A Note on “Disjunctive” Predicates

During my talk yesterday, somebody raised an important point when I discussed the Hoare rule for non-deterministic choice in the recursive semantics and claimed that the postcondition couldn’t be disjunctive: What if it was the negation of a conjunction?

I think I waved this away with a gesture towards the constructive Coq proof assistant, but that was in error (the Coq real number library actually posits the decidability of the reals).

In truth, propositions of the form A \vee B \neg (A \wedge B), or even Pr(b) \neq p are excluded from the post-condition too. The question at hand is to identify the class of propositions P such that P(\Theta_1) and P(\Theta_2) implies P(\Theta_1 \oplus_p \Theta_2).

In our expanded VPHL paper, we explicitly limited the propositions we were considering to those of the form Pr(b) \; \Box \; p, for \Box \in \{ \textless, \le, =, \ge, \textgreater \} (specifically excluding negation and \neq), and were therefore able to prove the following lemmas (from which the Hoare rule I introduced at PPS followed easily):

Lemma 4.1 For any non-disjunctive assertion P, P(\Theta_1) \wedge P(\Theta_2) implies that P(\Theta_1 \oplus_p \Theta_2) for any p \in (0,1).

Lemma 4.2 For any non-probabilistic assertion P, P(\Theta_1 \oplus_p \Theta_2) implies P(\Theta_1) and P(\Theta_2) for any p \in (0,1).

But I’d be interested in the broader case: Has any work been done on showing what class of mathematical propositions are closed under the combination of distributions?

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