Finite sequent calculi for PLTL

. Two sequent calculi for temporal logic of knowledge are presented: one containing invariant-like rule and the other containing looping axioms. Its proved that the calculi are equivalent, sound and complete.


Introduction
The considered logic KL (n) , a temporal logic of knowledge, is the fusion of linear time temporal logic with multi-modal logic S5 (n) . The temporal component is interpreted over a discrete linear model of time with finite past and infinite future. The modal component is the same as in S5 (n) which is often called logic of idealized knowledge. The logic KL (n) has been studied in detail [4]. Resolution-like proof search procedures for KL (n) and its application to security protocols was considered in [3].
The aim of this paper is to construct for considered logic KL (n) the sequent calculi with the invariant rule and with the looping axioms and to prove that they are equivalent. Hence we get that the calculus with the looping axioms is sound and complete, since the calculus with the invariant rule is sound and complete.
Formulas in KL (n) are defined in the traditional way; the formula A means "A is true at the next moment of time"; the formula A means "A is true now and in all moments of time in the future"; the formula K i A means "agent i knows A" A sequent is a formal expression A 1 , . . . , A n → B 1 , . . . , B m , where A 1 , . . . , A n (B 1 , . . . , B m ) is a finite set of formulas. In the definition of sequent the notion of set is used because it allows us to consider sequents without repeating of members.
A sequent S is a primary (quasi-primary) one, iff S = Σ 1 , is empty or consists of propositional symbols; Γ i (i ∈ {1, 2}) is empty or consists of formulas of the shape A, where A is an arbitrary formula; ∆ i (i ∈ {1, 2}) is empty or consists of formulas of the shape A, where A is an arbitrary formula; KΠ i (i ∈ {1, 2}) is empty or consists of formulas of the shape K l A, where A is an arbitrary formula and l ∈ {1, . . . , n}.
The sequent calculus KLG I for KL (n) with invariant like rule for temporal operator is obtained from traditional sequent calculus with invertible rules for propositional logic by adding: (a) rules for temporal operators: Here: the conclusion of ( ) is a primary sequent; Σ i (i ∈ {1, 2}) is empty or consists of arbitrary formulas, moreover is empty or consists of formulas of the type A. The formula I (called an invariant formula) is constructed from subformulas of formulas in the conclusion of rule (→ I ).
The rule (→ I ) corresponds to induction axiom used in temporal logic, namely, The rule (→ I ) was constructed using analogical rule of the sequent calculus for propositional dynamic logic (see, e.g. [5,12]).
A derivation V in KLG I is called atomic if every axiom occurring in V has the shape Γ, P → ∆, P , where P is a propositional symbol.
Remark 1. The sequent P → P is a simple example showing that KLG I does not possess the atomic derivation property.
(b) rules for modal operators: Here Σ j (j ∈ {1, 2}) is empty or consists of arbitrary formulas, moreover is empty or consists of formulas of the shape is empty or consists of formulas of the shape . . , C m . The formula K i A is the main formula of the rule (→ K i ). Though the rule (→ K i ) destroys the subformula property, the premise of this rule is constructed automatically from the conclusion and depends on the choice of the main formula of this rule. The rule (→ K i ) corresponds to distributivity, transitivity, and symmetry axioms for modal operators K i (see, e.g., [11]). For modal logic S5 the rule (→ K i ) has the following shape: The completeness and soundness of Hilbert-style version KLH of the calculus KLG I is presented in [4]. Using traditional proof-theoretical methods, we can prove that calculi KLG I and KLH are equivalent, therefore the calculus KLG I is sound and complete.

