One Sound and Complete R-Calculus with Pseudo-Subtheory Minimal Change Property *

The AGM axiom system is for the belief revision (revision by a single belief), and the DP axiom system is for the iterated revision (revision by a finite sequence of beliefs). Li [1] gave an R-calculus for R-configurations | , ∆ Γ where ∆ is a set of atomic formulas or the negations of atomic formulas, and Γ is a finite set of formulas. In propositional logic programs, one R-calculus N will be given in this paper, such that N is sound and complete with respect to operator ( , ) s t ∆ , where ( , ) s t ∆ is a pseudo-theory minimal change of t by ∆ .


Introduction
The AGM axiom system is for the belief revision (revision by a single belief) [2][3][4][5], and the DP axiom system is for the iterated revision (revision by a finite sequence of beliefs) [6,7].These postulates list some basic requirements a revision operator Γ Φ  (a result of theory Γ revised by Φ ) should satisfy.
The R -calculus ( [1]) gave a Gentzen-type deduction system to deduce a consistent one ′ Γ ∪ ∆ from an inconsistent theory , Γ ∪ ∆ where ′ Γ ∪ ∆ should be a maximal consistent subtheory of Γ ∪ ∆ which includes ∆ as a subset, where | ∆ Γ is an R-configuration, Γ is a consistent set of formulas, and ∆ is a consistent sets of atomic formulas or the negation of atomic formulas.It was proved that if ∆ The R -calculus is set-inclusion, that is, , Γ ∆ are ta- ken as belief bases, not as belief sets [8][9][10][11].In the following we shall take , ∆ Γ as belief bases, not belief sets.We shall define an operator ( , ), s t ∆ where ∆ is a set of theories and t is a theory in propositional logic programs, such that is provable.Let  be the pseudo-subtheory relation, ( ) P t be the set of all the pseudo-subtheories of , t and ∆ ≡ be an equivalence relation on ( ) P t such that for any is consistent; and is a minimal change of t by ∆ in the syntactical sense, not in the set-theoretic sense, i.e., ( , ) s t ∆ is a minimal change of t by ∆ in the theoretic form such that ( , ) s t ∆ is consistent with .∆ The paper is organized as follows: the next section gives the basic elements of the R-calculus and the definition of subtheories and pseudo-subtheories; the third section defines the R-calculus N; the fourth section proves that N is sound and complete with respect to the operator ( , ); s t ∆ the fifth section discusses the logical properties of t and ( , ), s t ∆ and the last section concludes the whole paper.

The R-Calculus
The R-calculus ( [1]) is defined on a first-order logical language.Let L′ be a logical language of the first-order logic; , ϕ ψ formulas and , Γ ∆ sets of formulas (the- ories), where ∆ is a set of atomic formulas or the negations of atomic formulas.
The R-calculus consists of the following axiom and inference rules: where in cut , means that ϕ occurs in the proof tree T of ψ from 1 Γ and ; ϕ and in , R t ∀ is a term, and is free in ϕ for x .
if there is a sequence {( , , , ) : ∆ Theorem 2.3(The soundness and completeness theorem of the R-calculus).For any theories , ′ Γ Γ and , The R-rules preserve the strong validity.Let L be the logical language of the propositional logic.A literal l is an atomic formula or the negation of an atomic formula; a clause c is the disjunction of finitely many literals, and a theory t is the conjunction of finitely many clauses.
Definition 2.5.Given a theory , t a theory s is a sub-theory of , t denoted by , s t . p q p q p p q p p p q

The R-Calculus N
The deduction system N: where ,t ∆ denotes a theory { }; and for each , is either by an a Nrule or by an N ∧ -,or N ∨ -rule.
An example is the following deduction for , .
For any theory set ∆ and theory , t there is a theory Proof.We prove the theorem by the induction on the structure of .
∧ then by the induction assumption, there are theories s 1 , s 2 such that is consistent, and so is

The Completeness of the R-Calculus N
For any theory t , define ( , ) s t ∆ as follows: About the inconsistence, we have the following facts: About the consistence, we have the following facts: and , ( , ) .s t t ∆ ∆  Proof.We prove the theorem by the induction on the structure of .t If = t l and l is consistent with ∆ then ( , ) = , s l l ∆ and the theorem holds for ; { , } t t ∆∪ is consistent, and by the induction assumption, , ( { }, ) , , ) .
is N-provable.We assume that for any < , i n the claim holds.
If = t l and the last rule is 1 ( ) By the induction assumption,

Conclusions and Further Works
We defined an R-calculus N in propositional logic programs such that N is sound and complete with respect to the operator ( , ). s t ∆ The following axiom is one of the AGM postulates: It is satisfied, because we have the following Proposition 7.1.If 1 and the last rule is