Fundamentals of Logic
The publisher of the book is ICPR:
Indian council of Philosophical Research (ICPR)
36, Tughlakabad Institutional Area, M. B. Road,
New Delhi - 110062
Phones: 91-11-29955405, 29956403, 29955129, 6078853
Book’s ISBN :81-85636-34-6
The distributor of the book is:
Publishers Pvt. Ltd.
PO Box 5715, 54 Rani Jhansi Road, New Delhi 110055, India
Tel: +91-11-23671668, Fax: +91-11-23612745
This was written in collaboration with Professor Chinmoy Goswami (Founder of the Cognitive Science department in University of Hyderabad), when I was in University of Hyderabad. We modified its many versions to bring it to a book form and finally it was published in 1998, three years after I left Hyderabad. Its flap cover says:
"The book addresses problems like
Can we prove all that is true?
Can symbolic manipulation capture everything?
Is there a general method to solve a class of solvable problems?
Is Mathematics contradictory?
To answer these fundamental questions, it comes up with results such as Deduction, reductio ad absurdum, Monotonicity, Compactness, Completeness, Undecidability, and Incompleteness as expounded in the works of Herbrand, Godel, Skolem, Lowenheim, Beth, Tarski, Post, Turing and others It deals with the logic of sentences and predicates as formal languages giving stress on formal semantics. It considers major styles of presenting these logics such as axiomatics, Gentzen systems, analytic tableaux, resolution refutation as various proof techniques. However, it requires nothing from the reader but a mere willingness to remain logical and have a fearless attitude towards precise use of symbols."
The contents are:
Chapter 0 -- Preliminaries:
0.0 Introduction   0.1 Language of Sets   0.2 NUmber System   0.3 Cardinality   0.4 Trees   0.5 Formal Languages
Chapter 1 -- Language of Sentential Logic:
1.0 Introduction   1.1 Syntax of SL   1.2 Semantics of SL   1.3 Consequences   1.4 Normal Forms   1.5 Compactness
Chapter 2 -- Language of Predicate Logic:
2.0 Introduction   2.1 Syntax of PL   2.2 Semantics of PL   2.3 Validity and Consequences   2.4 Standard Forms   2.5 Syntactic Interpretations
Chapter 3 -- Axiomatics:
3.0 Introduction   3.1 Axiom System SC   3.2 Adequacy of SC   3.3 Axiom System PC   3.4 Adequacy of PC   3.5 Gentzen’s Systems GSC and GPC
Chapter 4 -- Semantic Proofs:
4.0 Introduction   4.1 Natural Deduction   4.2 Sets of Models   4.3 Analytic Tablaux for SL   4.4 Analytic Tableaux for PL
Chapter 5 -- Resolution Refutation:
5.0 Introduction   5.1 Clauses and Clause Sets   5.2 Resolution in SL   5.3 Clause Sets and Substitutions   5.4 Unifiers and Factors   5.5 Resolution in PL
Chapter 6 -- Predicate Logic with Equality:
6.0 Introduction   6.1 Syntax of PLE   6.2 Semantics of PLE   6.3 Axiomatization of PLE   6.4 Semantic Proofs in PLE   6.5 Resolution in PLE
Chapter 7 -- Metalogic:
7.0 Introduction   7.1 First Order Theories   7.2 Model Isomorphism   7.3 Effective Procedures   7.4 Uncomputability   7.5 Arithmatic   7.6 About Arithmatic   7.7 Undecidability   7.8 Unprovability and Consistency
Appendix -- Natural Language and Reasoning:
A.0 Introduction   A.1 Symbolizing into SL   A.2 Symbolizing into PL/PLE   A.3 Other Applications
Index of Proper Names
Index of Named Theorems
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Logics for Computer Science
The publisher of the book is PHI Learning:
PHI Learning, India
’ Rimjhim House’
111, Patparganj Industrial Estate
Delhi - 110 092, India.
