Fooling the Parallel Or Tester with Probability 8/27
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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.
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