Deterministic dynamic logic is strictly weaker than dynamic logic

There is a language L and structures A1 and A2 for L such that, for each closed formula F of deterministic regular dynamic logic, the formula F is valid in A1 if and only if F is valid in A2. There is, however, a closed formula of nondeterministic regular dynamic logic is both valid in A1 and not valid in A2. Thus, nondeterminism adds to the expressive power even in the presence of quantifiers. This answers Meyer's question. Moreover, the proof here, unlike that of Berman, Halpern, and Tiuryn (1982, in “Automata, Language, and Programming,” Springer, Berlin), holds in the presence of first-order tests as well as quantifier-free tests