Phones: 91-11-22143311, 22143322, 22143344, 22143355, 22143377, 2143388
Book’s ISBN : 81-203-2284-3
Before 2002, I was using Fundamentals of Logic to teach a course on Logic to M.Sc. and B.Tech. students. However, I had to do a bit differently so that it would be more suitable to students of theoretical computert science. I found that they should be exposed to the calculational logic. There should also be a thorough introduction to program verification and modal logics. It was Professor M T Nair who suggested, infact, brought the publisher (Prentice Hall of India) to my door steps, to develop the class notes into another book. This resulted in Logics for Computer Science. The very approach is different, making it more suitable for self-study. Its flap cover says:
"Designed primarily as an introductory text on logic for and in Computer Science, this well-organized book deals with almost all the basic concepts and techniques that are pertinent to the subject. It provides an excellent understanding of the logics used in computer science today.
The book begins with the easiest of logics, the logic of propositions, and then it goes on to give a detailed coverage of first order logic and modal logics. The discussion revolves around logics from common sense as also formal syntax and semantics. Dr. Arindama Singh analyzes with consummate skill the various approaches to the proof theory of the logics, e.g. axiomatic systems, natural deduction systems, Gentzen systems, analytic tableau, and resolution. Along with the metaresults such as soundness, completeness and compactness, he deftly deals with an important application of logic, namely, verification of programs. The book gives the flavour of logic engineering through computation tree logic, a logic of model checking.
The book concludes with a fairly detailed discussion on nonstandard logics including intuitionistic logic, Lukasiewicz logics, default logic, autoepistemic logic, and fuzzy logic.
This student-friendly text, with an unusual clarity in the concepts and broad exposure to the subject, should prove to be a life-long companion for anyone who wants to understand the basic principles of logic and enjoy how logic works in Computer Science. Besides students of Computer Science, those offering courses in Mathematics and Philosophy would greatly benefit from this study."
The contents are:
Chapter 1 -- Propositional Logic:
1.1 Introduction   1.2 Syntax of PL   1.3 Semantics of PL   1.4 Calculations   1.5 Normal Forms   1.6 Some Applications $nbsp Summary   Problems
Chapter 2 -- First Order Logic:
2.1 Introduction   2.2 Syntax of FL   2.3 Preparing for Semantics   2.4 Semantics of FL   2.5 Some Useful Consequences   2.6 Calculations   2.7 Normal Forms & nbsp Herbrand Interpretation   Summary   Problems
Chapter 3 -- Resolution:
3.1 Introduction   3.2 Resolution in PL   3.3 Unification of Clauses   3.4 Extending Resolution   3.5 Resolution for FL   3.6 Horn Clauses in FL   Summary   Problems
Chapter 4 -- Proofs in PL and FL:
4.1 Introduction   4.2 Axiomatic System PC   4.3 Axiomatic System FC   4.4 Adequacy and Compactness   4.5 Natural deduction   4.6 Gentzen Systems   4.7 Analytic Tableau   Summary   Problems
Chapter 5 -- Program Verification:
5.1 Introduction   5.2 Issue of Correctness   5.3 The Core Language CL   5.4 Partial Correctness   5.5 Hoare Proof anf Proof Summary   5.6 Total Correctness   5.7 The Predicate Transformer wp   Summary   Problems
Chapter 6 -- Modal Logics:
6.1 Introduction   6.2 Syntax and Semantics of K   6.3 Axiomatic System KC   6.4 Other Proof Systems for KC   6.5 Other Modal Logics   6.6 Various Modalities   6.7 Computation Tree Logic   Summary   Problems
Chapter 7 -- Some Other Logics:
7.1 Introduction   7.2 Intuitionistic Logic   7.3 Lukasiewicz Logics   7.4 Probabilistic Logics   7.5 Possibilistic and Fuzzy Logic   7.6 Default Logic   7.7 Autoepistemic Logic   Summary   Problems
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I was teaching courses on Theory of Computation since 1991, in both University of Hyderabad and IIT Madras. My students at IIT Madras expressed their wish to see the class notes in book form. And that is the reason this book came into existence. When IIT Madras celebrated its Golden Jubilee year, a scheme was floated for encouraging book writing. As a result I got a semester off from teaching and also received a nominal financial help in preparing the manuscript. Like Logics for Computer Science, this book is also well suited for self-study. Its back cover says:
"As Computer Science progressively matures as an established discipline, it becomes increasingly important to revisit its theoretical foundations, learn the appropriate techniques for answering theory-based questions, and build one’s confidence in implementing this knowledge when building computer applications. Students well-grounded in theory and abstract models of computation can excel in computing’s many application arenas.
