# Fooling the Parallel Or Tester with Probability $8/27$

## TL;DR

This paper investigates a probabilistic version of PCF and demonstrates that the maximum success probability of fooling the parallel or tester is exactly 8/27, achieved by a simple probabilistic program.

## Contribution

It introduces a probabilistic variant of PCF and establishes a precise success probability bound for fooling the parallel or tester, extending understanding of operational semantics in probabilistic languages.

## Key findings

- Maximum success probability is exactly 8/27.
- A simple probabilistic program achieves this bound.
- The bound cannot be exceeded.

## Abstract

It is well-known that the higher-order language PCF is not fully abstract: there is a program - the so-called parallel or tester, meant to test whether its input behaves as a parallel or - which never terminates on any input, operationally, but is denotationally non-trivial. We explore a probabilistic variant of PCF, and ask whether the parallel or tester exhibits a similar behavior there. The answer is no: operationally, one can feed the parallel or tester an input that will fool it into thinking it is a parallel or. We show that the largest probability of success of such would-be parallel ors is exactly $8/27$. The bound is reached by a very simple probabilistic program. The difficult part is to show that that bound cannot be exceeded.