A method of marks and indices for linear modal logic

In the paper a method to check termination of history-free proof for linear modal logic S4.3 is proposed. This method improves the method proposed by the authors for modal logic S4. Analogously as for S4, instead of history we use marks and indices that allow us to eliminate loop checking. The method proposed in this paper specifies some kind of formulas which allow us to check termination of derivations in more effective way in comparison with S4.


Introduction
In [3] the notion of history to ensure termination of derivations in some non-classical logics was introduced. The history allows us to achieve efficient loop checking by means of an information about previous parts of a derivation. The history based method nowadays is widely used constructing derivations in non-classical logics. In [4] a method called marks and indices method (denoted M&I ) for modal logic S4 was proposed. In M&I , instead of history marks and indices that allow us to eliminate loop checking are used. In the present paper an improved version of M&I method for linear modal logic S4.3 is described. The logic S4.3 is obtained from modal logic S4 adding the linearity axiom ( A ⊃ B) ∨ ( B ⊃ A). The logic S4.3 is interpreted as discrete linear time logic. The aim of this paper is to construct an invertible sequent calculus for modal logic S4.3 without loop checking changing and extending the technique from [4].

Invertible calculus with specialized reflexivity rule
Formulas in the considered calculus are constructed in traditional way from propositional symbols using the classical logical connectives and the necessity modality . Along with modality a marked modality * is introduced. This marked modality has the same semantical meaning as non-marked modality and serves as a device to restrict a backward application of reflexivity rule. A formula of the shape A is called a modal one. The language considered does not contain the modality ♦ assuming that ♦ A = ¬ ¬A. We consider sequents, i.e., formal expressions For simplicity we consider sequents not containing branching formulas (see, e.g., [2]).
A sequent S is a primary one if (1) S has the shape 1 , * → 2 , , where i (i ∈ {1, 2}) is empty or consists of propositional symbols; * is empty or consists of formulas of the shape * M; is empty or consists of formulas of the shape M, and (2) antecedent and/or succedent of the sequent does not contain several occurrences of the same formula.
Cut-free sequent calculus with specialized reflexivity rule GS4.3 for modal logic S4.3 is defined by the following postulates (see, e.g., [1]): Axiom: , P → , P where multiset is permitted to contain some formulae of the shape * B, i.e., modality can be marked.

Modal rules:
where in the conclusion of the rule the outmost occurrence of modality in the formula A is not marked but some of modal formulas from can be marked. The rule ( * →) is called reflexivity rule because it corresponds to reflexivity axiom A ⊃ A.
where the conclusion of the rule is a primary sequent such that 1 ∩ 2 is empty. The rule ( ) is called linearity rule because it corresponds to linearity axiom From [1] it follows that the calculus GS4.3 is sound and complete. Using traditional proof-theoretical methods we get that each rule of the calculus GS4.3 is invertible in GS4.3.

