One Sound and Complete <i>R</i>-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 &Delta;|&Gamma;,?where?&Delta;?is a set of atomic formulas or the negations of atomic formulas, and?&Gamma;?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(&Delta;,t), where s(&Delta;,t)is a pseudo-theory minimal change of t by?&Delta;.

The Sound and Complete <i>R</i>-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 △|Γ, where Δ is a set of literals, and Γ is a finite set of formulas. We shall give two R-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 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 R-calculi are sound and complete with the pseudo-revision and the pre-revision, respectively.

The B4-valued propositional logic with unary logical connectives ～1 /～2/┐

A B4-valued propositional logic will be proposed in this paper which there are three unary logical connectives ～1, ～2, ┐ and two binary logical connectives A, v, and a Gentzen-typed deduction system will be given so that the system is sound and complete with B4-valued semantics, where B4 is a Boolean algebra.

Decomposition for a new kind of imprecise information system

In this paper, we first propose a new kind of imprecise information system, in which there exist conjunctions （∧＇s）, disjunctions （∨＇s） or negations （ ＇s）. Second, this paper discusses the relation that only contains ∧＇s based on relational database theory, and gives the syntactic and semantic interpretation for A and the definitions of decomposition and composition and so on. Then, we prove that there exists a kind of decomposition such that if a relation satisfies some property then it can be decomposed into a group of classical relations （relations do not contain ∧） that satisfy a set of functional dependencies and the original relation can be synthesized from this group of classical relations. Meanwhile, this paper proves the soundness theorem and the completeness theorem for this decomposition. Consequently, a relation containing ∧＇s can be equivalently transformed into a group of classical relations that satisfy a set of functional dependencies. Finally, we give the definition that a relation containing ∧＇s satisfies a set of functional dependencies. Therefore, we can introduce other classical relational database theories to discuss this kind of relation.

A sound and complete R-calculi with respect to contraction and minimal change

AGM postulates are for belief revision （revision by a single belief）, and DP postulates are for iterated revision （revision by a finite sequence of beliefs）. R-calculus is given for R-configurations △｜Г, where △ is a set of atomic formulas or the negations of atomic formulas, and Г is a finite set of formulas. We shall give two R-calculi C and M （sets of de- duction rules） such that for any finite consistent sets Г, △of formulas in the propositional logic, there is a consistent set ⊙ Г C of formulas such that △IГ → △, ⊙ is provable and⊙ is a contraction of F by A or a minimal change of F by A; and prove that C and M are sound and complete with respect to the contraction and the minimal change, respectively.

Variant quantifiers in L_(3)-valued first-order logic

Traditional first-order logic has four definitions for quantifiers,which are defined by universal and existential quantifiers.In L_(3)-valued(three-valued)first-order logic,there are eight kinds of definitions for quantifiers;and corresponding Gentzen deduction systems will be given and their soundness and completeness theorems will be proved.

Monotonicity and nonmonotonicity in L3-valued propositional logic

A sequent is a pair (Γ, Δ), which is true under an assignment if either some formula in Γ is false, or some formula in Δ is true. In L_(3)-valued propositional logic, a multisequent is a triple Δ∣Θ∣Γ, which is true under an assignment if either some formula in Δ has truth-value t, or some formula in Θ has truth-value m, or some formula in Γ has truth-value f. There is a sound, complete and monotonic Gentzen deduction system G for sequents. Dually, there is a sound, complete and nonmonotonic Gentzen deduction system G′ for co-sequents Δ: Θ: Γ. By taking different quantifiers some or every, there are 8 kinds of definitions of validity of multisequent Δ∣Θ∣Γ and 8 kinds of definitions of validity of co-multisequent Δ: Θ: Γ, and correspondingly there are 8 sound and complete Gentzen deduction systems for sequents and 8 sound and complete Gentzen deduction systems for co-sequents. Correspondingly their monotonicity is discussed.

Monotonic and nonmonotonic gentzen deduction systems for L_(3)-valued propositional logic

A sequent is a pair(Г,△),which is true under an as-signment if either some formula inГis false,or some formula in △ is true.In L3-valued propositional logic,a mulisequent is a triple △|Θ|Г,which is true under an assignment if either some formula in △ has truth-value t,or some formula in Θ has truth-value m,or some formula in Г has truth-value￡.Corre-spondingly there is a sound and complete Gentzen deduction system G for multisequents which is monotonic.Dually,a CO-multisequent is a triple △:Θ:Г,which is valid if there is an assignment v in which each formula in△has truth-value≠t,each formula in Θ has truth-value≠m,and each formula in Г has truth-value≠￡.Correspondingly there is a sound and com-plete Gentzen deduction system G-for co-multisequents which is nonmonotonic.

The M-computations induced by accessibility relations in nonstandard models M of Hoare logic

Hoare logic is a logic used as a way of specifying semantics of programming languages, which has been extended to be a separation logic to reason about mutable heap structure. In a model M of Hoare logic, each program α induces an M-computable function fα M on the universe of M; and the M-recursive functions are defined on M. It will be proved that the class of all the M-computable functions fα M induced by programs is equal to the class of all the M- recursive functions. Moreover, each M-recursive function is ∑ 1 NM -definable in M, where the universal quantifier is a num- ber quantifier ranging over the standard part of a nonstandard model M.

Nonmonotonic propositional logic

Propositional logic[1]is basic,based on which other logics are developed.The deduction system for propositional logic is monotonic.Nonmonotonic logics are a class of logics which deduction systems are nonmonotonic.Typical ones are default logic[2,3],R-calculus[4],autoepistemic logic[5],circumscription[5],etc.