Propositional Dynamic Logic is Weaker without Tests

Abstract We show that Test-free Propositional Dynamic Logic (PDL 0 ) is less expressive than Propositional Dynamic Logic, i.e. we show that there is a formula of PDL using the conditional program operator? which is not equivalent to any formula of PDL using only the regular program operators concatenation, branching and iteration. Extensions of the problem are discussed in the conclusion.