Through a deft interplay of rigor and intuitive motivation, Elements of Computation Theory comprehensively, yet flexibly provides students with the grounding they need in computation theory. The book is self-contained and introduces the fundamental concepts, models, techniques, and results that form the basic paradigms of computing. Readers will benefit from the discussion of the ideas and mathematics that computer scientists use to model, to debate, and to predict the behavior of algorithms and computation. Previous learning about set theory and proof by induction are helpful prerequisites.
Topics and features:
Undergraduate students of computer science, engineering, and mathematics will find this core textbook ideally suited for courses on the theory of computation, automata theory, formal languages, and computational models. Computing professionals and other scientists will also benefit from the work’s accessibility, plethora of learning aids, and motivated exposition."
- Contains an extensive use of definitions, proofs, exercises, problems, and other pedagogical aids
- Supplies a summary, bibliographical remarks, and additional (progressively challenging) problems in each chapter, as well as an appendix containing hints and answers to selected problems
- Reviews mathematical preliminaries such as set theory, relations, graphs, trees, functions, cardinality, Cantor’s diagonalization, induction, and the pigeon-hole principle
- Explores regular languages, covering the mechanisms for representing languages, the closure properties of such languages, the existence of other languages, and other structural properties
- Investigates the class of context-free languages, including context-free grammars, Pushdown automata, their equivalence, closure properties, and existence of non-context-free languages
- Discusses the true nature of general algorithms, introducing unrestricted grammars, Turing machines, and their equivalence
- Examines which tasks can be achieved by algorithms and which tasks can’t, covering issues of decision problems in regular languages, context-free languages, and computably enumerable languages
- Provides a concise account of both space and time complexity, explaining the main techniques of log space reduction, polynomial time reduction, and simulations
- Promotes students&apos confidence via interactive learning and motivational, yet informal dialogue
- Emphasizes intuitive aspects and their realization with rigorous formalization
The contents are:
Chapter 1 -- Mathematical Preliminaries:
1.1 Introduction   1.2 Sets   1.3 Relations and Graphs   1.4 Functions and Counting   1.5 Proof Techniques   1.6 Summary and Problems
Chapter 2 -- Regular Languages:
2.1 Introduction   2.2 Language Basics   2.3 Regular Expressions   2.4 Regular Grammars   2.5 Deterministic Finite Automata   2.6 Nondeterministic Finite Automata   2.7 Summary and Additional Problems
Chapter 3 -- Equivalences:
3.1 Introduction   3.2 NFA to DFA   3.3 Finite Automata and Regular Grammars   3.4 Regular Expression to NFA   3.5 NFA to Regular Expression   3.6 Summary and Additional Problems
Chapter 4 -- Structure of Regular Languages:
4.1 Introduction   4.2 Closure Properties   4.3 Non-regular Languages   4.4 Myhill-Nerode Theorem   4.5 State Minimization   4.6 Summary and Additional Problems
Chapter 5 -- Context-free Languages:
5.1 Introduction   5.2 Context-free Grammars   5.3 Parse trees   5.4 Ambiguity   5.5 Eliminating Ugly Productions   5.6 Normal Forms   5.7 Summary and Additional Problems
Chapter 6 -- Structure of CFLs:
6.1 Introduction   6.2 Pushdown Automata   6.3 CFG and PDA   6.4 Pumping Lemma   6.5 Closure Properties of CFLs   6.6 Deterministic Pushdown Automata   6.7 Summary and Additional Problems
Chapter 7 -- Computably Enumerable Languages:
