The Sound and Complete R-Calculi with Respect to Pseudo-Revision and Pre-Revision *

The AGM postulates ([1]) are for the belief revision (revision by a single belief), and the DP postulates ([2]) are for the iterated revision (revision by a finite sequence of beliefs). Li [3] gave an R-calculus for R-configurations ,   where Δ is a set of literals, and Γ is a finite set of formulas. We shall give two -calculi such that for any consistent set Γ and finite consistent set of formulas in the propositional logic, in one calculus, there is a pseudo-revision Θ of Γ by Δ such that R    is provable and and in another calculus, there is a pre-revision Ξ of Γ by Δ such that ;    ∪   is provable, and for some pseudo-revision Θ; and prove that the deduction systems for both the -calculi are sound and complete with the pseudo-revision and the pre-revision, respectively.    ,     R


Introduction
The AGM postulates ( [1], [4][5][6]) are for the revision K   of a theory by a formula K ;  and the DP postulates ( [2]) are for the iterated revision The -calculus ( [3]) gave a Gentzen-type deduction system to deduce a consistent theory from any theory where should be a maximal consistent subtheory of which includes  as a subset, where   is an -configuration, R  is a consistent set of formulas, and is a consistent sets of literals (atomic formulas or the negation of atomic formulas).It was proved that if R is introduced to deduce   into a consistent set  of formulas including ;   the soundness theorem holds, that is, if     is provable then  is a pseudo-revision of  by ;  and  the completeness theorem holds, that is, if  is a pseudo-revision of  by  then     is provable.
Because each rule in the -calculus consists of the statements of form R , , the -calculus is based on pseudo-revision, i.e., to contract is inconsistent, which makes the -calculus not preserve the minimal change principle.

R
Given two theories  and a pseudo-revision , ∪ is inconsistent; otherwise, ).    ∪ consistent formula set     ∪ such that     is provable and  is a pseudo-revision of  by  (the soundness theorem); and conversely, given any pseudo-revision  of  by , is provable (the completeness theorem);       in another R -calculus, say 2 , R for any consistent formula set  and finite formula set ,  there are consistent formula sets  and  such that and  according to whether 1


 is consistent with   ∪ or not.The reason is given as follows.
Given a consistent theory  and formulas where    ∪ is consistent and inconsistent, respectively.Therefore, we use In 2 , we shall give a deduction rule to reduce R ,


  to the atomic cases where , , with a cost that we cannot prove that if     is provable then  is a pseudo-revision of  by . Instead we shall prove that if     is provable then  is a pre-revision of  by that is, there is a consistent theory 2) and 3) no subformula


The paper is organized as follows: the next section gives the -calculus in [3] and basic definitions; the third section defines an -calculus 1 for the pseudorevision and proves that 1 is sound and complete with respect to the pseudo-revision; the fourth section defines another -calculus 2 for the pre-revision and prove that 2 is sound and complete with respect to the pseudo-revision, and the last section concludes the whole paper.

The R -Calculus
The -calculus is defined on a first-order logical language.Let R L be a logical language of the first-order logic; 1 2 3 , ,    formulas and sets of formulas (theories), where ,    is a set of atomic formulas or the negations of atomic formulas, and   is called an R-configuration.
The -calculus consists of the following axiom and inference rules:  and in is a term, and is free in

The
-calculus is in the first-order logic.In the following we discuss the -calculi in the propositional logic.

R R
Let be a logical language of the propositional logic which contains the following symbols: Formulas are defined as follows: Definition 2.1.Given a consistent set of formulas and a finite consistent set of formulas, a consistent set of formulas is a pseudo-revision of by Each pseudo-revision can be generated by the following procedure: given any consistent set Then, is a subset of such that and  is consistent. .
be the least such that of formulas and a finite consistent set of formulas, a consistent set ,    and 3) no subformula  of is contradictory to  . Each pre-revision  can be generated by the following procedure: given any consistent set and finite consistent set where  is the empty string.

Let
, n    and  be the pseudo-revision of  by  in the same ordering as Then, we have the following . Lemma 2.4.Let 0 i be the least such that i Then, for any 0 , ; and no subformula of is contradictory to We prove that for any with and i We prove by induction on the structure of  that ,     and , .
are consistent, and by the induction assumption, 1 1 , , ; , , ; , , ; and by the induction assumption, is inconsistent, and hence, for any formula

R
is inconsistent.Similarly we can prove that for any i with , .
i i 

The -Calculus R 1
In this section we give an -calculus 1 which is sound and complete with respect to the pseudo-revision, where the decision of whether and 3) for each         is either an axiom or de d from the previous statements by the deduction rules.
, the following duce For example Also, the following pr , p p q R p q q p q q p q R p q p q p q R   (1) p q is a proof and so q | p q p q p         is provable.
Theorem 3.3.For any consistent sets of for-,   mulas and formula , By the in ction as on, , and hence,
and  satisfies the Assume that the th there is a set because the ast form l ula | ,

The Calculu
In this section e an which is we giv -calculus R R 2 is sound and complete with respect to the pre-revision, where the ecision of whet r Then, by we have the following reduction: and by we shall have the following o 2 R ne: Copyright © 2013 SciRes.IJIS , , .p q r s     For the two reductions, we have

 
, , , , p q r s p q q r s  Let  be a consistent set of formulas and  a finite consistent set consists of two parts: which we use to The deductions for the inconsistent , q p p q r p q r p q r p q p r p q r ,   of formulas and formula , and there s , are formula 1 2 ∪ is consiste and   ∪ is consistent, and th re, λ and θ 2 ≠ λ then by the in umption, nsistent, and so is are co If θ ≠ λ and θ 2 ≠ λ then by the indu assumption, . and s is , , , , .
Henc e, we have by Lemma 2.5, we have By the induction assumption, , , Hence, by Lemma 2.5, we have Assume that φ is consistent with e prove the theorem by the induction on the structure of φ.
. Proof.We only prove that no subformula ξ of Ξ is contradictory to Δ.
Assume that there is a subformula ξ of some formula is consistent then by Lemma 3.

4 .
For any consistent sets of formulas and formula a pseudo-revision of  by , R R◊  is any set of formulas;◊ The cut-rule in the R -calculus is eliminated in the R -calculi; ◊ Because    -rule in the R -calculus is not 2013 SciRes.IJIS Correspondingly, if |    is irre ucible, that is, no deduction rule can be used to reduce | ,    then   may be a minimal change of  by .