Termination of Derivations in a Fragment of Transitive Distributed Knowledge Logic

Regimantas Pliuškevičius, Aida Pliuškevičienė
2008 Informatica  
A transitive distributed knowledge logic is considered. The considered logic S4nD is obtained from multi-modal logic S4n by adding transitive distributed knowledge operator. For a fragment of this logic loop-check-free sequent calculus is proposed. The considered fragment is such that it can be applied for specification and verification of safety properties of knowledge-based distributed systems. By relying on the constructed loop-check-free sequent calculus a PSPACE procedure to determine a
more » ... mination of backward derivation in considered fragment of the logic S4nD is presented. Keywords: logic of knowledge, distributed knowledge, safety and liveness properties of distributed systems, deduction-based decision procedure, sequent calculus, loop-check, backtracking, PSPACEcomplexity. Hilbert-Style and Gentzen-Style Calculi for the Logic S4 n D The logic S4 n D is obtained from the multi-modal logic S4 n by adding distributed knowledge operator D (see, e.g., (Fagin and Vardi, 1986; Fagin et al., 1992; Halpern and Moses, 1992; Fagin et al., 1995; Meyer and van der Hoek, 1995) ). A language of this logic contains: a set of propositional symbols P, P 1 , P 2 , . . . , Q, . . , n}); a set of knowledge operators K 1 , K 2 , . . . , K n ; the distributed knowledge operator D; the set of logical connectives ⊃, ∧, ∨, ¬. Formulas are defined in traditional way from propositional symbols using logical connectives, knowledge operators K i , i ∈ {1, . . . , n}, and distributed knowledge operator D. An atomic formula is defined as any propositional symbol. Each formula different from atomic formula is called a non-atomic one. A formula of the shape QA ( Q ∈ { K i , D}) is called a modal one. The formula K i A means "agent i knows A". The formula DA means "A is distributed knowledge of all set (group) of agents". Distributed knowledge is the knowledge that is implicitly present in a group of agents, and which might become explicit if the agents were able to communicate. For instance, it is possible that no agent knows the assertion Q, while at the same time DQ may be derived from K 1 P ∧ K 2 (P ⊃ Q). We have distributed knowledge of Q if, putting our knowledge together, Q may be deduced, even if none of us individually knows Q. Knowledge operators K i , i ∈ {1, . . . , n}, and distributed knowledge operator D for the logic S4 n D satisfy relations which comply with reflexivity and transitivity properties. The semantics of knowledge operators K i and distributed knowledge operator D is defined using Kripke structure (see, e.g., (Fagin and Vardi, 1986; Fagin et al., 1992; Halpern and Moses, 1992; Fagin et al., 1995; Meyer and van der Hoek, 1995) ). Let us remind the semantics of the formulas K i A and DA. Let S be a set of "states", s, t be some states from S, Π be an interpretation function, R i (i ∈ {1, . . . , n}) be the possibility (accessibility) relations, satisfying reflexivity and transitivity properties. Then in Kripke structure Following (Fagin et al., 1992) the intuition behind the definition of semantics of the formula DA is that if all the agents could "combine their knowledge" the only worlds they would consider possible are precisely those in the intersection of the set of worlds that each one individually considers possible. A Hilbert-style calculus HS4 n D for the logic S4 n D is defined by the following postulates:
doi:10.15388/informatica.2008.232 fatcat:tbpew6tncbcwvf6i2qcrt6pm6m