Combining Propositional Dynamic Logic with Formal Concept Analysis?

Propositional Dynamic Logic (PDL) has been introduced by M.J. Fischer and R.E. Ladner as an extension of the classical propositional logic and propositional modal logic K for reasoning about partial correctness of structured nondeterministic programs. Many variants of PDL have been studied extensively especially from viewpoint of decidability and axiomatizability. In particular, recently C. Lutz and D. Walther proved that PDL with complement (negation) of atomic programs is decidable in exponential time (while it is well known that in general case PDL with complement is undecidable). In this paper we introduce and study a new variant of PDL extended by modalities inspirited by Formal Concept Analyses (FCA) of R. Wille and B. Ganter. Formal semantics of these two new modalities is intent and, respectively, extent operations (two major algebraic constructions of FCA). Informal meaning and utility of these modalities can be illustrated by the following simple example. Assume that P and G are sets of states of a computer system, A is a binary relation between states: xAy means that y is reachable from x. Then sentence ‘From every state in P the system can reach some state in G’ can be expressed in PDL by P → 〈A〉G. Another sentence ‘From any state in P the system can not reach any state outside G’ can be expressed in PDL also as P → [A]G. But sentences ‘From every state in P the system can reach every state in G’ and ‘Every state in G is reachable by the system from every state in P ’ can not be expressed in PDL, but can be expressed in PDL extended by new modalities for intent and extent. We call PDL extended by new modalities for intent and extent by PDL for FCA. We denote this logic by PDL/FCA. We demonstrate in this paper that PDL/FCA is more expressive than PDL. Then we interpret a fragment of PDL/FCA without intent in PDL with complement. It implies decidability of PDL extended by extent of atomic programs with exponential upper bound. It remains an open problem whether PDL/FCA (without any restriction for intent and extent) is decidable and what the expressive power of this new logic is with respect to PDL with complement. ? This research is supported in parts by joint grant RFBR 05-01-04003-a DFG project COMO, GZ: 436 RUS 113/829/0-1, by grant RFBR 06-01-00464-a, and Integration Grant n.14 Siberia Branch, Russian Academy of Science. Proc. CS&P '06