On Axiomatic Products of PDL and S5: Substitution, Tests and Knowledge∗

Propositional dynamic logic (PDL for short) [5] is an expressive, powerful and convenient logical tool to reason about programs or actions. It has found applications in various fields of computer science, which range from program verification to multi-agent systems. In computer science applications, the modal logic S5 is generally accepted as an adequate representation of the notion of knowledge. For example, in agent based applications the S5 formula φ can be read to mean ‘the agent knows φ’. Thus, combinations of PDL and S5 are meaningful when we want to reason about dynamic and epistemic information. In modal logic different forms of combinations of logics have been investigated. The simplest form of combination of two (or more) logics is their fusion, or independent join. It is well-known [9, 6] that fusions of logics inherit many of the good properties of the individual logics, including soundness, completeness, the finite model property and decidability. Another form of combination of two logics is their product. With products the situation is more varied and complicated than with fusions. First of all, products can be defined in two ways: axiomatically and semantically [3]. Whereas in fusions there is no interdependence between the operators of the different modal dimensions, in products the modal operators are commuting. This complicates matters, so that for products there is no preservation theorem of the generality as for fusions. In fact, the particular type of interaction between the