Two complete ﬁnitary sequent calculi for reﬂexive common knowledge

. This paper discusses the use of complete sequent calculi for reﬂexive common knowledge logic. Description of language and complete inﬁnitary calculus for RCL is presented. Then ﬁnitary calculi RCL I and RCL L are introduced and completeness of ﬁnitary calculi RCL I and RCL L is proven.


Introduction
A reflexive common knowledge logic (RCL) containing individual knowledge operators, reflexive "common knowledge" and "everyone knows" operators is considered. Complete sequent calculi for reflexive common knowledge logic is discucced ,finitary calculi RCL I and RCL L are introduced and completeness of these calculi is obtained using completeness of the infinitary calculus for RCL.

Descpription of language and complete infinitary calculus for RCL
The language of considered RCL contains a set of propositional symbols P, P 1 , P 2 , . . . , Q, Q 1 , Q 2 , . . . the set of logical connectives ⊃, ∧, ∨, ¬; finite set of agent constants i, i 1 , i 2 , . . . ; multiple knowledge modality K(i), where i is an agent constant; everyone knows operator E; common knowledge operator C. A formula of RCL is defined inductively as follows: every propositional symbol is a formula; if A, B are formulas, then (A ⊃ B), (A ∧ B), (A ∨ B), ¬(A) are formulas; if i is an agent, A is a formula, then K(i)A is a formula; if A is a formula, then E(A) and C(A) are formulas. The operator K(i) behaves as modality of multi-modal logic K n [1].
The formula K(i)A means "agent i knows A". The formula E(A) means "every agent knows A", i.e. E(A) = ∧ n i=1 K(i)A (n is a number of agents). The formula C(A) means "A is common knowledge of all agents"; it is assumed that there is perfect communication between agents. The operator C and E behave as modalities of modal logic S5 . In addition these operators satisfy the following powerful properties: C(A) = A∧E(C(A)) (fixed point) and A∧C(A ⊃ E(A)) ⊃ C(A) (induction). Formal semantics of the formulas K(i), E(A), C(A) are defined as in the reflexive common knowledge logic [3].
Axiom: Γ, A → ∆, A. Rules consist of logical rules and modal ones. Logical rules consist of traditional invertible rules for logical symbols.
Modal rules: where m is number of agents.
It is known (see e.g. [3]) that calculus RCL ω is sound and complete.
The rule is derivable in RCL * where RCL * is obtained from RCL ω by dropping the rule (→ C ω ). Let Γ = A 1 , . . . , A n then derivability of (E) is carried out in the following way: It is easy to see that all rules of RCL ω , except (K i ) are invertible. Let us present a specialization of the rule (K i ) which is existential invertible.

Lemma 2.
By backward applications of rules, except (K i ) of RCL ω any sequent S can be reduced to a set of primary sequents S 1 , . . . , S n (n 1) such that if RCL ω ⊢S l then ∀l (l 1) RCL ω ⊢S l .
Proof. Follows from invertibility of rules RCL ω except (K i ).
Let RCL ′ ω be the calculus obtained from RCL ω replacing the rule (K i ) by the following one:

Lemma 3 [Existential invertability of the rule
be a primary sequent satisfying the condition of the conclusion of (K ′ l ) and let RCL ω ⊢ D S, then there exists a formula K s l (A) such that RCL ω ⊢Γ p → A. Proof. From Lemma 1 it follows that all axioms in D are atomic ones. Another hand, Σ 1 ∩ Σ 2 = ∅ therefore h(D) > 1. Therefore from the scope of the rule (K ′ i ) it follows that there exists a formula K s l (A) from the succedent of S such that RCL ω ⊢Γ p → A.

Finitary calculi RCL I and RCL L
Infinitary calculus RCL ω possesses the following beautiful property: it allows to present simple and evident completeness proof (see e.g. [3]). Despite of this property: all derivation containing infinitary rule (→ C ω ) are informal. To avoid this bad property several finitary complete sequent calculi for RCL can be presented [2].
(1) Calculus containing invariant-like rule. The finitary calculus RCL I is obtained from the calclus RCL ω replacing infinitary rule (→ C ω ) by following (cut) -like rule: where the formula I (called an invariant formula) is constructed from subformulas of formulas in the conclusion of the rule. There are some works in which constructive methods for finding invariant formulas in sequent calculi of epistemic logic are presented, e.g. [4,5]. Using these methods we can find invariant formulas and for the rule (→ C I ).
(2) Calculus containing weak-induction like rule and loop axiom. The finitary calculus RCL L is obtained from the calculus RCL I in the following way: (a) replacing the invariant rule (→ C I ) by the following rule: This rule corresponds to the so-called weak-induction axiom: A∧E(C(A)) ⊃ C(A).
(b) Adding loop-type axioms as follows: a sequent S ′ is a loop type axiom if (1) S ′ is above a sequent S on a branch of derivation tree, (2) S ′ is such that it subsumes S ′ (S ′ in notation), i.e. we can get S ′ from S using structural rules of weakening and contraction, in separate case S = S ′ .
(c) There is right premise of (→ C L ) between S and S ′ .
The completeness of finitary calculi RCL I and RCL L is obtained proving that the calculi RCL ω , RCL I and RCL L are equivalent to each other. The completeness of RCL ω is used.