Sequent calculus for logic of correlated knowledge

Sound and complete sequent calculi for general epistemic logic and logic of correlated knowledge are presented in this paper.


Introduction
Logic of correlated knowledge is obtained from epistemic logic by adding observational capabilities to agents. Traditionally, agents can perform logical inference, positive and negative introspection and their knowledge is truthful. Applications of epistemic logic cover fields such as distributed systems, knowledge base merging, robotics or network security in computer science and artificial intelligence. By adding observational capabilities to agents, in addition we can apply it for systems, where observations or measurements can be performed and results obtained.
Alexandru Baltag and Sonja Smets introduced general epistemic logic (GEL), logic of correlated knowledge (LCK) and Hilbert style proof systems for them in [1]. In this paper sequent calculi are presented for GEL and LCK to make possibilities for mechanizing proofs. We are using the ideas of semantic internalization of Sara Negri suggested in [3] and [2].
We'll start by defining syntax and semantics of general epistemic logic in Sections 1.1 and 1.2, then introduce Hilbert style proof system for GEL in Section 1.3 and present sound and complete sequent calculus for general epistemic logic in Section 1.4. In Section 2, we'll do the same for the logic of correlated knowledge.

Syntax
General epistemic logic is generalized epistemic logic, where knowledge operator K I is used for both individual agents and groups of agents. In traditional epistemic logic we have formulas like K i1 A, where i 1 represents agent. In GEL the corresponding formula is K {i1} A. When I = {i 1 , i 2 , i 3 }, we write K {i1,i2,i3} or K I A. Syntax of GEL is defined as follows: Definition 1 [Syntax of GEL]. The language of the logic GEL has the following syntax: where p is any atomic proposition and I ⊆ N , N = {i 1 , i 2 , i 3 , . . . , i n }.

Semantics
Imagine the system composed of components or locations. Agents can be associated to locations, where they perform observations. If observations are the same in different states, we have equivalence relation between them. Formally, it is written as:

Definition 2 [Observational equivalence].
If s and s ′ are two possible states and group of agents I can make exactly the same observations in these two states, then the states are observationally equivalent for I. We write this as s Semantics of general epistemic logic is multi-modal Kripke frames, where relations between states signify equivalence of observations of agent groups. • Observability principle: if s Satisfaction relation |= between states of S and formulas of GEL is defined inductively in a standard way. In particular, for formula K I A we have: K I A means that the group of agents I carries the information that A is the case.

Hilbert style calculus HS-GEL
Hilbert style proof system of general epistemic logic has the following axioms and rules: • Any axiomatization for propositional logic.
• Rules: where I, J ⊆ N and "→" stands for implication.
Alexandru Baltag and Sonja Smets have proved soundness and completeness of this calculus with respect to general epistemic frames in [1].

Gentzen style calculus GS-GEL
Using the ideas of semantic internalization of Sara Negri [3], we construct sequent calculus for general epistemic logic.
• Logical rules: • Knowledge rules: Rule (RK I ) requires, that I = N , I = ∅ and t is not in the conclusion.
• Rules for accessibility relations of reflexivity, emptiness, transitivity, euclideanness and monotonicity: Rule (M on) requires, that I ⊆ J.
• Structural rules are as in [2].
Γ, ∆ in the sequents are finite, possibly empty, multisets of labeled formulas s : A and relational atoms s I ∼ t. Labels s, t are ranging in the set of states S and s : A stands for s |= A.
Using soundness of axioms and rules of GS-GEL and completeness of HS-GEL, the following theorem can be proved.
Associating observations to results r ∈ R, new atomic formulas o r are obtained. Formally we have:

Definition 4 [Syntax of LCK].
The language of the logic LCK has the following syntax: where p is any atomic proposition and I ⊆ N , o = (o i ) i∈I ∈ O I , r ∈ R.

Semantics
States of models of LCK are functions s : O i1 × · · · × O in → R or s I : O I → R, where set of results R is in structure (R, Σ) together with abstract operation Σ : P(R) → R of composing results. For every e ∈ O I , local state s I is defined as:

Gentzen style calculus GS-LCK
Sequent calculus for logic of correlated knowledge contains: • Axioms: