Contraction-free calculi for modal logics S5 and KD45

. It is known that termination and backtracking are among the most important problems in constructing derivations in non-classical logics. In this paper contraction-free and backtracking-free sequent calculi for modal logics S5 and KD45 are presented and founded. These are index-style sequent calculi with some speciﬁc indexed axioms. The main new ideas in considered calculi are to use metavariables (along with natural numbers) as elements of indexes and to numerate by diﬀerent natural number all the positive occurrences of modality 2 .


Introduction
A proof that suitable logical calculus allows us to get a decision procedure is crucial but it is not enough.Check of termination of a derivation as well as backtracking are very important problems.In [4] termination is ensured using the notion of history.Contraction-free calculi were constructed for intuitionistic logic in [3,5] and for modal logic S4 in [6], however only for sequents in Mints normal form.In [9] a contraction-free and backtracking-free calculus for S5 is presented, however the reflexivity rule of the calculus includes repetition of the main formula and the second part of adequateness theorem is not proven.
In this paper the transitive logics S5 and KD45 are considered.These (and their multimodal counterparts) are extremely important in various applications in computer science and artificial intelligence.S5 is used in reasoning about knowledge and KD45 is the base for BDI logics.
The aim of this paper is to construct contraction-free and backtracking-free sequent calculi for considered modal logics.To construct such calculi some index-style invertible calculi with some specific axioms are constructed.The main new points in considered index-style modal calculi are (1) using metavariables (along with natural numbers) as elements of indexes and (2) numerating all the positive occurrences of modality 2 by different natural numbers.Metavariables and natural numbers as indexes are used in a similar way in free-variable tableaux calculi in [2].The modal rules of constructed calculi are contraction free (they do not have duplication of the main formula in the premises), moreover new modal rules are invertible.

S5 Case
The definition of sequent calculus for S5 used here is based on [8], where the cut rule is used.Definition 1. Gentzen-style calculus GS5 for S5 is obtained from traditional invertible Gentzen-style calculus for classical propositional logic by adding the following rules for modality 2: It is obvious that the rule (Cut2) is necessary for derivability of some sequents, for example, P → 2¬2¬P .The necessity to duplicate the main formula in application of rule (2 →) can be demonstrated by sequent 2¬(P ∨ 2¬P ) → , which is valid in S5.