7.1 Introduction   7.2 Unrestricted Grammars   7.3 Turing Machines   7.4 Acceptance and Rejection   7.5 Using Old Machines   7.6 Multitape TMs   7.7 Nondeterministic TMs and Grammars   7.8 Summary and Additional Problems
Chapter 8 -- A Non-computably Enumerable Language:
8.1 Introduction   8.2 Turing Machines as Computers   8.3 TMs as Language Deciders   8.4 How Many Machines?   8.5 Acceptance Problem   8.6 Chomsky Heirarchy   8.7 Summary and Additional Problems
Chapter 9 -- Algorithmic Solvability:
9.1 Introduction   9.2 Problem Reduction   9.3 Rice’s Theorem   9.4 About Finite Automata   9.5 About PDA   9.6 Post’s Correspondence Problem   9.7 About Logical Theories   9.8 Other Interesting problems   9.9 Summary and Additional Problems
Chapter 10 -- Computational Complexity:
10.1 Introduction   10.2 Rate of Growth of Functions   10.3 Complexity Classes   10.4 Space Complexity   10.5 Time Complexity   10.6 The Class NP   10.7 NP-completeness   10.8 Some NP-complete Problems   10.9 Dealing with NP-complete Problems   10.10 Summary and Additional Problems
Answers and Hints to Selected problems
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Tsing Hua University Press has translated the book "Elements of Computation Theory". I trust everything has gone well in the translation.
It contains an extra page which was not in the original English version. The extra page has been translated by Xian Hu of University of Arkansas from Chinese to English. It is as follows:
Words from the Translators
Due to the constant change in information technology, importing outstanding foreign achievements such as textbooks meet the needs of China’s corresponding discipline’s development. After reading this book, we really appreciate the author’s wisdom and rigorous approach. Meanwhile, the way the author describes and illustrates things are rare in Chinese authors today. It starts with intuitive approaches and speculations which bring up the students’ enthusiasm for learning. After that, the book uses rigorous mathematical language and reasoning to prove or disprove the previous speculations. This method can not only stimulate students’ enthusiasm, initiative and creativity, but also help them build a rigorous, strict and serious attitude. This serious and lively style is not only beneficial for improving the Chinese teaching methods, but also provides a good example to build our teaching model. This serious and lively spirit is not only good for academics, but also motivates a nation to move forward.
The imported textbook should be authoritative, systematic, advanced and popular. Meanwhile, it should be beneficial for improving the levels of our academics, teaching, and way of thinking. This book is outstanding in all of the above areas. In addition, this book is also easy to understand. It explains profound theories in simple language with rigorous reasoning. It covers a large range of material with many inspiring exercises which are worth revisiting.
In-depth study of this book can reinforce the mathematical foundations of computer science, but more importantly, it benefits you with good methods, skills and tricks. This book combines good theory with excellent exercises, which will improve the readers’ problem solving ability. Overall, this is an excellent textbook, the kinds of which are not easy to find.
This book was mainly translated by Aiwen Cao, Peng Ye and Shaoshuai Li. The following people also participated in the work: Kun Cao, Zhiyun Li, Xiaochun Li, Anhua Chen, Jiayi Hou, Wei Xu, Wenya Dai, Fanpeng Yu, Peng Liu, Jiajia Wang, Wei Deng, Fanping Deng, Bo Li, Yunjian Cheng, Xiaozhe Xu, Ke Zhu, Xiao Wei, Hong Sun, Teng Li, Lei Chen, Yu Wei, Jingping Zhou, Dong Xun, Zhe Feng, Fei Li, Qiang Li, Donghui Zhao, Gang Zhou, Yuehua Zhang, Yan Sun, Qiang Gao, Xin Liu, Hongliang Wang, Feng Zhou, Hui Xie, Lin Li, Xiangyang Sun, Yuanyuan Li, Zhipeng Zhao, Jia Feng, CaiE Lin, Lei Sun, Baitao Zhang, Nan Zhao and Henan Chen.
