A specialization of deﬁnitions in common knowledge logic

. It is known that one of main aims of specializations of derivations in non-classical logics is the various tools which allow us to simplify the searching of termination of derivations. The traditional techniques used to ensure termination of derivations in various non-classical logics are based on loop-checking. In this paper the reﬂexive common knowledge logic based on modal logic K 2 is considered. For considered logic a sequent calculus with specialized non-logical (loop-type) axioms is presented.


Introduction
The common knowledge logics are import class of non-classical logics and play a significant role in several areas of computer science, artificial intelligence, game theory, economics and etc.
Common knowledge logics are based on multi-modal logics extended with common knowledge operator. In this paper, a reflexive common knowledge logic (RCKL in short) based on multi-modal logic K 2 and reflexive common knowledge operator [3] is considered.
Common knowledge operator satisfies some induction like axioms. In derivation this induction-like tool is realized using loop-type axioms. Determination of these loop-type axioms involves creating a new "good loop" in contrast to "bad loops" and the new loop checking along with ordinary non-induction-type loop checking. Based on history method a method of determination of "good loops" for common knowledge logic is described in [1].
In this paper for reflexive common knowledge logic (based on multi-modal logic K 2 ), a sequent calculus with specialized non-logical (loop-type) axioms is presented. The specialization is achieved using some splitting rules.
Formulas of RCKL are defined as follows: every propositional symbol is a formula; The expression E(A) means "every agent knows A" and is used as abbreviation of formula . The formula C(A) means "A is common knowledge of all agents". The formula C(A) has the same meaning as the infinite conjunction Formal semantics of the formulas K l (A) and C(A) is defined as RCKL [3]. Hilbert-type calculus HRC is obtained from Hilbert-type calculus for propositional logic by adding the following postulates: (5) is called induction axiom. In [3] it is shown that the calculus HRC is sound and complete.
We consider two types of initial sequent calculi for RCKL.
1. Infinitary calculus G w RC is obtained from classical propositional calculus (with invertible logical rules) by adding the following rules: where K l Γ is empty or consists of formulas of the shape K l (B).
Analogously as in [2] one can prove that G w RC is sound and complete.
2. Loop-type calculus G L RC is obtained from the calculus G w RC by replacing the infinity rule (→ w ) by the following rule and adding loop-type axioms as follows: a sequent S is a loop-type axiom if S is above a sequent S ′ and on the same branch of a derivation, such that S ′ can be obtained from S using structural rules of weakening and contraction and there is the right premise of the rule (→ C) between S and S ′ .
Analogously as in [1] one can prove that G L RC is sound and complete. Therefore the calculi G w RC and G L RC are equivalent.
Let us introduce a canonical form of sequents. A sequent S is a primary sequent, if S is of the following shape Σ 1 , KΓ 1 , C∆ 1 → Σ 2 , KΓ 2 , C∆ 2 , where for every i (i ∈ {1, 2}) Σ i is empty or consists of propositional symbols; KΓ i is empty or consists of formulas of the shape K l (A) (l ∈ {1, 2}); C∆ i is empty or consists of formulas of the shape CA It is easy to see that bottom-up applying logical rules each sequent can be reduced to a set of primary sequents.
The primary sequent Σ 1 , KΓ 1 , C∆ 1 → Σ 2 , KΓ 2 is a K-primary one; the primary sequent Σ 1 , KΓ 1 , C∆ 1 → Σ 2 , C∆ 2 is a C-primary one. The primary sequent S is a non-splittable primary one if S is either a K-primary or C-primary sequent. Otherwise, the primary sequent S is a splittable primary one.
Let G S L RC be a calculus obtained from the calculus G L RC by following transformations: (1) Adding the following splitting rule where the conclusion of the rule (Sp) is splittable primary sequent; Σ 1 ∩ Σ 2 = ∅ (i.e. the sequent Σ 1 → Σ 2 is not an axiom); S 1 is K-primary sequent Σ 1 , KΓ 1 , C∆ 1 → Σ 2 , KΓ 2 ; S 2 is C-primary sequent; (2) Replacing a loop-type axiom by specialized loop-type axiom. A specialized loop-type axiom is a loop-type axiom, which is a C-primary sequent; (3) Using that the rule (Sp) is admissible in the calculus I ∈ {G w RC, G L RC} we can prove that the calculi G S L RC and I ∈ {G w RC, G L RC} are equivalent, therefore the calculus G S L RC is sound and complete.
In construction of derivation it is convenient to use the following (admissible in G S L RC) rule: Example 1. Let S = P, C(P ⊃ E(P )| → C(P ), K 1 (Q). Let's construct a derivation of S in G S L RC. Since S is splittable primary sequent let us backward apply to S the rule (Sp) and let us try to construct a derivation of C-primary sequent. S 1 = P 1 C(P ⊃ E(P )) → C(P ). S * 1 = P 1 C(P ⊃ E(P ) → C(P ) (E) P → P ; P 1 E(P ), E(C(P ⊃ E(P )) → E(C(P )) (⊃→) P 1 (P ⊃ E(P ), E(C(P ⊃ E(P )) → E(C(P )) (C →) P 1 C(P ⊃ E(P )) → P ; P, C(P ⊃ E(P )) → E(C(P )) (C →) S 1 = P 1 C(P ⊃ E(P )) → C(P ) Since S 1 = S * 1 , S * 1 is a loop type axiom. Therefore G S L RC ⊢ S.