Termination of derivations for minimal tense logic

It is known that loop checking and backtracking are extensively used in various non-classical logics. An efficient loop checking is obtained using a technique based on histories. In the paper a method for elimination of loop checking in backward proof search for minimal tense logic Kt is proposed. To obtain termination of derivation indices and marks are used instead of history.


Introduction
In the paper minimal tense logic K t is considered. Minimal tense logic K t was extensively studied by various authors. In [4] Hilbert-style axiomatization of K t was described, completeness and finite model property (decidability) were proved. In [5,3] Gentzen-style calculi with some restricted cut rule for K t are presented. In [2] a graph-style calculus for K t and efficient loop-check with histories is described. In [1] a labeled sequent calculus with histories is presented. Since the logic K t can be regarded as some prototype for temporal logic with past operator and for dynamic logic with inverse operator, this logic is interesting to Computer Science and Artificial Intelligence.
It is known that constructing derivations loop checking and backtracking are extensively used in various non-classical logics. An efficient loop checking is obtained using a technique based on histories. In the paper indexed loop-check-free and cut-free invertible sequent calculus for K t is presented. Instead of histories marking of modal operators and indexation of propositional variables are used. The proposed technique is based on invertible and decidable sequent calculus which has some similarity with Kanger-style invertible indexed calculus for modal logic S5. The soundness, completeness, termination and decidability of constructed calculus are stated.

Initial Hilbert-style and Gentzen-style calculi for K t
A language of K t consists of a set of propositional symbols, logical symbols ⊃, ∧, ∨, ¬ and two modalities 1 and 2 . Formulas are constructed in traditional way from propositional symbols using logical symbols and modalities i (i ∈ {1, 2}). The language does not contain the modality ♦ i (i ∈ {1, 2}) assuming that ♦ i A = ¬ i ¬A. The modality 1 ( 2 ) may be read as "at all future times" ("at all previous times", correspondingly). Sequent is a formal expression A 1 , . . . , A k → B 1 , . . . , B m , where A 1 , . . . , A k (B 1 , . . . , B m ) is a multiset of formulas.
A Hilbert-style calculus H K t for K t is obtained from Hilbert-style calculus H for classical propositional logic by adding the following postulates: and the rule of inference A/ A ( ).
In [4] it is proved that H K t is sound and complete. A Gentzen-style calculus GK t for K t is obtained from Gentzen-style calculus G for classical propositional logic by adding the following rules: i.e., the calculus GK t is sound and complete.

Cut-free sequent calculus for K t
To obtain cut-free sequent calculus for K t along with the set of propositional symbols we introduce a set of indexed propositional symbols of the shape P γ where γ is a list (possibly empty) of the shape σ 1 a 1 , σ 2 a 2 , . . . , σ n a n , where σ i ∈ {+, −} and a i is a parameter.
A cut-free sequent calculus G 1 K t for K t is obtained from GK t replacing the modal rules ( i ), (→ i ) and (cut i ) where i ∈ {1, 2} by the following rules: where i ∈ {1, 2} and if i = 1 then σ = + else σ = −; in the rule (→ i ) b does not enter in the conclusion; in the rule ( i →) α is metaparameter such that if the conclusion does not contain any indices, then value of α is an arbitrary parameter otherwise value of α is chosen from parameters entering in the conclusion.
The cut rule is admissible in G 1 K t .
THEOREM 3. All rules of the calculus G 1 K t are invertible in G 1 K t .
Relying on Theorem 1 we get THEOREM 4. The calculus G 1 K t is a conservative extension of GK t , i.e., the calculus G 1 K t is sound and complete. Remark 1. The duplication of the main formula in the premise of the rule ( i →) is necessary. The calculus G d 1 K t obtained from the calculus G 1 K t replacing the rule ( i →) by the rule ( d i →) in premise of which the main formula is not preserved is incomplete. Let S = 1 (P ∧ ¬ 1 P ) → . Then it is easy to verify that G 1 K t S but G d 1 K t S. The sequent S is derivable in the defined in [3] calculus GK t using cut rule.

Loop-check-free sequent calculus for K t
From the shape of the rule ( i →) it follows that to ensure termination of derivations in backward proof search a loop-check (with histories or without them) is necessary. With the aim to eliminate a loop-check instead of the rules ( i →), (→ i ) marked modal rules are used. Along with non-marked modalities i marked modalities * i are used. The marked modalities have the same semantical meaning as non-marked modalities i and serves as a stopping device for a backward application of the rules (→ i ) and ( i →).
To define the marked modal rules let us introduce operation * which is applied to any formula and defined in the following way: 1. (P γ ) * = P γ , where index γ can be empty. {1, 2}). Let G * 1 K t be a calculus obtained from the calculus G 1 K t replacing the modal rules (→ i ) and ( i →) by the following marked rules: where j (j ∈ {1, 2}) is empty or consists of propositional symbols, moreover, 1 ∩ 2 is empty; in the conclusion of the rules the modality i in the main formula i A is not marked; in the premises of the rules the indices σ b and σ α are defined as in the rules (→ i ) and ( i →), correspondingly; in the premise of the rule (→ * i ), if i = j (j ∈ {1, 2}) then * • j j = j j otherwise * • j j = * j j . Using invertibility of the rules of G * 1 K t we can prove THEOREM 5. G 1 K t S if and only if G * 1 K t S, i.e., the calculus G * 1 K t is sound and complete.
A sequent S of the shape 1 , * 1 1 , * 2 2 → 2 , * 1 1 , * 2 2 is a final one if 1 ∩ 2 is empty. A derivation V of a sequent S in G * 1 K t is successful if each branch of V ends with an axiom. In this case G * 1 K t S. A derivation V of a sequent S in G * 1 K t is unsuccessful if V contains a branch ending with a final sequent. In this case G * 1 K t S. From the shape of the rules of G * 1 K t we get THEOREM 6. Any backward derivation in calculus G * 1 K t of a sequent S terminates. Example 1. (a) Let S be a sequent 1 P → 1 ¬ 2 ¬ 1 P from Example in [5]. Then a successful derivation of S in G * 1 K t is as follows: Choosing α = c we get that the sequent S * is an axiom of the shape * 1 P , P +c , * 2 ¬ * 1 P +b → P +c . It is obvious that in the sequent S the formula 1 P in the succedent is obtained from 1 (P +b ) −b using indexation procedure.
(b) Let us construct a derivation of the sequent P → ¬ 2 ¬ 1 P from Example in [5].
Since sequent S * is a final sequent we get G * 1 K t S. It is obvious that in the sequent S * the formula P −a,+b in the succedent is obtained from (P −a ) +b using indexation procedure.