During the translation, we tried our best to analyze the information in each word and sentence, not to guess. We respect the style and way of thinking of the original book and try to keep it. Due to the limitation of the translators’ knowledge and skill, it is inevitable to have errors and imperfections in the translation. We will highly appreciate the readers’ forgiveness and generous correction.
There ends the translator’s notes.
Introduction to Matrix Theory
The publisher of the book is Ane Books:
Ane Books Pvt. Ltd.
4821, Parwana Bhawan, 24, Ansari Road,
Daryaganj, New Delhi - 110 002, India
Phones: 91-11-2327 6863-44
Fax: 91-11-2327 6863
Book’s ISBN : 978-93-8676-121-7
Perhaps the best description about the book is the following extract from its preface.
Practising scientists and engineers feel that calculus and matrix theory form
the minimum mathematical requirement for their future work. Though it is
recommended to spread matrix theory or linear algebra over two semesters in
an early stage, the typical engineering curriculum allocates only one semester
for it. In addition, I found that science and engineering students are at a loss
in appreciating the abstract methods of linear algebra in the first year of their
undergraduate programme. This resulted in a curriculum that includes a thorough study of system of linear equations via Gaussian and/or Gauss-Jordan
elimination comprising roughly one month in the first or second semester. It
needs a follow-up of one semester work in matrix theory ending in canonical
forms, factorizations of matrices, and matrix norms.
Initially, we followed the books such by Leon, Lewis, and Strang as possible texts, referring occasionally to papers and other books. None of these could be used as a text book on its own for our purpose. The requirement was a single text containing development of notions, one leading to the next, and without any distraction towards applications. It resulted in
creation of our own material. The students wished to see the material in a
book form so that they might keep it on their lap instead of reading it off the
laptop screens. Of course, I had to put some extra effort in bringing it to this
form; the effort is not much compared to the enjoyment in learning.
The approach is straight forward. Starting from the simple but intricate
problems that a system of linear equations presents, it introduces matrices
and operations on them. The elementary row operations comprise the basic tools in working with most of the concepts. Though the vector space
terminology is not required to study matrices, an exposure to the notions is
certainly helpful for an engineer’s future research. Keeping this in view, the
vector space terminology are introduced in a restricted environment of subspaces of finite dimensional real or complex spaces. It is felt that this direct
approach will meet the needs of scientists and engineers. Also, it will form a
basis for abstract function spaces, which one may study or use later.
Starting from simple operations on matrices this elementary treatment of
matrix theory characterizes equivalence and similarity of matrices. The other
tool of Gram-Schmidt orthogonalization has been discussed leading to best
approximations and least squares solution of linear systems. On the go we
discuss matrix factorizations such as rank factorization, QR-factorization,
Schur triangularization, diagonalization, Jordan form, singular value decomposition and polar decomposition. It includes norms on matrices as a means to deal with iterative solutions of linear systems and exponential of a matrix.
Keeping the modest goal of an introductory text book on matrix theory, which
may be covered in a semester, these topics are dealt with in a lively manner.
Though the earlier drafts were intended for use by science and engineering
students, many mathematics students used those as supplementary text for
learning linear algebra. This book will certainly fulfil that need.
Each section of the book has exercises to reinforce the concepts; and problems have been added at the end of each chapter for the curious student. Most of these problems are theoretical in nature and they do not fit into the running text linearly. Exercises and problems form an integral part of the book. Working them out may require some help from the teacher. It is hoped that the teachers and the students of matrix theory will enjoy the text the same way I and my students did.
