R-CALCULUS: A Logic of Belief Revision (Perspectives in Formal Induction, Revision and Evolution)

Perspectives in Formal Induction, Revision and Evolution is a book series focusing on the logics used in computer science and artificial intelligence, including but not limited to formal induction, revision and evolution. It covers the fields of formal representation, deduction, and theories or meta-theories of induction, revision and evolution, where the induction is of the first level, the revision is of the second level, and the evolution is of the third level, since the induction is at the formula stratum, the revision is at the theory stratum, and the evolution is at the logic stratum. In his book "The Logic of Scientific Discovery", Karl Popper argues that a scientific discovery consists of conjecture, theory, refutation, and revision. Some scientific philosophers do not believe that a reasonable conjecture can come from induction. Hence, induction, revision and evolution have become a new territory for formal exploration. Focusing on this challenge, the perspective of this book series differs from that of traditional logics, which concerns concepts and deduction. The series welcomes proposals for textbooks, research monographs, and edited volumes, and will be useful for all researchers, graduate students, and professionals interested in the field. Preface to the Series Preface Contents 1 Introduction 1.1 Belief Revision 1.2 R-Calculus 1.3 Extending R-Calculus 1.4 Approximate R-Calculus 1.5 Applications of R-Calculus References 2 Preliminaries 2.1 Propositional Logic 2.1.1 Syntax and Semantics 2.1.2 Gentzen Deduction System 2.1.3 Soundness and Completeness Theorem 2.2 First-Order Logic 2.2.1 Syntax and Semantics 2.2.2 Gentzen Deduction System 2.2.3 Soundness and Completeness Theorem 2.3 Description Logic 2.3.1 Syntax and Semantics 2.3.2 Gentzen Deduction System 2.3.3 Completeness Theorem References 3 R-Calculi for Propositional Logic 3.1 Minimal Changes 3.1.1 Subset-Minimal Change 3.1.2 Pseudo-Subformulas-Minimal Change 3.1.3 Deduction-Based Minimal Change 3.2 R-Calculus for subseteq-Minimal Change 3.2.1 R-Calculus S for a Formula 3.2.2 R-Calculus S for a Theory 3.2.3 AGM Postulates Asubseteq for subseteq-Minimal Change 3.3 R-Calculus for preceq-Minimal Change 3.3.1 R-Calculus T for a Formula 3.3.2 R-Calculus T for a Theory 3.3.3 AGM Postulates Apreceq for preceq-Minimal Change 3.4 R-Calculus for vdashpreceq-Minimal Change 3.4.1 R-Calculus U for a Formula 3.4.2 R-Calculus U for a Theory References 4 R-Calculi for Description Logics 4.1 R-Calculus for subseteq-Minimal Change 4.1.1 R-Calculus SDL for a Statement 4.1.2 R-Calculus SDL for a Set of Statements 4.2 R-Calculus for preceq-Minimal Change 4.2.1 Pseudo-Subconcept-Minimal Change 4.2.2 R-Calculus TDL for a Statement 4.2.3 R-Calculus TDL for a Set of Statements 4.3 Discussion on R-Calculus for vdashpreceq-Minimal Change References 5 R-Calculi for Modal Logic 5.1 Propositional Modal Logic 5.2 R-Calculus SM for subseteq-Minimal Change 5.3 R-Calculus TM for preceq-Minimal Change 5.4 R-Modal Logic 5.4.1 A Logical Language of R-Modal Logic 5.4.2 R-Modal Logic References 6 R-Calculi for Logic Programming 6.1 Logic Programming 6.1.1 Gentzen Deduction Systems 6.1.2 Dual Gentzen Deduction System 6.1.3 Minimal Change 6.2 R-Calculus SLP for subset-Minimal Change 6.3 R-Calculus TLP for preceq-Minimal Change References 7 R-Calculi for First-Order Logic 7.1 R-Calculus