Loop-check-free calculus for S4.3
To construct backward proof search without loop checking a notion of an indexed modality is introduced and sequents containing occurrences of the indexed modality are considered. Let us introduce the following indexation technique.
A positive occurrence of modality in a sequent S is a special one if it occurs within the scope of a negative occurrence of modality in S. A special occurrence α of modality in a sequent is an isolated one if within the scope of α there is a negative occurrence of modality . We distinguish two sorts of isolated occurrences of the modality. An isolated occurrence α of modality in a sequent is strongly special if within the scope of α there are no isolated occurrences of . A special occurrence of which is not strongly special is simply special.
Let us introduce two sorts of indexes used only for special occurrences of modality, namely, index i where i ∈ {1, . . . , n} and n is the number of simply special occurrences of modality in a sequent, and index •k, where k ∈ {1, . . . , m} and m is the number of strongly special occurrences of modality in a sequent. The modality σ (σ ∈ {i, •k}) is an indexed modality.
For example, let S = ¬ (¬ Q∨ ( ¬ ¬P ∨ ¬ ¬ P )) → , then S ind has the shape ¬ 1 (¬ Q ∨ ( •1 ¬ ¬P ∨ 2 •2 ¬ ¬ 3 P )) → ; the occurrences of 1 , 2 , •1 , •2 in S ind are isolated ones, and the occurrence of 3 in S ind is not isolated one; there are no isolated occurrences of the modality within the scope of the occurrences of •1 , •2 , but there are isolated occurrences of the modality within the scope of the occurrences of 1 , 2 .
Along with the marked modality * (introduced in the previous section and used only for negative occurrences of modality) let us introduce one more marked modality, namely, + . The marked modality + serves as a stopping device for a backward application of the linearity rules. The marked and indexed modalities have the same semantical meaning as non-marked and non-indexed modality . Let A be a formula from a sequent S, then an indexed formula A ind is a formula obtained from A by replacing any simply (strongly) special occurrence of in A by the indexed modality i ( •k , correspondingly) in such a way that different special occurrences of get different indices. Let S be a sequent, then an indexed sequent S ind is a sequent obtained from S by replacing every formula in S by appropriate indexed formula in such a way that different special occurrences of in an indexed sequent S ind get different indices.
A simply special occurrence of modality , i.e., indexed modality of the shape i , in S ind is dependent if within the scope of i there is at least one occurrence of some indexed modality σ (σ ∈ {i, •k}). In opposite case the occurrence of i in S ind is independent.
For example, let S = ¬ ( P ∨ ¬ P ) → , then S ind has the shape ¬ 1 2 ( 3 P ∨ 4 •1 ¬ P ) → ; the occurrence of 3 in S ind is independent one, and occurrences of 1 , 2 , 4 in S ind are dependent ones.
Let us introduce an operation σ + (σ ∈ {i, •k}). Let A be any indexed formula from an indexed sequent S ind . Then application of the operation σ + to A is denoted as A σ + and the result of this application is a formula obtained from A by replacing the occurrence of σ in A by marked modality + . If A does not contain occurrences of σ then A σ + = A. The notation σ + means A σ + 1 , . . . , A σ + k , where k 1 and is a sequence of indexed formulas A 1 , . . . , A k .
Let us note that only positive occurrences of the modality may get indexes or the mark + and only negative occurrences of may get the mark * .
Let us introduce the following notions which allow us to check termination of derivations in more effective way in comparison with checking for S4 described in [4].
Let B be a formula entering in a sequent S. A subformula of B is a modal one if it has the shape µ M, where µ ∈ {∅, i, •k, +, * }.
A modal formula B is a passive formula if • B occurs in a sequent S positively and has the shape i 1 . . . i n M (n 1), where M is a formula containing at least one occurrence of index-free modality (probably, marked modality) and does not contain any occurrences of indexed modality; B is called a passive formula of the first type; • B occurs in a sequent S positively, has the shape τ 1 . . . τ n M (n 1), τ j ∈ {i, +}, j ∈ {1, . . . , n} and there exists j such that τ j = +; B is called a passive formula of the second type; • B occurs in a sequent S negatively and has the shape * m times . . . M (m 0), where M is a formula composed of the first and/or the second kind passive formulas using logical symbols; B is called an passive formula of the third type. Any modal formula that is not passive one is active formula. For example, let S be a sequent * ¬ 1 P , * ¬ 2 + Q → 1 P , 2 + Q, 4 ( + P 1 ∨ + Q 1 ), R. Then the formula 4 ( + P 1 ∨ + Q 1 ) is the passive formula of the first kind; the formula 2 + Q is the passive formula of the second kind; the formula * ¬ 2 + Q is the passive formula of the third kind. Formulas * ¬ 1 P , 1 P and R are active formulas. An indexed sequent S is a primary one if (1) S has the shape 1 , * → 2 ,˜ , where i (i ∈ {1, 2}) is empty or consists of propositional symbols; * is empty or consists of formulas of the shape * M;˜ is empty or consists of formulas of the shape µ M (µ ∈ {∅, i, •k, +}, and for any formulas A and B, if σ A ∈˜ (σ ∈ {i, •k}) and A = B then for the same index σ, σ B / ∈˜ , and (2) antecedent and/or succedent of the sequent S does not contain several occurrences of the same formula.
Taking into account the introduced notions of active and passive formulas let us specify the shape of the succedent part of a primary sequent. Namely, the part˜ of primary sequent has the shape ∇, λ , + where ∇ is empty or consists of active non-indexed formulas, λ is empty or consists of active indexed formulas, and + is empty or consists of the first and/or the second kind passive formulas.
Let G 1 S4.3 be a calculus obtained from the calculus GS4.3 replacing the rule ( ) by the following linearity rule: where the conclusion is a primary sequent such that 1 ∩ 2 is empty, + is empty or consists of passive formulas of the first or second type; σ 1 A 1 , . . . , σ j A j , . . . , σ n A n , where σ j ∈ {∅, i, •k} (1 j n), consists of active formulas; σ in the notation of the rule ( σ p ) denotes the sequence σ * 1 , . . . , σ * j , . . . , σ * n , where σ * j ∈ {∅, σ j , σ j +}. For every j (j ∈ {1, . . . , n}), the shape of the j th premise of this rule and the meaning of σ * j in σ depend on the shape of the j th main formula σ j A j in the conclusion of this rule. For the sake of simplicity, we can imagine that each premise of the rule ( σ p ) is obtained applying one-in-three following rules depending on the shape of the main formula: Non-indexed rule: Weak indexed rule: where in the conclusion of this rule contains a dependent occurrence of i or contains an occurrence of •k for some k. Strong indexed rules: where λ ∈ {i, •k} and if λ = i then in the conclusion of this rule contains an independent occurrence of i and does not contain an occurrence of •k for some k, i.e., conditions indicated in the rule ( i p ) does not hold. It is important that, as it follows from the shape of the linearity rule ( σ p ), this rule satisfies the following conditions: • the passive formula can not be the main formula of the linearity rule ( σ p ) and passive formulas entering in the conclusion of the rule are not preserved in any premise.
• if the j th main formula of the linearity rule ( σ p ) is an indexed formula σ j A j such that σ j = i (but not σ j = •i) and in the conclusion of this rule contains a dependent occurrence of i or contains an occurrence of •k for some k, then in the premise S j the operation σ j + is not applied to in * .