Most engineering colleges in India allocate only one semester for Linear
Algebra or Matrix Theory. In such a case, the first two chapters of the book
can be covered in a rapid pace with proper attention to elementary row operations. If time does not permit, the last chapter on matrix norms may be omitted, or covered in numerical analysis under the veil of iterative solutions of linear systems.
The contents are:
Chapter 1 -- Matrix Operations:
1.1 Examples of linear equations   1.2 Basic matrix operations   1.3 Transpose and adjoint   1.4 Elementary row operations   1.5 Row reduced echelon form   1.6 Determinant   1.7 Computing inverse of a matrix & nbsp 1.8 Problems for Chapter 1
Chapter 2 -- Systems of Linear Equations:
2.1 Linear independence   2.2 Determining linear independence   2.3 Rank of a matrix   2.4 Solvability of linear equations   2.5 Gauss-Jordan elimination   2.6 Problems for Chapter 2
Chapter 3 -- Matrix as a Linear Map:
3.1 Subspace and span   3.2 Basis and dimension   3.3 Linear transformations   3.4 Coordinate vectors   3.5 Coordinate matrices   3.6 Change of basis matrix   3.7 Equivalence and similarity   3.8 Problems for Chapter 3
Chapter 4 -- Orthogonality:
4.1 Inner products   4.2 Gram-Schmidt orthogonalization   4.3 QR-factorization   4.4 Orthogonal projection   4.5 Best approximation and least squares solution   4.6 Problems for Chapter 4
Chapter 5 -- Eigenvalues and Eigenvectors:
5.1 Invariant line   5.2 The characteristic polynomial   5.3 The spectrum   5.4 Special types of matrices   5.5 Problems for Chapter 5
Chapter 6 -- Canonical Forms:
6.1 Schur triangularization   6.2 Annihilating polynomials   6.3 Diagonalizability   6.4 Jordan form   6.5 Singular value decomposition   6.6 Polar decomposition   6.7 Problems for Chapter 6
Chapter 7 -- Norms of Matrices:
7.1 Norms   7.2 Matrix norms   7.3 Contraction mapping   7.4 Iterative solution of linear systems   7.5 Condition number   7.6 Matrix exponential   7.7 Estimating eigenvalues   7.8 Problems for Chapter 7
P.43, Line 25: Theorem 3.12 to be replaced with Theorem 1.1
P.109, Last two lines: a to be replaced with c, and (a,0,0) to be replaced with (0,0,c).
Extracts from its preface reads as follows:
In this revised version, the circularity in presenting logic via formal semantics is brought to the fore in a very elementary manner. Instead of developing everything from semantics, we now use an axiomatic system to model reasoning. Other proof methods are introduced and worked out later as alternative models.
Elimination of the equality predicate via equality sentences is dealt with semantically even before the axiomatic system for first order logic is presented. The replacement laws and the quantifier laws are now explicitly discussed along with the necessary motivation of using them in constructing proofs in mathematics. Adequacy of the axiomatic system is now proved in detail. An elementary proof of adequacy of Analytic Tableaux is now included.
Special attention is paid to the foundational questions such as decidability, expressibility, and incompleteness. These important and difficult topics are dealt with briefly and in an elementary manner.
The material on Program Verification, Modal Logics, and Other Logics in Chapters 9, 11 and 12 have undergone minimal change. Attempt has been made to correct all typographical errors pointed out by the readers. However, rearrangement of the old material and the additional topics might have brought in new errors. Numerous relevant results, examples, exercises and problems have been added. The correspondence of topics to chapters and sections have changed considerably, compared to the fist edition. A glance through the contents page will give you a comprehensive idea.
