A derivation-loop method for temporal logic

Various types of calculi (Hilbert, Gentzen sequent, resolution calculi, tableaux) for propositional linear temporal logic (PLTL) have been considered in the literature. Cutfree Gentzen-type sequent calculi are convenient tools for backward proof-search search of formulas and sequents. In this paper we present a cut-free Gentzen type sequent calculus for PLTL with the operator “until”. We show that the calculus is sound and complete for the considered logic.


5.
A cut-free and invariant-free sequent calculus for PLTL is presented in [5].
This calculus has the new operator "unless", and do not retain the sub-formula property.
To our knowledge, the loop-type sequent calculus introduced in the present paper has not been considered in the literature before.
The present paper is organized as follows. In Section 2, we recall the syntax and semantics of PLTL. The calculus LTSC is introduced in Section 3. In this section, we present also the definition of derivation loops and prove some propositions concerning them. The soundness and completeness of LTSC with respect to PLTL is proved in Section 4. Some concluding remarks are in Section 5.
Propositional symbols and ⊤ are called atomic formulas. The formulas φ of PLTL are inductively defined as follows: We use the Greek letters φ and ψ, possibly with subscripts, to denote arbitrary formulas.

Semantics
We assume that time is linear, discrete, and ranges over the set of natural numbers. The formula φ intuitively means "φ is true at the next point of time"; the formula φ Uψ intuitively means "either ψ is true now or φ is true now and in all future time points until the one at which ψ is true".
An interpretation M = (T, I) for propositional linear tense logic consists of the set T = {t i : i 0}, where t i < t j , if i < j, and the function I : T → 2 P , where 2 P is the set of subsets of P. The semantics of PLTL formulas is provided by the satisfaction relation |=: M, t i |= φ Uψ, iff there is m ≥ i such that M, t m |= ψ and for all i ≤ j < m, M, t j |= φ.
(|= for the propositional operators is defined in the usual way.) A formula φ (sequent S) is called valid, |= φ (|= S) in notation, iff every PLTL interpretation is a model for φ (S). For example, it is true that |= (p ∧ q) ⊃ (p Uq).
Given a sequent S, a LTSC proof-search tree with the sequent S at the root is constructed in usual way by subsequently applying backwards the LTSC derivation rules to S and the sequents obtained in the course of the tree construction. A proof search tree is denoted by V . The expression V (S) denotes that S is the root of V .
We say that a sequent S ′ subsumes S (S ′ S in notation), iff S ′ can be inferred from S by the structural rule of weakening. For example, the sequent Γ, Π ⇒ ∆, Λ subsumes Γ ⇒ ∆.

Definition 1.
Given a proof-search tree, the upward path p from some sequent S in the tree to S ′ inclusive is called a (strong) derivation loop, [S − S ′ ] in notation, iff: 1) the length of p is greater than 0, 2) S ′ S (S ′ ⊒ S), and 3) there is no other sequent in p, except S, which (strongly) subsumes S. The nodes marked with S and S ′ are called the base and terminal of [S − S ′ ], respectively. The sequents S and S ′ are called the base and terminal sequents of [S − S ′ ], respectively. It is true that λ(S) λ(S ′ ).

Proposition 1. In any derivation loop with an eventuality formula φ Uψ, there is an application of (φ Uψ ⇒) between any two applications of ( ).
Proof. The proof follows from the fact that the succedent of the base of the loop has the member φ Uψ or (φ Uψ), from item 3 of Definition 2, and from the shape of the rule ( ). ⊓ ⊔ Definition 3. Any maximal connected graph Υ in a backward proof-search tree V such that each edge of Υ is in some derivation loop with an eventuality formula, is called a connected component.
It is said that a connected component Υ has a common eventuality formula, iff all derivation loops in Υ have a common eventuality formula φ Uψ. Such a formula is called the eventuality formula of Υ .
A connected component is called strong, iff all derivation loops in it are strong.

Definition 4.
Let Υ be a connected component. The path between sequents S 1 and . . , k}) is obtained by merging Υ into a single path containing all the edges of Υ . It is assumed that any two adjacent p(S l , S ′ m ) in the loop structure are connected with an application of an unnamed rule.

Definition 5.
A sequent S is called axiomatically derivable in LTSC, iff there exists a backward proof-search tree V (S) such that each leaf of V (S) is an axiom.

Definition 6.
A sequent S is called derivable in LTSC ( ⊢ S in notation), iff it is axiomatically derivable or there exists a backward proof-search tree V (S) such that: 1) each leaf of V (S) is an axiom or a terminal sequent of a derivation loop with some eventuality formula and 2) each connected component in V (S) has a common eventuality formula.
Such a tree V (S) is called a derivation of S or a derivation tree. We use the notation ⊢ V S to say that V (S) is a derivation of S.

Soundness and completeness of LTSC
be an arbitrary instance of an application of any LTSC derivation rule, except ( ).
Proof. The proof of the lemma is straightforward. ⊓ ⊔ Lemma 2. For any instance of an application of ( ) with the conclusion S and premise S ′ , if M, t i |= S, then M, t i+1 |= S ′ .
Proof. The proof of the lemma is straightforward. Proof. We prove that ⊢ S implies |= S. ⊓ ⊔

Concluding remarks
In the present paper, we have introduced and considered the Gentzen-type sequent calculus LTSC. This calculus is sound (Theorem 1) and complete (Theorem 2) for the considered logic PLTL. Hence an arbitrary sequent is derivable in LTSC if and only if it is valid in PLTL. Let us consider, for example, the sequent S : p, ⊤ U¬p ⇒ ⊤ U¬(p ⊃ p).