A labeled sequent calculus for propositional linear time logic

. A labeled sequent calculus LSC for propositional linear discrete time logic PLTL is introduced. Its sub-calculus LSC − TL is proved to be complete for some class of PLTL sequents.


Introduction
Temporal logic is a special type of modal logic. It provides a formal system for qualitatively describing and reasoning about how the truth values of assertions change over time. Propositional linear discrete time logic PLTL with temporal operators "next" and "always" is considered in the present paper.
Various syntactical proof-search systems are used for PLTL. Some of them are: • Sequent calculi with the invariant rule Γ → ∆, I; I → I; I → A Γ → ∆, A (→ I ), [10,11]. There are some interesting works in which invariant-free (and cut-free) calculi for PLTL are constructed [3,6].
• Proof procedures containing loop-type axioms for logics sub-logic of which is propositional temporal branching time logic [9].
• Resolution-type proof procedures based on formulas in some normal form, see, e.g., [5].
In the present paper a labeled sequent calculus LSC is presented. Its sub-calculus LSC − TL is proved to be complete for some easily defined but large class of PLTL sequents. Unlike the other deductive systems mentioned above, calculi LSC and LSC − TL are loop-axiom and invariant and infinitary rule free, which allows to construct effective proof-search procedures based on the calculi.

Syntax
Formulas are defined in the traditional way.
Formulas of the shape x k : A, where k ∈ {0} ∪ N (in particular, x 0 = x) and A is a formula, are called labeled formulas, l-formulas for short; x is called a label or a variable and k its power. Labels/variables are denoted by u, x, y, z, w and the corresponding powered labels by u k , x k , y k , z k , and w k . The intended meaning of 'x : A' is "A holds at some moment of time x" and that one of 'x k : A' is "A holds at the k-th from x moment of time".
Expressions m n : A, where m, n ∈ {0} ∪ N , are called fixed-label formulas and 'm n ' fixed labels.
One more type of formulas is x m y m , where m 0. Such formulas are called order atoms.
Sequents are objects of the type Γ → ∆, where Γ and ∆ are some finite multisets of formulas.
Labeled sequents, l-sequents for short, are objects of the type Γ → ∆, where Γ is some finite multiset of labeled formulas and order atoms; the same for ∆ except that order atoms do not occur in it.

Semantics
Kripke semantics of PLTL is defined as follows.
where F is the set of formulas and φ is defined in the following way.
where L is the set of labels. The stable sequent S ν is obtained from S by substituting every label x i by ν(x i ). S ν ς → {⊤, ⊥}, where S ν is the class of stable sequents and ς is defined as follows: A stable sequent S ν is valid, denoted by |= S ν , iff ς(S ν ) = ⊤ for any τ . A stable sequent S ν is an axiom if it is of the shape Γ, l k : E → m n : E, ∆, where l + k = m + n.
A labeled sequent S is valid, |= S in notation, iff every stable sequent obtained from S is valid.

TL
The labeled sequent calculus LSC for PLTL is defined as follows: where E is an atomic formula.
Here A and B arbitrary formulas.
Here x, y, z are unequal in pairs in Trans and Lin; x x does not occur in Γ in Ref; x k+1 y k+1 does not occur in Γ in Fwd; x z does not occur in Γ in Trans; In Lin, neither y z nor z y occur in Γ neither can be obtained by some backward applications of Trans.
The calculus LSC − TL is obtained from LSC by dropping Trans and Lin. A formula F is called derivable in the labeled sequent calculus The Hilbert-style calculus HSC for PLTL is defined by axioms: where p and q are arbitrary PLTL formulas. It is well known that this calculus is sound and complete for PLTL, see, e.g. [7].

Some Properties of LSC and LSC
, where S(w/u) is obtained from S by substituting the label w for the label u.
A rule is height-preserving admissible if, whenever its premiss(es) is (are) derivable, also its conclusion is derivable with the same bound on the derivation height.

Lemma 2. The rule of weakening
is height-preserving admissible in LSC and LSC − TL .
A rule is height-preserving invertible if, whenever its conclusion is derivable, also its premiss(es) is (are) derivable with the same bound on the derivation height. A sequent S is called proper if the fact that x k y k occurs in S, where k 0, implies that x 0 y 0 occurs in S.  By Theorem 2 and invertibility of the rules (→ ∨), (¬ →), and (∧ →), LSC − TL is complete for sequents S = Γ → ∆ such that F (S) is derivable in HSC without using the axiom A 6 .

Theorem 1. The rule of cut
An example of non-derivable in LSC − TL formula is Theorem 2 implies that this formula is not derivable in HSC without the axiom A 6 . This formula is derivable in LSC. Some examples of non-derivable in LSC formulas are We get by Theorem 2 that these formulas are not derivable in HSC without A 6 .