Cut-elimination for knowledge logics with interaction

. In the article, multimodal logics K4n and S4n with the central agent axiom are analysed. The Hilbert type calculi are presented,then the Gentzen type calculi with cut are derived, and the proofs of the cut-eliminationtheoremsare outlined.The work shows thatit is possibleto constructan analyticalGentzen type calculi for these logics.


Introduction
In earlier research we analysed multimodal logic T n enriched with the central agent axiom. In [6] we showed that it is possible to construct an analytical calculus for this logic. In this arcticle we continue our work by presenting the same results for modal logics K4 n and S4 n with the central agent axiom.
We define a propositional formula in a standard recursive way, including operators ¬ (negation), ∨ (disjunction), ∧ (conjunction), ⊃ (implication) and a modal operator K l (meaning "agent l knows"). We say that l can be either c, meaning the central agent, or an agent number starting from 1. Capital Latin letters (A, B, . . . ) denote any modal logic formula, capital Greek letters ( , , 1 , * , ) denote a (possibly empty) multiset of modal formulas (the order of the formulas in a multiset does not matter). In the prooftrees we will also use capital Latin letters in square brackets ([P ], [Q]), which denote some prooftree.

Hilbert type calculus
First we define a Hilbert type propositional calculus. DEFINITION 2.1. The propositional Hilbert type calculus (HPC) consists of traditional postulati for propositional logic (see [5] or [2]).
Next, we extend the calculus to cover multimodal logics K4 n and S4 n . DEFINITION 2.2. The Hilbert type calculus for logic K4 n (HK4 n ) consists of all the rules and axioms of HPC and: • axiom k: • rule of necessity: A K l A , called K l , where l is any agent. DEFINITION 2.3. The Hilbert type calculus for logic S4 n (HS4 n ) consists of all the rules and axioms of HK4 n and axiom t: K l A ⊃ A, where l is any agent.
Finally we add the central agent axiom. DEFINITION 2.4. The Hilbert type calculus for logic K4 n (respectively S4 n ) with the central agent axiom (HK4 I n , respectively HS4 I n ) consists of all the rules and axioms of HK4 n (respectively HS4 n ) and the axiom K i A ⊃ K c A, where i is any agent, except the central one.

Gentzen type calculus with cut
We use the standard Gentzen type propositional calculus, which can be found in [4], but we do not include structural rules of exchange, because the order of the formulas in antecedent and succedent of a sequent is not important. Similarly we extend GPC to include rules for modal logics K4 n and S4 n . DEFINITION 3.2. The Gentzen type calculus for logic K4 n (GK4 n ) consists of an axiom A → A, all the rules of GPC and modal rule for the operator K l : where l is any agent. DEFINITION 3.3. The Gentzen type calculus for logic S4 n (GS4 n ) consists of an axiom A → A, all the rules of GPC and modal rules for the operator K l : where l is any agent. Then, to model the behaviour of the central agent axiom, we add the rule for the central agent: DEFINITION 3.4. The Gentzen type calculus with cut for logic K4 n (respectively S4 n ) with the central agent axiom (GK4 I n cut, respectively GS4 I n cut) consists of an axiom A → A, all the rules of GK4 n (respectively GS4 n ) and the rule of interaction: where i is any agent, except the central one.
In a traditional way (see [3]) we can prove the following: 2) all the rules of GS4 I n cut, except the cut rule; 3) the rule of interaction: where i is any agent, except the central one and K ? 1 consists of formulas, that begins with K l , but l can be different for different formulas in K ? 1 .
The proof of the cut-elimination theorem for logic S4 n is similar to the proof for logic T n (see [6]). At the begining we replace the cut rule by the mix rule: DEFINITION 4.2. The Gentzen type calculus with mix for logic S4 n with the central agent axiom (GS4 I n mix) is equivalent to GS4 I n cut, except that the cut rule is replaced by the mix rule: where * and * are obtained from and , respectively, by deleting all the occurrences of formula A (which is called the mix formula).
It is not hard to prove the equivalency:

LEMMA 4.3. A sequent is provable in GS4 I n mix, if and only if it is provable in GS4 I
n cut.

THEOREM 4.4 (the cut-elimination). A sequent is provable in GS4 I n , if and only if it is provable in GS4 I n mix.
Proof. The "only if" part is trivial. For the "if" part we analyse only those proofs, which have only one mix, occurring as their last inference. By induction on applications of the mix rule we can extend this reasoning to all the proofs. Than we define the height, the rank and the grade of the proof, which has only one mix, occurring as its last inference. We say that the height of such a proof will be larger by one than the sum of heights of the proofs of its left and right sequents. The rank of such a proof is the sum of ranks of the left and right sequents of the mix and the rank of the left (right) sequent is the maximum number of consecutive sequents in all the threads of the proof of that sequent, which contain the mix formula. The grade is the number of logical symbols in the mix formula. However an important difference from the logic T n case is that K i adds two to the grade of the mix formula, if i is any agent except the central one. So the grade of K i A is considered larger than the grade of K c A. For more formal definitions refer to [4].
Another difference from the logic T n case is that we need one additional lemma, which can be proved easily. LEMMA 4.5. If a sequent is provable in GS4 I n , then it is provable in GS4 I n without an application of the rule (→ c) to the formula K i A.
And the final difference is that we need three variables for induction. Firstly, we try to lower the grade of the proof. Secondly, we try to keep the grade unchanged and lower the rank of the proof. And finally, if other methods fail, we keep the grade and the rank unchanged and lower the height of the proof. The details of the proof are left to the reader.
The following theorem follows imediately: THEOREM 4.6. A sequent is provable in GS4 I n cut, if and only if it is provable in GS4 I n .

Gentzen type calculus without cut for K4 n
In this case the situation is very simillar to the logic S4 n case.
DEFINITION 5.1. The Gentzen type calculus without cut for logic K4 n with the central agent axiom (GK4 I n ) consists of: 1) axiom A → A; 2) all the rules of GK4 I n cut, except the cut rule; 3) the rules of interaction: where i is any agent, except the central one.
The proof of the cut-elimination for logic K4 n is similar to the proof for logic S4 n . We will show just one example. Suppose we have the proof, which has only one mix as its last inference. Suppose the rank of the right upper sequent of the mix is larger than 1. Then we must analyse all the possible forms of the mix formula. Suppose the mix formula is of the form K c B. And suppose that the last inference in the right upper sequent of the mix is an application of the (→ K ?, c ) rule. Then we must analyse all the possible last inferences in the left upper sequent of the mix. Suppose, it is an Other cases are left to the reader. The following theorem follows imediately from the latter proof.

THEOREM 5.2. A sequent is provable in GK4 I
n cut if and only if it is provable in GK4 I n .

Conclusions
In this paper, we showed that it is possible to find an analytical calculi for multimodal logics K4 n and S4 n with central agent axiom. However, further analysis could concentrate on other interaction rules (some of them are presented in [1]). What is more, the central agent axiom could have different properties in different logics (e.g., K45 n , KD45 n ) and consequently add different rules for cut-elimination.