This second volume of the book series shows R-calculus is a combination of one monotonic tableau proof system and one non-monotonic one. The R-calculus is a Gentzen-type deduction system which is non-monotonic, and is a concrete belief revision operator which is proved to satisfy the AGM postulates and the DP postulates. It discusses the algebraical and logical properties of tableau proof systems and R-calculi in many-valued logics.

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.

Auteur

Wei Li, is a Professor in the School of Computer Science and Engineering, Beihang University, Beijing, China and is a member of the Chinese Academy of Sciences. Prof. Li is mostly engaged in the applied research of Computer Software and Theory, and the Internet, including programming languages, software development, artificial intelligence, and integrated circuit design.

Yuefei Sui, is a Professor in the Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China. His main interests include knowledge representation, applied logic and the theory of computability.

Contenu

1 Introduction 111.1 Belief revision . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 R-calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3 Contents in the first-volume . . . . . . . . . . . . . . . . . . . 141.4 Contents in this volume . . . . . . . . . . . . . . . . . . . . . 171.5 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2 R-Calculus For Propositional Logic 242.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Monotonic tableau proof systems . . . . . . . . . . . . . . . . 262.2.1 Tableau proof system Tf . . . . . . . . . . . . . . . . 262.2.2 Tableau proof system Tt . . . . . . . . . . . . . . . . 292.3 Nonmonotonic tableau proof systems . . . . . . . . . . . . . . 312.3.1 Tableau proof system St . . . . . . . . . . . . . . . . . 322.3.2 Tableau proof system Sf . . . . . . . . . . . . . . . . . 342.4 R-calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.4.1 R-calculus Rt . . . . . . . . . . . . . . . . . . . . . . . 362.4.2 R-calculus Rf . . . . . . . . . . . . . . . . . . . . . . . 402.5 Projecting R-calculi to tableau proof systems . . . . . . . . . 412.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3 R-Calculus For L3-Valued Propositional Logic 453.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . 463.2 Monotonic tableau proof systems . . . . . . . . . . . . . . . . 493.2.1 Tableau proof system Tt . . . . . . . . . . . . . . . . 493.2.2 Tableau proof system Tm . . . . . . . . . . . . . . . . 503.2.3 Tableau proof system Tf . . . . . . . . . . . . . . . . 513.3 Nonmonotonic tableau proof systems . . . . . . . . . . . . . . 523.3.1 Tableau proof system St . . . . . . . . . . . . . . . . . 543.3.2 Tableau proof system Sm . . . . . . . . . . . . . . . . . 553.3.3 Tableau proof system Sf . . . . . . . . . . . . . . . . . 553.4 R-calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4.1 R-calculus Rt . . . . . . . . . . . . . . . . . . . . . . . 573.4.2 R-calculus Rm . . . . . . . . . . . . . . . . . . . . . . . 603.4.3 R-calculus Rf . . . . . . . . . . . . . . . . . . . . . . . 633.5 Satisfiability and unsatisfiability . . . . . . . . . . . . . . . . 653.5.1 t-satisfiability and t-unsatisfiability . . . . . . . . . . 653.5.2 m-satisfiability and m-unsatisfiability . . . . . . . . . . 673.5.3 f-satisfiability and f-unsatisfiability . . . . . . . . . . 683.6 Projecting R-calculi to tableau proof systems . . . . . . . . . 703.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4 R-Calculus For L3-Valued PL,II 754.1 Monotonic tableau proof systems . . . . . . . . . . . . . . . . 