Saturated calculus (with loop-type axioms)
The calculus KLG I contains one serious problem: the invariant problem. The fact that temporal, dynamic and other induction-like logics contain a form of induction necessitates a departure from classical Gentzen systems. The basic closure axiom A → A is not sufficient. In 1985 some extension of Gentzen's branch closure was realized for temporal tableaux calculus [14]. Starting from 1993, inspired by prof. G. E. Mints, saturation method was proposed in several works, e.g., [6,7,8,9,10,1,13]. The terms "saturation method", "saturated derivation", "saturated sets" and so on were used despite of the fact that these terms were widely used (in various senses) in proof-theory and in model-theory. Saturation intuitively corresponds to certain type regularity in proof search. Saturation suggests that "essentially nothing new" can be obtained by continuing the proof search process. The saturated calculus KLG L is obtained from the calculus KLG I by: (a) replacing the invariant rule (→ I ) by the weak-induction rule and (b) adding loop-type (or looping) axioms defining as follows.
A quasi-primary sequent S ′ is a looping sequent with respect to S, if (1) S ′ is not a logical axiom, (2) S ′ is above a sequent S on a branch of a derivation tree, (3) S is such that it subsumes S ′ (S S ′ in notation), i. e., S ′ coincides with S or S ′ can be obtained from S by using the structural rule of weakening.
A sequent S ′ is called a degenerated sequent (d-sequent, in short), if the one of the following two conditions is satisfied: (1) either S ′ is a looping sequent with respect to S such that there is no the right premiss of any application of (→ L ) between S and S ′ , and in the case when S ′ does not coincide with S any rule, except the rule (→ L ), cannot be backward applied to S ′ , or S ′ consists of only propositional variables and is not a logical axiom; (2) S ′ is a looping sequent with respect to S and there is the right premiss of an application of (→ L ) between S and S ′ but S is an ancestor of some d-sequent in the derivation.
A looping sequent S ′ with respect to S is called a loop-type (or looping) axiom if it is not a d-sequent, the sequent S has the shape Γ → ∆, A and S is such that in the derivation tree there exists at least one looping sequent S * with respect to S (S * can coincide with S ′ ) such that there is only one application of the rule ( ) between S and S * . In this case the sequent S is called a quasi-looping axiom. From the definition of looping axiom it follows that there is the right premiss of (→ L ) between S and S ′ , and S is not an ancestor of some d-sequent in the derivation, i.e. S ′ is never subsumed by any d-sequent in the derivation.
The loop rule (→ L ) corresponds to the temporal fixed point axiom A, A → A. The temporal loop rule possesses subformula property and is more effectual than invariant-like rule. Unfortunately this rule is not sufficient to derive rather trivial sequents. To get complete calculus it is necessary to add loop-type axioms. The looping axioms allows us to stop derivation when a "good" loop is obtained. The "good" loop indicates that some regularity of a derivation is obtained and nothing new can be obtained continuing the proof-search process.
A sequent S is derivable in the calculus KLG L (KLG L ⊢ S, in notation) if it is possible to construct a derivation each leaf of which is either a logical axiom or a looping axiom. Otherwise the sequent S is non-derivable in the calculus KLG L (KLG L ⊢ S, in notation). In this case there exists the leaf with a sequent S ′ , such that S ′ is a d-sequent.
The sequent calculus with looping axioms was constructed in [5] for BDI logic. Efficient loop-check for this logic was constructed in [2].
Defining that A is a sub-formula of A and that A and A have the same complexity, we get that all rules of the calculus KLG L have the sub-formula property, and complexity of the premiss of any application of the rule is not greater than that of the conclusion.
Applying the rule (→ ∧) to S 1 and S 2 we get Let us construct the derivation of the sequent The derivation of the sequent is obvious. Applying (→ I ) to (1), (2), and (3) with the invariant formula I = A ∧ A, we get the conclusion of the rule (→ L ), i. e., the sequent Γ → ∆, A.
Using, for example [6,7], we can get  Proof. The lemma is proved by induction on the ordered pair |A|, h(S 1 ) + h(S 2 ) , where |A| is the complexity of A, assuming that | A| = | A|; h(S i ) stands for the height of a derivation of the left (right) premiss of (cut), here i ∈ {1, 2}.