Partial cut elimination for combinations of propositional multi-modal logics with past time

. We consider combinations of nine propositional multi-modal logics with propositional discrete linear time temporal logic with past time. For these combinations, we present sound and complete Gentzen-type sequent calculi with a restricted cut rule.


Introduction
Recently, there has been a significant interest in multi-modal logics combining operators of knowledge and time [1,2,3,5,7,8,9,12]. With only a few exceptions, this literature deals with future time temporal operators. As indicated in [3] the logic of future time axioms is too weak to fully express the unique initial states, synchrony and some other properties of computer systems. Another reason to consider knowledge in combination with past time operators is that knowledge-based programs behave better with past-time operators than with future-time ones.
In [9] we consider logic of knowledge and past time introduced in [3]. In this paper we consider combinations of nine familiar propositional multi-modal logics with propositional discrete linear time temporal logic with past-time operators. Such a combination is denoted by LT . As far as we know such combinations of logics (except logic of knowledge and past time) has not been considered earlier.
For each formula ϕ of LT we introduce sequent calculus G LT (ϕ) with a restricted cut rule. We call a cut rule restricted if cut formulas used in the rule are taken from some finite set (say, Π L (ϕ)). We denote such a rule (Π L (ϕ)-cut).
We prove soundness and weak completeness (with respect to the certain class of models) for the presented calculi. Decidability of provability in G LT (ϕ) is the consequence of restricted cut rule.

Syntax and semantics
Let L be a logic from the set of nine well-known propositional multi-modal log- [4]). Let LTL − be the propositional discrete linear time temporal logic with past operators (see e.g. [6,9]). As above we denote the combination of logic L and LTL − by LT .
The combinations of logics we are considering are all propositional and share a common syntax. Let P be a nonempty set of primitive propositions. Let {1, . . . , m} be a set of agents, m 1. The language L is given by the abstract syntax where p ∈ P , and 1 i m is the index of an agent. The operators are, respectively not, or, next (tomorrow), weak yesterday, until, since, and necessity for i with their usual meanings. We use the familiar propositional abbreviations (true, ∧, ⊃).
We recall the definition of models for LT (introduced in [12]). Let S be a nonempty set of states. A timeline r is an infinitely long to the future, bounded to the past, linear, discrete sequence of states, indexed by natural numbers. Let Tlines be the set of all timelines. A point is a pair (r, n), where r is a timeline and n ∈ N is a temporal index into r. Let the set of all points (over S) be Points.
We say that formula φ is valid in a model M iff (M, (r, n)) |= φ for every point (r, n) ∈ M . Let C be a class of models. We say that φ is C-valid iff φ is valid in every model from C.
We get the class of models (denoted by C L ) for LT by imposing the familiar corresponding conditions on the accessibility relations R 1 , . . . , R m (see e.g. [4]). For example, C Km is the class of models with no conditions on each accessibility relation; C KD45m is the class of models such that each accessibility relation is serial, transitive and Euclidean.

Construction of closure sets FL L (ϕ) and Π L (ϕ)
In the introduction we have presented the notion of (Π L (ϕ)− cut) rule. At we end of this subsection we construct the finite set of formulas Π L (ϕ).

Remark 1.
We get the set Π L (ϕ) by looking through the proofs of statements which are used to prove the truth theorem and Lemma 2 (see below) (similar as in [9]). These statements are omitted here due to the lack of space.

Gentzen-type sequent calculi for LT , soundness
Let GLTL − be a Gentzen type calculus for temporal logic LTL − introduced in Section 3.3 of [10]. Gentzen-type sequent calculi G LT (ϕ) for the logic LT and a formula ϕ is defined as follows: GLTL − + inference rules for operators of necessity for logic L + (Π L (ϕ)-cut) rule.
The notion of a proof of a given sequent in the introduced calculi is defined in the usual way. By the induction on the height of the proof of a sequent Γ ⇒ ∆ we can verify the following:

Completeness with the restricted cut rule
We give a sketch of a proof of the completeness theorem. We say that a finite set of formulas Γ is Π L (ϕ)-consistent if the sequent Γ ⇒ is not provable in the calculus G LT (ϕ). We call a maximal Π L (ϕ)-consistent subset of the set FL L (ϕ) an atom(of FL L (ϕ)). We prove completeness theorem by contraposition (similar as in [9] and [10]): Lemma 1. If ¬ϕ is Π L (ϕ)-consistent formula then there exists a model M ∈ C L and a point (r, n) such that (M, (r, n)) |= ¬ϕ.
By induction on the complexity of formula ψ we prove the following Using definition of canonical model one can prove the following Lemma 2. If ψ ∈ FL L (ϕ) and ψ is a Π L (ϕ)-consistent formula then there exists a point (r, n) ∈ M c L (ϕ) such that ψ ∈ r(n).
Lemma 1 follows by Lemma 2 and Theorem 3. Theorem 2 is proved.