Example 1. (a) Let S be the indexed sequent of the shape
to S we get two premises: The sequent S 1 is the weak indexed premise and S 2 is the non-indexed premise.
(b) Let S be an indexed sequent of the shape is the passive formula of the third type and 1 + (P ∨ Q) is the passive formula of the second type, backward applying ( 3+ p ) to S we get the strong indexed premise ¬ + (R ⊃ R), * ¬ + (R ⊃ R) → (R ⊃ R).
From the shape of the linearity rule ( σ p ) it follows that there is the one way to construct the premises of this rule. From this fact we get that the rule ( σ p ) is invertible. A primary sequent of the shape 1 , * → 2 , + , where 1 ∩ 2 is empty and * ( + ) is empty or consist of formulas of the shape * M (passive formulas of the first and/or second type, correspondingly), is a final one. It is impossible to apply any rule to a final sequent.
A derivation V of a sequent S in the calculus G 1 S4.3 is a successful one, if each branch of V ends with an axiom. In this case a sequent S is derivable in G 1 S4.3. A derivation V of S in the calculus G 1 S4.3 is an unsuccessful one if V contains a branch ending with a final sequent. In this case a sequent S is non-derivable.
Let us note that in calculus G 1 S4.3 derivation of indexed sequent is constructed and indexed end-sequent S ind of a derivation is obtained from arbitrary sequent S which does not contain any indices and marks. Thus, end-sequent S ind does not contain marked modalities * , + .
Since using invertibility of the rules of G 1 S4.3 and technique from [4] we can prove that the calculi GS4.3 and G 1 S4.3 are equivalent, we get THEOREM 1. The calculus G 1 S4.3 is sound and complete.
Analogously as in [4] we can show that complexity of each sequent constructing backward derivation of any indexed sequent S in G 1 S4.3 decreases. Thus, backward proof search in G 1 S4.3 terminates.