754.1.1 Tableau proof system Tt . . . . . . . . . . . . . . . . 76 4.1.2 Tableau proof system Tm . . . . . . . . . . . . . . . . 774.1.3 Tableau proof system Tf . . . . . . . . . . . . . . . . 784.2 Nonmonotonic tableau proof systems . . . . . . . . . . . . . . 794.2.1 Tableau proof system St . . . . . . . . . . . . . . . . . 794.2.2 Tableau proof system Sm . . . . . . . . . . . . . . . . . 804.2.3 Tableau proof system Sf . . . . . . . . . . . . . . . . . 814.3 R-calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.3.1 R-calculus Rt . . . . . . . . . . . . . . . . . . . . . . . 824.3.2 R-calculus Rm . . . . . . . . . . . . . . . . . . . . . . . 854.3.3 R-calculus Rf . . . . . . . . . . . . . . . . . . . . . . . 874.4 Validity and invalidity . . . . . . . . . . . . . . . . . . . . . . 904.4.1 t-invalidity and t-validity . . . . . . . . . . . . . . . . 904.4.2 m-invalidity and m-validity . . . . . . . . . . . . . . . . 924.4.3 f-invalidity and f-validity . . . . . . . . . . . . . . . . 944.5 Projecting R-calculi to tableau proof systems . . . . . . . . . 96 5 R-Calculus For B22-Valued PL 985.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . 1005.2 Monotonic tableau proof systems . . . . . . . . . . . . . . . . 1015.2.1 Tableau proof system Tt . . . . . . . . . . . . . . . . 1035.2.2 Tableau proof system Ttop . . . . . . . . . . . . . . . . 1045.2.3 Tableau proof system T . . . . . . . . . . . . . . . . 1055.2.4 Tableau proof system Tf . . . . . . . . . . . . . . . . 1075.3 Nonmonotonic tableau proof systems . . . . . . . . . . . . . . 1085.3.1 Tableau proof system St . . . . . . . . . . . . . . . . . 1105.3.2 Tableau proof system Stop . . . . . . . . . . . . . . . . 1125.3.3 Tableau proof system S . . . . . . . . . . . . . . . . 1135.3.4 Tableau proof system Sf . . . . . . . . . . . . . . . . . 1145.4 R-calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.4.1 R-calculus Rt . . . . . . . . . . . . . . . . . . . . . . . 1175.4.2 R-calculus Rtop . . . . . . . . . . . . . . . . . . . . . . 1215.4.3 R-calculus R . . . . . . . . . . . . . . . . . . . . . . 1255.4.4 R-calculus Rf . . . . . . . . . . . . . . . . . . . . . . . 1285.5 Projecting R-calculi to tableau proof systems . . . . . . . . . 1325.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6 R-Calculus For B22-Valued PL,II 1386.1 Monotonic tableau proof systems . . . . . . . . . . . . . . . . 1406.1.1 Tableau proof system Tt top . . . . . . . . . . . . . . . . 142 6.1.2 Tableau proof system Tt . . . . . . . . . . . . . . . . 143 6.2 Tableau proof systems . . . . . . . . . . . . . . . . . . . . . . 1456.2.1 Tableau proof system Tt top . . . . . . . . . . . . . . . . 1466.2.2 Tableau proof system Tt . . . . . . . . . . . . . . . 1486.3 Nonmonotonic tableau proof systems . . . . . . . . . . . . . . 1496.3.1 Tableau proof system St top . . . . . . . . . . . . . . . . 1506.3.2 Tableau proof system St . . . . . . . . . . . . . . . . 1516.3.3 Tableau proof system St top . . . . . . . . . . . . . . . . 1526.3.4 Tableau proof system St . . . . . . . . . . . . . . . . 1536.4 R-calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.4.1 R-calculus Rt top . . . . . . . . . . . . . . . . . . . . . . 1556.4.2 R-calculus Rt . . . . . . . . . . . . . . . . . . . . . . 1576.4.3 R-calculus Rf top . . . . . . . . . . . . . . . . . . . . . . 1596.4.4 R-calculus Rf. . . . . . . . . . . . . . . . . . . . . . 1616.5 Projecting R-calculi to tableau proof systems . . . . . . . . . 1636.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7 Complementary R-Calculus For PL 1687.1 Co-R-calculi in propositional logic . . . . . . . . . . . . . . . 1697.1.1 Co-R-calculus …