure termination

Abstract

We introduce a system of monadic affine sized types, which substantially generalise usual sized types, and allows this way to capture probabilistic higher-order programs which terminate almost surely. Going beyond plain, strong normalisation without losing soundness turns out to be a hard task, which cannot be accomplished without a richer, quantitative notion of types, but also without imposing some affinity constraints. The proposed type system is powerful enough to type classic examples of probabilistically terminating programs such as random walks. The way typable programs are proved to be almost surely terminating is based on reducibility, but requires a substantial adaptation of the technique.