Proc. CS&P '06 Combining Propositional Dynamic Logic with Formal Concept Analysis (extende

Combining Propositional Dynamic Logic

with Formal Concept Analysis

(extended abstract)

N.V.Shilov,N.O.Garanina,and I.S.Anureev

A.P.Ershov Institute of Informatics Systems,

Lavren’ev av.,6,Novosibirsk630090,Russia,

{shilov,garanina,anureev}@iis.nsk.su,

Abstract.Propositional Dynamic Logic(PDL)has been introduced by

M.J.Fischer and dner as an extension of the classical proposi-

tional logic and propositional modal logic K for reasoning about par-

tial correctness of structured nondeterministic programs.Many variants

of PDL have been studied extensively especially from viewpoint of de-

cidability and axiomatizability.In particular,recently C.Lutz and D.

Walther proved that PDL with complement(negation)of atomic pro-

grams 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 im-

plies decidability of PDL extended by extent of atomic programs with ex-

ponential 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 comple-

ment.

This research is supported in parts by joint grant RFBR05-01-04003-a-DFG project COMO,GZ:436RUS113/829/0-1,by grant RFBR06-01-00464-a,and Integration Grant n.14Siberia Branch,Russian Academy of Science.

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