for subseteq-Minimal Change 7.1.1 R-Calculus SFOL for a Formula 7.1.2 R-Calculus SFOL for a Theory 7.2 R-Calculus for preceq-Minimal Change 7.2.1 R-Calculus TFOL for a Formula 7.2.2 R-Calculus TFOL for a Theory References 8 Nonmonotonicity of R-Calculus 8.1 Nonmonotonic Propositional Logic 8.1.1 Monotonic Gentzen Deduction System G'1 8.1.2 Nonmonotonic Gentzen Deduction System Logic G2 8.1.3 Nonmonotonicity of G2 8.2 Involvement of ΓA in a Nonmonotonic Logic 8.2.1 Default Logic 8.2.2 Circumscription 8.2.3 Autoepistemic Logic 8.2.4 Logic Programming with Negation as Failure 8.3 Correspondence Between R-Calculus and Default Logic 8.3.1 Transformation from R-Calculus to Default Logic 8.3.2 Transformation from Default Logic to R-Calculus References 9 Approximate R-Calculus 9.1 Finite Injury Priority Method 9.1.1 Post's Problem 9.1.2 Construction with Oracle 9.1.3 Finite Injury Priority Method 9.2 Approximate Deduction 9.2.1 Approximate Deduction System for First-Order Logic 9.3 R-Calculus Fapp and Finite Injury Priority Method 9.3.1 Construction with Oracle 9.3.2 Approximate Deduction System Fapp 9.3.3 Recursive Construction 9.3.4 Approximate R-Calculus Frec 9.4 Default Logic and Priority Method 9.4.1 Construction of an Extension Without Injury 9.4.2 Construction of a Strong Extension with Finite Injury Priority Method References 10 An Application to Default Logic 10.1 Default Logic and Subset-Minimal Change 10.1.1 Deduction System SD for a Default 10.1.2 Deduction System SD for a Set of Defaults 10.2 Default Logic and Pseudo-subformula-minimal Change 10.2.1 Deduction System TD for a Default 10.2.2 Deduction System TD for a Set of Defaults 10.3 Default Logic and Deduction-Based Minimal Change 10.3.1 Deduction System UD for a Default 10.3.2 Deduction System UD for a Set of Defaults References 11 An Application to Semantic Networks 11.1 Semantic Networks 11.1.1 Basic Definitions 11.1.2 Deduction System G4 for Semantic Networks 11.1.3 Soundness and Completeness Theorem 11.2 R-Calculus for subseteq-Minimal Change 11.2.1 R-Calculus SSN for a Statement 11.2.2 Soundness and Completeness Theorem 11.2.3 Examples 11.3 R-Calculus for preceq-Minimal Change 11.3.1 R-Calculus TSN for a Statement 11.3.2 Soundness and Completeness Theorem of TSN References Index This Book Introduces New Models Based On R-calculus And Theories Of Belief Revision For Dealing With Large And Changing Data. It Extends R-calculus From First-order Logic To Propositional Logic, Description Logics, Modal Logic And Logic Programming, And From Minimal Change Semantics To Subset Minimal Change, Pseudo-subformula Minimal Change And Deduction-based Minimal Change (the Last Two Minimal Changes Are Newly Defined). And It Proves Soundness And Completeness Theorems With Respect To The Minimal Changes In These Logics. To Make R-calculus Computable, An Approximate R-calculus Is Given Which Uses Finite Injury Priority Method In Recursion Theory. Moreover, Two Applications Of R-calculus Are Given To Default Theory And Semantic Inheritance Networks. This Book Offers A Rich Blend Of Theory And Practice. It Is Suitable For Students, Researchers And Practitioners In The Field Of Logic. Also It Is Very Useful For All Those Who Are Interested In Data, Digitization And Correctness And Consistency Of Information, In Modal Logics, Non Monotonic Logics, Decidable/undecidable Logics, Logic Programming, Description Logics, Default Logics And Semantic Inheritance Networks.