Its contents page reads as follows:
Chapter 1 -- Propositional Logic
1.1 Introduction   1.2 Syntax of PL   1.3 Is It a Proposition?   1.4 Interpretations,   1.5 Models   1.6 Equivalences and Consequences   1.7 More About Consequence   1.8 Summary and Problems
Chapter 2 -- A Propositional Calculus
2.1 Axiomatic System PC   2.2 Five theorems about PC   2.3 Using the metatheorems   2.4 Adequacy of PC to PL   2.5 Compactness of PL   2.6 Replacement Laws   2.7 Quasi-proofs in PL   2.8 Summary and Problems
Chapter 3 -- Normal Forms and Resolution
3.1 Truth Functions   3.2 CNF and DNF   3.3 Logic Gates   3.4 Satisfiability Problem   3.5 2SAT and Horn-SAT   3.6 Resolution in PL   3.7 Adequacy of resolution in PL   3.8 Resolution Strategies   3.9 Summary and Problems
Chapter 4 -- Other Proof Systems for PL
4.1 Calculation   4.2 Natural Deduction   4.3 Gentzen Sequent Calculus   4.4 Analytic Tableaux   4.5 Adequacy of PT to PL   4.6 Summary and Problems
Chapter 5 -- First Order Logic
5.1 Syntax of FL   5.2 Scope and Binding   5.3 Substitutions   5.4 Semantics of FL   5.5 Translating into FL   5.6 Satisfiability and Validity   5.7 Some Metatheorems   5.8 Equality Sentences   5.9 Summary and Problems
Chapter 6 -- A First Order Calculus
6.1 Axiomatic System FC   6.2 Six theorems about FC   6.3 Adequacy of FC to FL   6.4 Compactness of FL   6.5 Laws in FL   6.6 Quasi-proofs in FL   6.7 Summary and Problems
Chapter 7 -- Clausal Forms and Resolution
7.1 Prenex form   7.2 Quantifier-free forms   7.3 Clauses   7.4 Unification of clauses   7.5 Extending resolution   7.6 Factors and Pramodulants   7.7 Resolution for FL   7.8 Horn clauses in FL   7.9 Summary and Problems
Chapter 8 -- Other Proof Systems for FL
8.1 Calculation   8.2 Natural Deduction   8.3 Gentzen sequent calculus   8.4 Analytic Tableaux   8.5 Adequacy of FT to FL   8.6 Summary and Problems
Chapter 9 -- Program Verification
9.1 Debugging a Program   9.2 Issue of Correctness   9.3 The Core Language CL   9.4 Partial Correctness   9.5 Axioms And Rules   9.6 Hoare Proof   9.7 Proof Summary   9.8 Total Correctness   9.9 A Predicate Transformer   9.10 Summary and Problems
Chapter 10 -- First Order Theories
10.1 Structures and Axioms   10.2 Set Theory   10.3 Arithmetic   10.4 Herbrand Interpretation   10.5 Herbrand Expansion   10.6 Skolem-Lowenheim Theorems   10.7 Decidability   10.8 Expressibility   10.9 Provability predicate   10.10 Summary and Problems
Chapter 11 -- Modal Logic K
11.1 Introduction   11.2 Syntax and Semantics of K   11.3 Validity and Consequence in K   11.4 Axiomatic System KC   11.5 Adequacy of KC to K   11.6 Natural Deduction in K   11.7 Analytic Tableau for K   11.8 Other Modal Logics   11.9 Various Modalities   11.10 Computation Tree Logic   11.11 Summary and Problems
Chapter 12 -- Some Other Logics
12.1 Introduction   12.2 Intuitionistic Logic   12.3 Lukasiewicz Logics   12.4 Probabilistic Logics   12.5 Possibilistic and Fuzzy Logic   12.5.1 Crisp sentences and precise information   12.5.2 Crisp sentences and imprecise information   12.5.3 Crisp sentences and fuzzy Information   12.5.4 Vague sentences and fuzzy information   12.6 Default Logic   12.7 Autoepistemic Logics   12.8 Summary
I will be happy to receive suggestions from you for improving the books.
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