KD45 Case
Definition 2. Gentzen-style calculus GKD45 for KD45 is obtained from traditional invertible Gentzen-style calculus for classical propositional logic by adding the single rule: where Θ contains at most one formula.
The repetition of the main formula in the premise of the rule (2) is needed as can be demonstrated by sequent → 2(2P ⊃ P ), which is valid in KD45.
Calculus GKD45 is based on the one provided in [10].
It is well known that both considered Gentzen-style calculi are sound and complete.Moreover in both considered Gentzen-style calculi the (Cut) rule is admissible.It is easy to verify that both considered calculi contain non-invertible rule.
In this article notation A γ is used, where A is some formula, therefore it is important to define operations with indexes: (1) 2A, which can be part of succedent of any sequent of any possible derivation, must be numbered.However in order to make the numbering differ from formula indexes, all the positive occurrences of modality 2 are numbered instead.It should be stressed that in the presented calculi the numeration of positive occurrences of the modality 2 allows us to numerate in a new way succedental modal rule.Definition 3. The contraction-free and backtracking-free calculus CF S5 for modal logic S5 consists of traditional invertible propositional rules with single premise and the rules: Here γ is any index, if γ is simple metavariable, then γ ′ is new branching metavariable (the one, that is not part of the derivation yet), otherwise γ ′ = γ and α is a simple metavariable. 2The rule (2 α →) is called α-rule, and the rule (→ 2 l ) is called l-rule.
An axiom of the calculus is sequent Γ, p γ1 → p γ2 , ∆ and if one of the indexes γ 1 and γ 2 is natural number then either the other one is natural number and γ 1 = γ 2 , or the other one is a metavariable. 3Propositional symbols p γ1 and p γ2 are main formulas of an axiom.Furthermore, the traditional concept of sucesfull derivation is further limited by Condition ( * ).As usual, if at least one branch of the derivation is not an axiom and no rule of the calculus can be applied, then the derivation is not succesful.However, if every branch ends with an axiom, then the compatibility of those axioms must be checked.First of all, a pair of main formulas must be selected in every axiom.If p β1 and p β2 are the main formulas of some axiom, then it is said that β 1 and β 2 are equivalent and denoted β 1 ∼ β 2 .The relation ∼ is reflexive, symetric and transitive.Now, let S 1 and S 2 be two axioms of the same derivation.Additional requirements for compatibility apply only, if in both axioms one of the two main formulas is indexed by branching metavariable, another one by natural number and the branching metavariables of those axioms are equivalent.Otherwise, S 1 and S 2 are compatible without further check.Now let p β1 and p l1 be the main formulas of S 1 , q β2 and q l2 be the main formulas of S 2 and β 1 ∼ β 2 .Then S 1 and S 2 are compatible, if: Otherwise it is part of succedent of S 2 .And if q β2 is part of antecedent of S 2 , then it is part of antecedent of S 1 .Otherwise, it is part of succedent of S 1 .
The Condition ( * ) simply states that any two axioms must be compatible.Propositional rules (→ ∧), (∨ →) and (⊃→) have two presumptions and therefore are called the branching rules.It should be noted, that their only difference from the traditional ones is that if simple metavariable is part of the main formula, than it is changed to new branching metavariable in the premises.
Moreover, in the initial sequent of the derivation in CF S5 every positive occurence of 2 must be indexed.This ensures the correct application of the l-rule.
Using traditional proof-theoretical technique it can be proved that all the rules of CF S5 are invertible and that the (Cut) rule is admissible in CF S5 .Since modal rules do not contain duplication of the main formula, CF S5 is contraction-free.Moreover, because of ivertibility, the rules of CF S5 can be applied in any order and no backtracking is needed to check whether application of different rule to the same sequent would yield different result.Therefore, CF S5 is backtracking-free, however it should be stressed that some elements of backtracking are involved in axiom searches.Definition 4. The contraction-free and backtracking-free calculus CF KD45 for modal logic KD45 is the same as calculus CF S5 , except that one more restriction is added to the axiom.An axiom of the calculus is sequent Γ, p γ1 → p γ2 , ∆ and: 1.If one of indexes γ 1 and γ 2 is natural number then either the other one is number and γ 1 = γ 2 , or the other one is a metavariable.
2. If one of indexes γ 1 and γ 2 is metavariable then the other one is either natural number or metavariable.
It should be noticed, that the only difference between CF KD45 and CF S5 definitions is that sequents Γ, p α → p, ∆ and Γ, p → p α , ∆ (in this case α is either simple or branching metavariable) are axioms of CF S5 , but not axioms of CF KD45 .

Foundation of the calculi
Using traditional proof-theoretical technique (see e.g.[1,10]) we can prove that all structural rules (including (Cut)) are admissible in both CF S5 and CF KD45 .Using admissibility of (Cut), the following can be proved.

Lemma 1.
For each sequent S we have that if GI ⊢ S then CF I ⊢ S, where I ∈ {S5, KD45}.
To prove inverse statement let us introduce an auxiliary hybrid calculi CF ′ I , where I ∈ {S5, KD45} obtained from GI by adding the α-rule and branching rules of the calculus CF I , correspondingly to each logic I. However sequents are allowed to contain any indexes.Additional requirements for axiom of CF I and Condition ( * ) is also part of CF ′ I .It is easy to prove admissibility of (Cut) in CF ′ I . Let I ⊢ S and let the derivation of S be V .Let formula A be a member of S. We say that A is essential in V if at least one propositional symbol from A is main formula of an axiom of V .Proof.The lemma is proved by induction on the number of applications of the rule (→ 2 l ) in given the derivation of sequent S, using Lemma 2. ⊓ ⊔ From Lemmas 1 and 3 we get: Theorem 1.The calculi CF S5 and CF KD45 are sound and complete.

Lemma 2 [Replacing indexes by modality]. Let's say we have a sequent
Finally, the following theorem can be easily proved.
Theorem 2. The calculi CF S5 and CF KD45 are decidable.That is, for any sequent S it is possible to say that either S is derivable or not derivable after finite number of rule applications.